[ { "id": "anthropics/fermats-last-theorem/formalization.yaml", "repo": "anthropics/fermats-last-theorem", "path": "formalization.yaml", "directory": "", "origins": [ "cited" ], "url": "https://github.com/anthropics/fermats-last-theorem/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Fermat's Last Theorem in Lean 4", "description": "A complete, machine-checked Lean 4 proof of Fermat's Last Theorem, built on Mathlib: for every natural number n >= 3 there are no positive natural numbers a, b, c with a^n + b^n = c^n (declaration fermat_last_theorem). The default build target also derives Mathlib's own statement FermatLastTheorem. The argument is that of Frey, Serre, Ribet, Wiles and Taylor-Wiles; the named classical inputs (irreducibility of the Frey representation, Langlands-Tunnell, modularity lifting, modularity of semistable curves, level lowering) are proved within the development, in the strength the argument needs, ra", "authors": [ "Anthropic" ], "maintainers": [], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Modular elliptic curves and Fermat's Last Theorem", "id": "doi:10.2307/2118559", "authors": [ "Andrew Wiles" ], "type": "article", "relationship": "adapts", "endorsement": "not-contacted" }, { "title": "Ring-theoretic properties of certain Hecke algebras", "id": "doi:10.2307/2118560", "authors": [ "Richard Taylor", "Andrew Wiles" ], "type": "article", "relationship": "adapts", "endorsement": "not-contacted" }, { "title": "Fermat's Last Theorem", "id": "In: Current Developments in Mathematics, 1995 (International Press), 1-154", "authors": [ "Henri Darmon", "Fred Diamond", "Richard Taylor" ], "type": "article", "relationship": "adapts", "endorsement": "not-contacted" } ], "related": [ { "id": "https://github.com/ImperialCollegeLondon/FLT", "relationship": "builds-on", "note": "The Imperial College London FLT project led by Kevin Buzzard (Apache-2.0). Material adapted from it (Frey package, Galois representations, deformation theory, patching and more) is listed file by file in ATTRIBUTION.md and credited in NOTICE." }, { "id": "https://github.com/leanprover-community/flt-regular", "relationship": "builds-on", "note": "flt-regular (Apache-2.0): Kummer's theorem. Adapted files are listed in ATTRIBUTION.md and credited in NOTICE." } ], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11D41" ] }, "automation": { "methods": [ { "method": "agent", "models": [], "framework": "" } ], "strongest": "agent", "models": [], "spend": "", "notes": "" }, "scope": "The full statement of Fermat's Last Theorem for natural-number exponents n >= 3, proved from Lean's three standard axioms with every intermediate result proved in the development or taken from Mathlib; nothing is assumed from the literature.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "fermat_last_theorem", "file": "Theorems/Thm_fermat_last_theorem.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "flt_mathlib", "file": "FinalCheck.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The development follows the strategy of the sources, not their text. Named classical theorems are proved in the strength the argument needs, as set out under \"Exact strength of the named steps\" in PROOF-PATH.md: irreducibility of E[p] is proved for Frey curves rather than via Mazur's general theorems; Langlands-Tunnell in the octahedral case only; modularity lifting under level conditions at p = 3 and p in {3, 5}; modularity for semistable integral Weierstrass models in the sense of matching a_l; level lowering for the Frey representation as a congruence of traces. The top-level statement is the standard one and is checked identical to a Mathlib-only challenge file.", "alignment": false, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "The proof is machine-checked: by the Lean kernel in a from-scratch build whose default target fails unless fermat_last_theorem depends on exactly propext, Classical.choice and Quot.sound; by leanprover/comparator against the Mathlib-only challenge statement in verification/comparator/Challenge.lean (statement and constants identical, no other axiom, full kernel replay including Mathlib); and by the independent kernel nanoda, which accepted every declaration of the exported environment. See README.md, \"How it was verified\"." }, "canonical": { "repo": "anthropics/fermats-last-theorem", "directory": "" }, "nodes": [ { "id": "mathlib:FermatLastTheorem", "join": "artifact", "page": "s/mathlib-fermatlasttheorem.html", "label": "Fermat's Last Theorem" } ], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "anthropics/formal-math/zeta23/formalization.yaml", "repo": "anthropics/formal-math", "path": "zeta23/formalization.yaml", "directory": "zeta23", "origins": [ "cited", "search" ], "url": "https://github.com/anthropics/formal-math/blob/HEAD/zeta23/formalization.yaml", "version": "v0.4", "missing": [], "name": "Zeta23 — more than two thirds of the zeta zeros are simple and on the critical line", "description": "A complete, sorry-free Lean 4 / Mathlib formalization of the headline theorems of Alpöge–Furman, \"More than two thirds of the zeta zeros are simple and on the critical line\" (arXiv:2608.13637). Writing N(T) for the number of nontrivial zeros ρ of the Riemann zeta function with 0 < Im ρ ≤ T, counted with multiplicity, the formalization proves unconditionally that, as T → ∞, at least 2/3 of them are simple and lie on the critical line Re ρ = 1/2 and at least 5/6 are distinct (liminf bounds on the ratios), and that with the optimal Montgomery–Taylor window the constants improve to 2 − 1/c₁* = 0.6", "authors": [ "Claude" ], "maintainers": [ "Ralph Furman" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "More than two thirds of the zeta zeros are simple and on the critical line", "id": "arXiv:2608.13637", "authors": [ "Levent Alpöge", "Ralph Furman" ], "type": "paper", "relationship": "formalizes", "endorsement": "participated" } ], "related": [ { "id": "https://github.com/AlexKontorovich/PrimeNumberTheoremAnd", "relationship": "adapts", "note": "Files under Zeta23/FromPNTPlus/ are derived from the PrimeNumberTheoremAnd project with local modifications; each carries its upstream notice (file, commit, copyright, licence). See NOTICE." } ], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11M06", "11M26", "15A42" ] }, "automation": { "methods": [ { "method": "autonomous", "models": [ "Claude" ], "framework": "n/a" } ], "strongest": "autonomous", "models": [ "Claude" ], "spend": "not tracked", "notes": "The formalization was entirely automated: Claude produced all Lean code in the repository, including Zeta23/LinAlg/ (the §3 linear algebra, produced first as a standalone development accompanying the paper, under the paper authors' direction, and incorporated unchanged). The paper's authors wrote the mathematics being formalized, directed the formalization and reviewed its outputs, and wrote no Lean by hand; Eric Easley orchestrated the Lean work (paper Acknowledgments); the paper author read the main challenge module against the paper (review.notes). See AUDIT.md for the checks run at each re" }, "scope": "Complete, sorry-free Lean 4 proofs over Mathlib alone (axioms propext/Classical.choice/Quot.sound) of the paper's headline theorems. The submitted configuration comparator.json covers the 17 statements of the Mathlib-only challenge module Challenge.lean: Theorems A and B at the constants stated in the paper, including the Montgomery–Taylor-window forms and the Dirichlet analogues, with their on-the-line companion statements. Also in the repository, each checked but not part of the submission: comparator-xiprime.json (Challenge/XiPrime.lean, 6 statements on the zeros of ξ′, Remark 7.1 of the pa", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "two_thirds_simple_on_critical_line", "file": "Challenge.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "five_sixths_distinct", "file": "Challenge.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "montgomery_taylor_simple_on_critical_line_mult", "file": "Challenge.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "two_thirds_on_critical_line", "file": "Challenge.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "dirichlet_two_thirds_simple_on_critical_line", "file": "Challenge.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "liminf bounds are rendered as: for all ε > 0 there is T₀ such that for all T ≥ T₀, (c − ε)·N ≤ X. \"Nontrivial zero\" is rendered as a zero with 0 < Re ρ < 1. Windows are T₁ < Im ρ ≤ T₂ (positive ordinates). Left sides count with multiplicity; N₀*, N₀ˢ, N_d count distinct points (the strong direction). Theorem B of the paper is formalized for primitive characters of modulus q > 1. The repository states the theorems at the paper's constants; the weaker Cauchy–Schwarz-form variants that earlier revisions also certified are implied by these and are no longer separately stated. The 5/6 constant is obtained from the rank–trace inequality with parameter c = 3 where the paper's text uses Proposition 4.5(iii) with c = 2. (In the non-submitted ξ′ configuration the proportion statements carry fixed decimal constants rather than ε-forms.) See README.md, \"Reading notes for the statements\".", "alignment": true, "original": false, "review": { "status": "author-verified", "bucket": "author-verified", "reviewers": [ "Ralph Furman" ], "notes": "Ralph Furman (paper author) read the challenge module Challenge.lean (the 17 statements and their definition layer) and confirmed it matches the paper; the ξ′ challenge module (Challenge/XiPrime.lean, not submitted) is agent-reviewed. Recorded in AUDIT.md at the pinned commit: `#print axioms` checks for all 23 statements (exactly [propext, Classical.choice, Quot.sound]) and Comparator + NanoDa kernel runs for both configurations. The underlying paper was reviewed and endorsed by a human analytic number theorist, who also produced an independent condensed proof of the main result; journal peer review is pending. The Lean statements were additionally cold-read by Claude instances with no prior project context." }, "canonical": { "repo": "anthropics/formal-math", "directory": "zeta23" }, "nodes": [ { "id": "zeta:critical-line-proportion", "join": "artifact", "page": "t/zeta-critical-line-proportion.html", "label": "Proportion of zeta zeros proved simple and on the critical line" } ], "container": false, "anchors": {}, "palomar": [], "checks": [ { "by": "qed.bot", "verdict": "verified", "url": "https://github.com/anthropics/zeta-23-lean", "commit": "cec57f919ccf34e5fa5372b4ba332f7c848bbb6e", "theorems": 23, "nonstandard_axioms": [] } ], "checked_by": [ "qed.bot" ], "contradictions": [], "confirmations": 1 }, { "id": "AxiomMath/TanArctan/formalization.yaml", "repo": "AxiomMath/TanArctan", "path": "formalization.yaml", "directory": "", "origins": [ "cited", "search" ], "url": "https://github.com/AxiomMath/TanArctan/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "TanArctan", "description": "`TanArctan` is a formal proof of the divisibility barrier and the bound on the exceptional set in *Integer values of $\\tan(\\arctan 1+\\arctan 2+\\cdots+\\arctan n)$ are rare* (arXiv:2607.05739). It proves Theorem 1.1, the proximity estimate (12) in the proof of Lemma 3.1, and Lemma 3.1 itself. The Lean files were generated by AxiomProver, Axiom Math's in-house theorem proving system.", "authors": [ "Kenny Lau" ], "maintainers": [ "Kenny Lau" ], "license": "MIT", "role": "substantive-development", "substantive": "", "sources": [ { "title": "Integer values of $\\tan(\\arctan 1+\\arctan 2+\\cdots+\\arctan n)$ are rare", "id": "https://arxiv.org/abs/2607.05739", "authors": [ "Ken Ono" ], "type": "article", "relationship": "formalizes", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.NT", "math.CO" ], "msc2020": [ "11D09", "11N37" ] }, "automation": { "methods": [ { "method": "autonomous", "models": [ "AxiomProver" ], "framework": "" } ], "strongest": "autonomous", "models": [ "AxiomProver" ], "spend": "", "notes": "" }, "scope": "Theorem 1.1 for every $n \\ge 1$ with $A_n \\ne 0$, equation (12) for every $n \\in E$, and Lemma 3.1. `problem.lean` defines $E$ as $\\{n \\ge 5 : A_n = 0 \\text{ or } |x_n| > n/2 + 1\\}$, so an index where $x_n$ is undefined counts as exceptional.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "thm_main", "file": "output/solution.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "lem_proximity", "file": "output/solution.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "cor_density", "file": "output/solution.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "", "alignment": true, "original": false, "review": { "status": "author-verified", "bucket": "author-verified", "reviewers": [ "Kenny Lau" ], "notes": "The formal challenge in `output/problem.lean` was checked against the source paper by an author of that paper." }, "canonical": { "repo": "axiommath/tanarctan", "directory": "" }, "nodes": [ { "id": "arxiv:2607.05739/TanArctanSum", "join": "artifact", "page": "s/arxiv-2607-05739-tanarctansum.html", "label": "Integer values of $\\tan(\\arctan 1 + \\arctan 2 + \\cdots + \\arctan n)$" } ], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "benkeene/erdos266/formalization.yaml", "repo": "benkeene/erdos266", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/benkeene/erdos266/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Erdős problem #266: the Kovač–Tao disproof of Stolarsky's conjecture", "description": "A Lean 4 formalization, against Mathlib, of the Kovač–Tao resolution of Erdős problem #266 (a conjecture of Stolarsky): it is not true that every sequence of positive integers with convergent reciprocal sum admits an integer shift t >= 1 making the shifted Ahmes series irrational. The stronger construction is also formalized: a strictly increasing sequence of positive integers whose shifted series has a rational sum for every admissible rational shift.", "authors": [ "Ben Keene" ], "maintainers": [ "Ben Keene" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "On several irrationality problems for Ahmes series", "id": "https://arxiv.org/abs/2406.17593", "authors": [ "Vjekoslav Kovač", "Terence Tao" ], "type": "article", "relationship": "formalizes", "endorsement": "" }, { "title": "Erdős problem #266 (erdosproblems.com)", "id": "https://www.erdosproblems.com/266", "authors": [ "Thomas F. Bloom (site); problem due to Kenneth Stolarsky, recorded by Erdős and Graham" ], "type": "web page", "relationship": "background", "endorsement": "" }, { "title": "Formal Conjectures: ErdosProblems/266.lean", "id": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/266.lean", "authors": [ "The Formal Conjectures Authors" ], "type": "code", "relationship": "adapts", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.NT", "math.CA" ], "msc2020": [ "11J72", "40A05", "11B83" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Claude Fable 5 (Anthropic)" ], "framework": "Claude Code" } ], "strongest": "agent", "models": [ "Claude Fable 5 (Anthropic)" ], "spend": "", "notes": "" }, "scope": "COMPLETE. Both compared theorems are fully proved. The only two `sorry`s in the repository are the deliberate holes in Challenge.lean (the Comparator convention); Solution.lean and the entire Erdos266 development library are sorry-free, and both compared theorems depend on exactly propext, Classical.choice and Quot.sound.", "sorry_count": 2, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The arXiv abstract of [KoTa24] states the final construction for integer shifts (t in Z) only; the body of the paper proves more. Checked against arXiv v4 (14 Jul 2025): Erdos266.erdos_266_rational_shifts is Theorem 2.11 verbatim — a strictly increasing sequence of positive integers whose shifted series converges to a rational for every t in Q avoiding {-a_n} — and Section 2.3 states explicitly that the conjecture is disproved \"not only when t ranges over the integers\" but for rational t. Erdos266.erdos_266 is the negative answer to Stolarsky's conjecture as recorded by Erdős–Graham (integer shifts t >= 1), implied by Theorem 2.11. \"Converges to a rational\" is rendered as HasSum toward the real embedding of a rational; for these series (all but finitely many terms positive) unconditional and ordered convergence agree.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Ben Keene" ], "notes": "The author directs and reviews the development as it is produced. No external or independent review has been performed. Every step is checked by Lean." }, "canonical": { "repo": "benkeene/erdos266", "directory": "" }, "nodes": [ { "id": "erdos:266", "join": "names", "page": "s/erdos-266.html", "label": "Erdős Problem 266" } ], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-20-000008", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-20-000008", "commit": "aa0cd43fb45d4f8a3019bf8925424c06b1e67874", "trust": "high", "theorems": 2, "date": "2026-08-20" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "deancureton/MovingSofa/formalization.yaml", "repo": "deancureton/MovingSofa", "path": "formalization.yaml", "directory": "", "origins": [ "cited", "search" ], "url": "https://github.com/deancureton/MovingSofa/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Optimality of Gerver's sofa", "description": "A Lean 4 proof of Jineon Baek's theorem (arXiv:2411.19826) that Gerver's sofa solves the moving sofa problem. A moving sofa is a nonempty closed connected set in the horizontal arm of the closed L-shaped hallway of width one, together with a continuous path of plane isometries, starting at the identity, that keeps the set inside the hallway and ends with it in the vertical arm. The main theorem says that the supremum of the areas of moving sofas equals the area of Gerver's sofa, which is defined as the intersection of the rotated and translated hallways along Gerver's rotation path. The suprem", "authors": [ "Dean Cureton" ], "maintainers": [ "Dean Cureton" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Optimality of Gerver's Sofa", "id": "arXiv:2411.19826v1", "authors": [ "Jineon Baek" ], "type": "paper", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "On moving a sofa around a corner", "id": "Geometriae Dedicata 42 (1992), 267-283", "authors": [ "Joseph L. Gerver" ], "type": "paper", "relationship": "adapts", "endorsement": "not-contacted" }, { "title": "Differential equations and exact solutions in the moving sofa problem", "id": "Experimental Mathematics 27 (2018), 316-330", "authors": [ "Dan Romik" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" } ], "related": [ { "id": "https://github.com/google-deepmind/formal-conjectures/tree/ddfbaf90f4482030d88aae5233fe933874296a23", "relationship": "adapts", "note": "The source of the statement; it contains no proof. MovingSofaSubmission/Challenge.lean copies the moving sofa definitions and the three compared statements, unchanged, from FormalConjectures/Wikipedia/MovingSofa.lean at this commit. Four things differ: the plane notation and instances are inlined, the rigid-motion topology instance is given a name, docstrings are added to the definitions that had " }, { "id": "https://github.com/dawidmtrela-dotcom/GerverSofaLean/releases/tag/v1.1.0", "relationship": "builds-on", "note": "Dawid Trela's Lean development (MIT). Used here: its kernel-checked proof that Gerver's constants exist and are unique, its proof that Gerver's sofa moves around the corner, and its certified enclosures of the constants and of the transcendental functions. MovingSofa/Canonical/GerverDefinitions.lean and MovingSofa/Gerver/Motion.lean adapt these to the Challenge definitions. The v1.1.0 release arch" }, { "id": "https://github.com/rkirov/jordan_pick/tree/b3c9b7cf7358bf81a077d78ad67e6e8247869ddd", "relationship": "builds-on", "note": "The Jordan curve theorem, used here through MovingSofa.jordan_separation. Registered with Palomar as PALOMAR-2026-08-19-000001." }, { "id": "https://github.com/Vilin97/lean-pool/tree/bb74ee07fc23bc81358d75a9c40303e5e27fced8", "relationship": "builds-on", "note": "Jonathan Ho's Brunn-Minkowski inequality with its Prekopa-Leindler prerequisites. It keeps the area above the threshold along convex combinations of convex bodies. The three files used are copied unchanged into vendor/lean-pool (Apache-2.0) and built as a library of this package, because Lake records the package name lean-pool in the manifest as «lean-pool», which Palomar's verifier rejects." }, { "id": "https://github.com/TauCetiProject/TauCeti/tree/c52a81811e4626d2e9769fe9b0c701168c211332", "relationship": "builds-on", "note": "A Jordan curve built from two arcs that meet only at their endpoints, the filled hull, and lemmas on bounded variation. The ten files used are copied unchanged into vendor/tauceti (Apache-2.0) and built as a library of this package, because at this commit Tau Ceti pins a Mathlib revision for Lean v4.34.0-rc2 and this project uses Mathlib v4.35.0-rc1." } ], "classification": { "arxiv": [ "math.MG" ], "msc2020": [ "52A40", "52A10", "49Q10" ] }, "automation": { "methods": [ { "method": "agent", "models": [], "framework": "Codex" }, { "method": "agent", "models": [], "framework": "Claude Code" } ], "strongest": "agent", "models": [], "spend": "", "notes": "AI agents wrote the proofs, with the work split between Codex and Claude Code. The maintainer directed the work throughout. All builds and commits were run locally. No human expert has reviewed the mathematics." }, "scope": "The three theorems named in comparator.json are proved: MovingSofa.GerversSofa.ABφθSpec.existsUnique (Gerver's four constants exist and are unique), MovingSofa.isMovingSofa_gerversSofa (Gerver's sofa is a moving sofa) and MovingSofa.sofaConstant_eq_volume_gerversSofa (the moving sofa constant equals the area of Gerver's sofa). No proof has hypotheses beyond those stated in the Challenge. No classical proposition is assumed, and the Green-type and Minkowski-type area facts that the paper quotes are proved here. The counts below cover everything outside the vendored Gerver certificate. The three", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "Compared with formal-conjectures: MovingSofaSubmission/Challenge.lean copies the formal-conjectures definitions and both statements unchanged and imports only Mathlib. It inlines the plane notation and instances, names the rigid-motion topology instance so that Comparator sees the same name in Challenge and Solution, and leaves out the tests, metadata and open shape-uniqueness conjecture found there. Compared with the paper: an AI review of the statement found the following differences, none of which weakens the theorem. Sofas satisfy m 0 = id and start inside the horizontal arm, where the paper allows any initial translation; canonical_paper_motion_bridge proves that a sofa of either kind is a translate of one of the other kind. Sofas are closed and connected, as in the paper's Definition 1.2, so the theorem says nothing about disconnected sets. Motions lie in the full isometry group E(2), where the paper uses SE(2); orientation preservation is proved. Boundedness and measurability are not assumed; every moving sofa is proved compact. The supremum is taken in the extended nonnegative reals and is proved finite, at least 11/5 and attained. ABφθSpec uses non-strict inequalities, whi", "alignment": true, "original": false, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [ "Codex and Claude subagent reviewers (mathematical review of each proof, and statement checks against the paper)", "Claude subagent auditors (final audit, statement audit, whole-paper scope audit)" ], "notes": "No human expert has reviewed the mathematics, and this repository is not peer reviewed. AI reviewers, each starting from a fresh context, reviewed every proof and checked its statement against the paper. Three further AI audits were run: a final audit of the repository, which passed with minor items; an audit of the statements of the main theorem and of ABφθSpec.existsUnique against the paper, which found them correct with the differences listed under fidelity; and a scope audit of the whole paper, which found the 20 unformalized environments listed under status.known_gaps." }, "canonical": { "repo": "deancureton/movingsofa", "directory": "" }, "nodes": [ { "id": "wikipedia:MovingSofa", "join": "artifact", "page": "s/wikipedia-movingsofa.html", "label": "Moving Sofa Problem" } ], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "Dishah3241/Erdos1007/formalization.yaml", "repo": "Dishah3241/Erdos1007", "path": "formalization.yaml", "directory": "", "origins": [ "cited", "palomar" ], "url": "https://github.com/Dishah3241/Erdos1007/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Unit-distance graphs of dimension four with nine edges", "description": "A graph has dimension n when n is the least number such that its vertices can be placed injectively in ℝⁿ with every edge of length one (Erdős, Harary and Tutte). House proved that a graph of dimension four has at least nine edges, and that nine is attained only by K_{3,3}. This project proves the second half in Lean 4, in the form google-deepmind/formal-conjectures states it as erdos_1007.variants.dimension_four_extremal: a graph of dimension four with nine edges and no isolated vertex is isomorphic to K_{3,3}. The proof follows Chaffee and Noble's proof of House's theorem. On the way it prov", "authors": [ "Dishant Shah" ], "maintainers": [ "Dishant Shah" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Dimension 4 and dimension 5 graphs with minimum edge set", "id": "https://ajc.maths.uq.edu.au/pdf/64/ajc_v64_p327.pdf", "authors": [ "Joe Chaffee", "Matt Noble" ], "type": "paper", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "A 4-dimensional graph has at least 9 edges", "id": "https://doi.org/10.1016/j.disc.2013.05.005", "authors": [ "Roger F. House" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "On the dimension of a graph", "id": "https://doi.org/10.1112/S0025579300005222", "authors": [ "Paul Erdős", "Frank Harary", "William T. Tutte" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Embedding graphs in Euclidean space", "id": "https://doi.org/10.1016/j.jcta.2019.105146", "authors": [ "Nóra Frankl", "Andrey Kupavskii", "Konrad J. Swanepoel" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Erdős Problem #1007", "id": "https://www.erdosproblems.com/1007", "authors": [ "Thomas F. Bloom" ], "type": "other", "relationship": "background", "endorsement": "" } ], "related": [ { "id": "https://github.com/google-deepmind/formal-conjectures/blob/2a46c7bd74505b85f4967475bb733ded0ef8d348/FormalConjectures/ErdosProblems/1007.lean", "relationship": "builds-on", "note": "The formal statement is this file's erdos_1007.variants.dimension_four_extremal, reproduced with its two supporting definitions, SimpleGraph.UnitDistanceEmbeddable and SimpleGraph.HasDimension, inlined so that the statement depends on Mathlib alone." }, { "id": "https://github.com/plby/lean-proofs", "relationship": "independent", "note": "Boris Alexeev's formalization, made with Aristotle according to his comment on the problem's forum, proves the other half of problem 1007: formal-conjectures' erdos_1007, that the least number of edges of a graph of dimension four is nine. That statement requires a nine-edge graph of dimension four and the representability in ℝ³ of every graph with at most eight edges, both of which are also prove" } ], "classification": { "arxiv": [ "math.CO", "math.MG" ], "msc2020": [ "05C62", "52C10" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "glm-5.3-flash" ], "framework": "pi coding agent 0.85.1" }, { "method": "agent", "models": [ "claude-opus-5" ], "framework": "Claude Code" }, { "method": "agent", "models": [ "gpt-6-astra" ], "framework": "Codex CLI 0.155.1" }, { "method": "agent", "models": [ "grok-4.7", "glm-5.3-flash" ], "framework": "Grok CLI 1.0.41; pi coding agent 0.85.1" }, { "method": "agent", "models": [ "claude-opus-5-5", "grok-4.7", "glm-5.3-flash" ], "framework": "Claude Code 2.1.280; Grok CLI 1.0.41; pi coding agent 0.85.1" } ], "strongest": "agent", "models": [ "claude-opus-5", "claude-opus-5-5", "glm-5.3-flash", "gpt-6-astra", "grok-4.7" ], "spend": "subscription-based usage", "notes": "The owner chose the target, signed off on what each statement-level declaration must mean (docs/compass.md), and approved publication; the owner wrote no Lean. Every Lean change reached this repository through a manager that re-ran the kernel checks, and the model names above are taken from the run logs. Model identifiers are as the harnesses report them; the Grok CLI records its usage as grok-4.7-build." }, "scope": "Complete for its target, the extremal half of Erdős problem 1007 exactly as formal-conjectures states it. The other half, that the least number of edges of a graph of dimension four is nine, is not stated here, although both of its ingredients are proved. The dimension-five results of the same paper are out of scope.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "Erdos1007.Palomar.target", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Erdos1007.Standalone.Mathlib.InlineErdos1007.DimensionFourExtremal.proof", "file": "Erdos1007/Standalone/Mathlib/InlineErdos1007Proof.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "Erdos1007.Standalone.Mathlib.InlineErdos1007.DimensionFourExtremal.witness.proof", "file": "Erdos1007/Standalone/Mathlib/InlineErdos1007Proof.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "None in meaning. The statement is formal-conjectures' erdos_1007.variants.dimension_four_extremal with its supporting definitions inlined and its ℝ^n notation written out as EuclideanSpace ℝ (Fin n). With both builds loaded together, the inlined statement is equal to the upstream declaration's type by rfl, and the proof elaborates against that type using only propext, Classical.choice and Quot.sound (docs/upstream-check-2026-09-22.md). Two earlier reviews checked the same equality against copies of the upstream definitions (docs/red-team-2026-09-20.md, docs/review-2026-09-22.md). The statement carries the upstream hypothesis that no vertex is isolated. It is necessary: adding isolated vertices to K_{3,3} changes neither the dimension nor the edge count, and a kernel-checked companion proves the statement false without it.", "alignment": true, "original": false, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [ "gpt-6-astra (Codex CLI): adversarial review of the statement", "claude-opus-5-5 (Claude Code): independent review of the proof", "Dishant Shah: sign-off on the meaning of the statement-level declarations" ], "notes": "Kernel-checked from a clean tree: lake build with warnings as errors; lake exe axioms (only propext, Classical.choice, Quot.sound); the fidelity battery, including separating examples, a non-vacuity witness and hypothesis-drop counterexamples; module-system, layering, standalone-mathlib, proof-links, style, documentation and palomar-compatibility audits; Mathlib's environment linters; leanblueprint checkdecls; and scripts/audit-probes.sh, which checks that each audit rejects what it should. Comparator accepted Solution against Challenge with both the Lean kernel and NanoDa (docs/comparator-2026-09-22.md). No independent human has reviewed the proof. Its correctness rests on the kernel checks above, and the statement's meaning on the reviews listed here." }, "canonical": { "repo": "dishah3241/erdos1007", "directory": "" }, "nodes": [ { "id": "erdos:1007", "join": "anchor", "page": "s/erdos-1007.html", "label": "Erdős Problem 1007" } ], "container": false, "anchors": { "erdos:1007": "https://github.com/google-deepmind/formal-conjectures/blob/2a46c7bd74505b85f4967475bb733ded0ef8d348/FormalConjectures/ErdosProblems/1007.lean" }, "palomar": [ { "id": "PALOMAR-2026-09-23-000003", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-23-000003", "commit": "43f89415a6663848a1445effbda8b3abe77b052b", "trust": "high", "theorems": 1, "date": "2026-09-23" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "Dishah3241/Erdos1007Dim5/formalization.yaml", "repo": "Dishah3241/Erdos1007Dim5", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/Dishah3241/Erdos1007Dim5/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Erdős 1007, dimension five: fifteen edges, attained by K6 and K1,3,3", "description": "The least number of edges of a graph of Euclidean dimension five (Erdős–Harary–Tutte) is 15, and both K6 and K1,3,3 attain it (Chaffee and Noble 2016). These are formal-conjectures' erdos_1007.variants.dimension_five and dimension_five_extremal. A graph has dimension n when n is the least dimension in which its vertices can be placed injectively with every edge a unit segment. The lower bound is Chaffee and Noble's Theorem 10: every graph with at most fourteen edges has such a placement in ℝ⁴. Its proof removes a vertex of degree at most three, joins that vertex's neighbours to one another, pl", "authors": [ "Dishant Shah" ], "maintainers": [ "Dishant Shah" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Dimension 4 and dimension 5 graphs with minimum edge set", "id": "https://ajc.maths.uq.edu.au/pdf/64/ajc_v64_p327.pdf", "authors": [ "Joe Chaffee", "Matt Noble" ], "type": "paper", "relationship": "formalizes", "endorsement": "not-contacted" } ], "related": [ { "id": "https://github.com/google-deepmind/formal-conjectures/blob/2a46c7bd74505b85f4967475bb733ded0ef8d348/FormalConjectures/ErdosProblems/1007.lean", "relationship": "builds-on", "note": "The formal statement is this file's erdos_1007.variants.dimension_five and erdos_1007.variants.dimension_five_extremal, reproduced with its supporting definitions inlined so that it depends on Mathlib alone." } ], "classification": { "arxiv": [ "math.CO", "math.MG" ], "msc2020": [ "05C62", "52C10" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "glm-5.3-flash" ], "framework": "pi coding agent 0.85.1" }, { "method": "agent", "models": [ "glm-5.3-flash", "grok-4.7-build" ], "framework": "pi coding agent 0.85.1; Grok CLI 1.0.41" }, { "method": "agent", "models": [ "grok-4.7-build", "glm-5.3-flash", "Grok 4.7 500K Extra High Fast" ], "framework": "Grok CLI 1.0.41; pi coding agent 0.85.1; Cursor Agent 2026.09.18-9a7762b" }, { "method": "agent", "models": [ "claude-opus-5-5" ], "framework": "Claude Code 2.1.280" } ], "strongest": "agent", "models": [ "Grok 4.7 500K Extra High Fast", "claude-opus-5-5", "glm-5.3-flash", "grok-4.7-build" ], "spend": "subscription-based usage", "notes": "The owner chose the target, signed off on what each statement-level declaration must mean (docs/compass.md), and approved publishing this repository and the library; the owner wrote no Lean. Every Lean change reached these repositories through a manager that re-ran the kernel checks. The model names above are taken from the run logs with the workspace's harness/run-model, and the harness versions from the installed tools, which predate every run." }, "scope": "Only the two upstream statements. Chaffee and Noble's Theorem 11, that K6 and K1,3,3 are the only extremal graphs, is out of scope (finding A2 in the Math workspace).", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "Erdos1007Dim5.Palomar.target", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "None in meaning. The statements are formal-conjectures' erdos_1007.variants.dimension_five and erdos_1007.variants.dimension_five_extremal, with their supporting definitions inlined (UnitDistanceEmbeddable, HasDimension and K133) and the ℝ^n notation written out as EuclideanSpace ℝ (Fin n). With both builds loaded together, each inlined statement equals the upstream declaration's type by rfl. A model of a different lineage from the one that wrote the statements ran that check (docs/upstream-check-stage1.md). Both proofs also elaborate against upstream's declaration types in one environment (docs/upstream-check-2026-09-23.md). The proof works with the library's copies of the two definitions, which have the same bodies, and rfl identifies them (docs/review-2026-09-23.md, check 2). Neither statement has a hypothesis, so there is none to drop. Kernel-checked separating examples show that each definition differs from its nearest plausible misreading, and satisfiability witnesses show that neither claim is vacuous.", "alignment": true, "original": false, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [ "grok-4.7-build (Grok CLI): independent check of the statements against upstream, by rfl", "claude-opus-5-5 (Claude Code): independent review of the proof", "Dishant Shah: sign-off on the meaning of the statement-level declarations" ], "notes": "Kernel-checked from a fresh clone of this repository, with the library fetched from GitHub at its pinned revision: lake build with warnings as errors; lake exe axioms (only propext, Classical.choice and Quot.sound); the fidelity battery of separating examples and satisfiability witnesses; the module-system, layering, standalone-mathlib, proof-links, style, documentation and palomar-compatibility audits; the Challenge check; Mathlib's environment linters; leanblueprint checkdecls; and scripts/audit-probes.sh, which confirms that each audit rejects what it should (10 of 10). Comparator accepted Solution against Challenge with both the Lean kernel and NanoDa (docs/comparator-2026-09-23.md). The review found nothing that blocks the proof. It asked for this file's placeholders to be filled and " }, "canonical": { "repo": "dishah3241/erdos1007dim5", "directory": "" }, "nodes": [ { "id": "erdos:1007", "join": "anchor", "page": "s/erdos-1007.html", "label": "Erdős Problem 1007" } ], "container": false, "anchors": { "erdos:1007": "https://github.com/google-deepmind/formal-conjectures/blob/2a46c7bd74505b85f4967475bb733ded0ef8d348/FormalConjectures/ErdosProblems/1007.lean" }, "palomar": [ { "id": "PALOMAR-2026-09-25-000003", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-25-000003", "commit": "85032a43ca6db7d92b492efd05deeea0739c4570", "trust": "high", "theorems": 1, "date": "2026-09-25" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "elliotglazer/erdos501/formalization.yaml", "repo": "elliotglazer/erdos501", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/elliotglazer/erdos501/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Erdős Problem #501 in Lean 4: the closed case and the independence of the first question", "description": "Erdős problem #501 (Erdős 1961; Erdős–Hajnal 1971, Problem 38): for every real x let A_x be a bounded set of reals of Lebesgue outer measure < 1; must there be an infinite independent set, i.e. an infinite X ⊆ ℝ with x ∉ A_y for all distinct x, y ∈ X? And if the sets A_x are closed of measure < 1, must there be an independent set of size 3? This repository formalizes both answers. Second question: yes — closed sets of measure < 1 admit an infinite independent set (Newelski–Pawlikowski–Seredyński 1987; no boundedness is needed), hence one of size 3. First question: independent of ZFC. Under CH ", "authors": [ "Elliot Glazer", "Sol" ], "maintainers": [ "Elliot Glazer" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Some unsolved problems (Problem II.9)", "id": "Magyar Tud. Akad. Mat. Kutató Int. Közl. 6 (1961), 221–254", "authors": [ "Paul Erdős" ], "type": "paper", "relationship": "background", "endorsement": "n/a" }, { "title": "Unsolved problems in set theory (Problem 38)", "id": "Proc. Sympos. Pure Math. XIII, Part I (Axiomatic Set Theory), AMS 1971, 17–48", "authors": [ "Paul Erdős", "András Hajnal" ], "type": "paper", "relationship": "background", "endorsement": "n/a" }, { "title": "Erdős problem #501 (erdosproblems.com)", "id": "https://www.erdosproblems.com/501", "authors": [ "Thomas F. Bloom (ed.)" ], "type": "web discussion", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Erdős problem #501 — forum discussion (erdosproblems.com)", "id": "https://www.erdosproblems.com/forum/thread/501", "authors": [ "Sungchul Lee", "Nat Sothanaphan", "Elliot Glazer", "Sol" ], "type": "web discussion", "relationship": "background", "endorsement": "participated" }, { "title": "Elliot Glazer — proof claims for #501 (erdosproblems.com)", "id": "https://www.erdosproblems.com/forum/user/ElliotGlazer/proof-claims", "authors": [ "Elliot Glazer" ], "type": "web discussion", "relationship": "background", "endorsement": "participated" }, { "title": "Infinite free set for small measure set mappings", "id": "Proc. Amer. Math. Soc. 100 (1987), 335–339", "authors": [ "Ludomir Newelski", "Janusz Pawlikowski", "Wiesław Seredyński" ], "type": "paper", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "Directed graphs over topological spaces: some set theoretical aspects", "id": "Israel J. Math. 11 (1972), 231–248", "authors": [ "Stephen H. Hechler" ], "type": "paper", "relationship": "formalizes", "endorsement": "n/a" }, { "title": "Erdős Problem 501 after adding ω₂ random reals (draft, revision 10)", "id": "docs/paper/erdos501_random_profiles_rev10.pdf (unpublished draft, 2026-08-16)", "authors": [ "Elliot Glazer", "Sol" ], "type": "paper", "relationship": "adapts", "endorsement": "participated" }, { "title": "Some remarks on set theory VIII", "id": "Michigan Math. J. 7 (1960), 187–191", "authors": [ "Paul Erdős", "András Hajnal" ], "type": "paper", "relationship": "background", "endorsement": "n/a" } ], "related": [ { "id": "https://github.com/flypitch/flypitch", "relationship": "builds-on", "note": "Han–van Doorn, \"A formal proof of the independence of the continuum hypothesis\" (CPP 2020), Lean 3: first-order logic, the theory ZFC, Boolean-valued models, forcing, the completeness theorem and the collapse algebra Col(ω₁, 𝒫(ω)) with independence_of_CH. The vendored library Flypitch4/ is derived from it (see below); its axiomatization of ZFC is the one mirrored, axiom by axiom, in the Mathlib-si" }, { "id": "https://github.com/ianklatzco/flypitch", "relationship": "builds-on", "note": "Lean 4 port of Flypitch (Ian Klatzco with Claude; commit ad649f8, directory flypitch4/), vendored here as Flypitch4/ and forward-ported to the pinned Mathlib, then extended with measure algebras, the random algebra and the forcing development for #501 (Flypitch4/Erdos501/); see third_party/flypitch4/ and docs/PORTING-NOTES.md" }, { "id": "https://github.com/google-deepmind/formal-conjectures", "relationship": "other", "note": "FormalConjectures/ErdosProblems/501.lean states the first question (answer(sorry)); the Mathlib-level statements here (hypotheses, independence predicate) follow it verbatim, and the faithfulness target erdos501_sentence_faithful ties the first-order sentence to that proposition. No code is shared; the relationship is \"same statement shape\"" }, { "id": "https://github.com/leanprover-community/mathlib4", "relationship": "builds-on", "note": "Mathlib.ModelTheory (languages, bounded formulas, structures, Theory.ModelsBoundedFormula, Löwenheim–Skolem via Theory.Model.isSatisfiable) is the vocabulary of the Challenge; Mathlib measure theory, ZFSet and cardinals supply the rest of the trusted statements" } ], "classification": { "arxiv": [ "math.LO", "math.CO" ], "msc2020": [ "03E35", "03E75", "03E40", "03E50", "05D10", "28A05" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Claude (Anthropic) — Fable 5 (claude-fable-5) and Opus 4.8 (claude-opus-4-8); the final integration and the Mathlib ModelTheory bridge (Erdos501/FOL/, Challenge.lean, Solution.lean) were produced by Fable 5" ], "framework": "Claude (Anthropic) agent sessions — Claude Cowork / Claude Code" }, { "method": "agent", "models": [ "GPT-5.6 (\"Sol\")" ], "framework": "" }, { "method": "manual", "models": [], "framework": "" } ], "strongest": "agent", "models": [ "Claude (Anthropic) — Fable 5 (claude-fable-5) and Opus 4.8 (claude-opus-4-8); the final integration and the Mathlib ModelTheory bridge (Erdos501/FOL/, Challenge.lean, Solution.lean) were produced by Fable 5", "GPT-5.6 (\"Sol\")" ], "spend": "subscription-based usage", "notes": "Full disclosure: the mathematical argument of the positive direction (docs/paper/) was produced with GPT-5.6 (\"Sol\"), tasked by the author; every line of Lean in this repository was written by Claude (Fable 5 and Opus 4.8) under the author's direction, no line by hand. The Lean 4 port of Flypitch that is vendored (Flypitch4/) was itself produced with Claude by Ian Klatzco. All proofs are checked by Lean's kernel; the comparator (leanprover/comparator) accepts both challenge configurations with the standard axioms only, and no declaration in the repository depends on sorryAx (docs/audits/, docs" }, "scope": "Both questions of Erdős #501: the closed case (second question, infinite independent set, hence size 3) and the independence of the first question from ZFC (Hechler's CH counterexample at Mathlib level and in the collapse model; the positive answer after 𝔠⁺ random reals), plus the faithfulness of the first-order rendering. Complete: no proof-development sorry remains.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "erdos501_closed_infinite", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "erdos501_closed_size3", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "erdos501_hechler_of_CH", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "erdos501_not_refutable", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "erdos501_not_provable", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "erdos501_independent", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "erdos501_sentence_faithful", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "(1) Independence is proved for the first-order sentence Erdos501 (\"every complete ordered field has the Erdős property\"), i.e. for the first question rendered inside ZFC; the target erdos501_sentence_faithful certifies that in Mathlib's ZFSet this sentence is equivalent to the Mathlib statement of the first question, so nothing weaker is being claimed. Inside the sentence, \"outer measure < 1\" is rendered as the existence of a cover by countably many open intervals of total length < 1 (the definition of Lebesgue outer measure), \"bounded\" as bounded above and below, \"infinite\" as \"ω injects into X\". (2) The theory ZFC is Flypitch's axiomatization (extensionality, empty set, ordered pairs, union, power set, infinity, regularity, Zorn's lemma, strong collection), which is equivalent to the usual ZFC. (3) Independence is stated semantically (Mathlib has no proof calculus): ¬ (ZFC ⊨ᵇ φ) means that some model of ZFC (with carrier in Type 0; by Löwenheim–Skolem this is no restriction) satisfies ¬φ. The underlying Flypitch results are the syntactic ¬ (ZFC ⊢ₛ' Erdos501_f) and ¬ (ZFC ⊢ₛ' ∼Erdos501_f) (comparator-flypitch.json). (4) The consistency of a positive answer is obtained from the ran", "alignment": true, "original": false, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [ "Claude (Anthropic) audit sessions: statements of the seven targets against erdosproblems.com/501 and formal-conjectures (docs/audits/2026-08-16-audit-formal-conjectures-501-statements.md), the paper (docs/audits/2026-08-16-audit-rev10-profile-certificate.md), the vendored port (third_party/flypitch4/), and the forcing development (docs/audits/2026-08-17-erdos501-forcing-audit-355bc1e.txt)", "Elliot Glazer (review of the trusted statements and of the mathematics)" ], "notes": "No independent human peer review has been performed yet. Mechanical checks: lake build of the whole repository, #print axioms of every target and of the forcing tree (docs/audits/), and the comparator on both configurations (docs/COMPARATOR.md), all at the pinned toolchain (Lean v4.34.0-rc1, Mathlib 355bc1e0ed1d36e49525121e1a280ca13a058a92)." }, "canonical": { "repo": "elliotglazer/erdos501", "directory": "" }, "nodes": [ { "id": "erdos:501", "join": "names", "page": "s/erdos-501.html", "label": "Erdős Problem 501" } ], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-19-000002", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-19-000002", "commit": "218d1c1e46f77d4db80e566d1721782e85b94a17", "trust": "high", "theorems": 7, "date": "2026-09-06" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "gotrevor/lean-gallery/formalization.yaml", "repo": "gotrevor/lean-gallery", "path": "formalization.yaml", "directory": "", "origins": [ "cited" ], "url": "https://github.com/gotrevor/lean-gallery/blob/HEAD/formalization.yaml", "version": "v0.3", "missing": [], "name": "LeanGallery", "description": "", "authors": [ "Trevor Morris" ], "maintainers": [], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "On the restricted ordinal theorem", "id": "https://doi.org/10.2307/2268019", "authors": [ "R. L. Goodstein" ], "type": "paper", "relationship": "", "endorsement": "" }, { "title": "Accessible independence results for Peano arithmetic", "id": "https://doi.org/10.1112/blms/14.4.285", "authors": [ "Laurie Kirby", "Jeff Paris" ], "type": "paper", "relationship": "", "endorsement": "" }, { "title": "Erdős Problem #403 (only finitely many powers of two are sums of distinct factorials)", "id": "https://www.erdosproblems.com/403", "authors": [ "P. Frankl", "Shen Lin" ], "type": "problem", "relationship": "", "endorsement": "" }, { "title": "On consecutive sums in sequences", "id": "https://doi.org/10.1007/BF01949064", "authors": [ "Norbert Hegyvári" ], "type": "paper", "relationship": "", "endorsement": "" }, { "title": "On the irrationality of ∑ 1/(qⁿ + r)", "id": "https://www.cecm.sfu.ca/~pborwein/PAPERS/P59.pdf", "authors": [ "Peter Borwein" ], "type": "paper", "relationship": "", "endorsement": "" }, { "title": "Answer to the Burr–Erdős question on restricted addition (HHP07)", "id": "https://doi.org/10.1017/S0963548306008224", "authors": [ "Norbert Hegyvári", "François Hennecart", "Alain Plagne" ], "type": "paper", "relationship": "", "endorsement": "" }, { "title": "A Fancy Way to Generate the Binary Digits of √2 / Graham–Pollak-type recurrences", "id": "https://arxiv.org/abs/0902.4168", "authors": [ "Thomas Stoll" ], "type": "preprint", "relationship": "", "endorsement": "" } ], "related": [], "classification": { "arxiv": [], "msc2020": [] }, "automation": { "methods": [ { "method": "agent", "models": [], "framework": "Claude Code" }, { "method": "autonomous", "models": [], "framework": "lean-treadmill (custom autonomous grind harness over Claude Code)" } ], "strongest": "autonomous", "models": [], "spend": "subscription-based usage (Claude Max), not metered API spend -- so there is no meaningful dollar figure to report. The real cost was usage-window, not currency.", "notes": "Models: Anthropic Claude, driven through Claude Code. The autonomous harness defaults to the Opus tier, with Sonnet available as an adaptive floor on cheaper laps. Exact per-result model VERSIONS were not logged at the time and are not reconstructable, so they are not listed here rather than guessed; development spans roughly 2026-05-30 to 2026-07-06, across several Opus releases. Wall-clock, per result, is reported as the CALENDAR SPAN between the first and last commit of the corresponding private development repo. This OVERSTATES effort: the repos stayed open for polish after the mathematics" }, "scope": "A publish-only showcase: seven finished, axiom-clean formalizations, promoted here from private development repos once complete. Every headline is machine-certified twice over -- `scripts/AxiomCheck.lean` pins each theorem's axiom set with `#guard_msgs`, and `leanprover/comparator` (see `comparator_config` below, and .github/workflows/comparator.yml) independently certifies that each headline really proves the statement written in a Mathlib-only challenge file, replayed through BOTH the Lean kernel and the independent `nanoda` kernel.", "sorry_count": 1, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "LeanGallery.Logic.Goodstein.goodstein_terminates", "file": "LeanGallery/Logic/Goodstein/Statement.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "LeanGallery.Logic.Hydra.hydra_terminates", "file": "LeanGallery/Logic/Hydra/Statement.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "LeanGallery.NumberTheory.Erdos403.erdos_403_finite", "file": "LeanGallery/NumberTheory/Erdos403/Statement.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "LeanGallery.Combinatorics.Erdos1213.erdos_1213", "file": "LeanGallery/Combinatorics/Erdos1213/Statement.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "LeanGallery.NumberTheory.Erdos1050.erdos_1050", "file": "LeanGallery/NumberTheory/Erdos1050/Basic.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "LeanGallery.Combinatorics.Erdos880.erdos_880", "file": "LeanGallery/Combinatorics/Erdos880/Statement.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "LeanGallery.NumberTheory.Erdos482.graham_pollak", "file": "LeanGallery/NumberTheory/Erdos482/Statement.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "Erdős #482 is resolved in GREATER generality than the source: `erdos482_resolution` proves that for every real w > 0 and base g >= 2 an explicit Graham-Pollak-type recurrence reads the base-g digits of w, not merely the binary digits of √2. Erdős #880 is likewise extended beyond the headline question (exact order, and the HHP07 Theorem 3/4/8/9 companions are included). Erdős #1050: the general engine (`borwein_thm1_abs`) proves Borwein's Theorem 1 in full -- ∑ 1/(qⁿ+c) is irrational for any integer base of magnitude >= 2 and any nonzero rational c -- which is stronger than the q=2, c=-3 instance the problem asks for. In the comparator challenges, both k >= 3 headlines of #880 are stated EXISTENTIALLY (\"there is a basis of order h whose restricted-sum set has unbounded gaps\"), which is what HHP07 Thm 1(ii) claims. The proof supplies the explicit Hegyvári-Hennecart-Plagne witness, but naming that construction in the challenge would drag its recurrence into the trusted surface for no gain.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Trevor Morris" ], "notes": "No external peer review. What IS machine-enforced, on every push: (1) `scripts/AxiomCheck.lean` -- one `#guard_msgs`-pinned `#print axioms` per headline, asserting the exact standard triple. Any drift (new axiom, a `sorry`, a renamed theorem) fails elaboration. Note this runs inside our own Lean environment. (2) `leanprover/comparator` -- for each result, a Mathlib-only `Challenge.lean` states the headlines; comparator checks the solution proves THOSE statements (every declaration used in a statement must be identical in both environments), replays the proofs through the Lean kernel AND the independent `nanoda` kernel, and enforces the axiom whitelist. The solution is built under a `landrun` sandbox, so a reader can check this repo WITHOUT trusting it or us. (3) warnings-as-errors, and Mat" }, "canonical": { "repo": "gotrevor/lean-gallery", "directory": "" }, "nodes": [ { "id": "erdos:1050", "join": "artifact", "page": "s/erdos-1050.html", "label": "Erdős Problem 1050" } ], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "ibrahimmian36/palomar-decanus/formalization.yaml", "repo": "ibrahimmian36/palomar-decanus", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/ibrahimmian36/palomar-decanus/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "A kernel-checked exclusion certificate for Erdős Problem 647, to 10^8", "description": "Erdős Problem 647 asks whether there is any n > 24 with max_{m < n} (m + tau(m)) <= n + 2, where tau counts divisors. The value n = 24 satisfies it and the question is open. This entry certifies a bounded exclusion: no such n exists in (24, 10^8]. It does not resolve the problem and asserts nothing about n > 10^8. The exclusion is certified rather than searched at check time. A witness at a cover position is an offset together with prime-power pairs dividing m, certifying tau(m) >= prod (e + 1); every prime in a pair is below 1024 and is checked prime by eleven trial divisions, so no primality", "authors": [ "Ibrahim Mian", "Shayaan Siddique" ], "maintainers": [ "Ibrahim Mian" ], "license": "Apache-2.0", "role": "thin-wrapper", "substantive": "ibrahimmian36/decanus", "sources": [ { "title": "A Kernel-Checked Exclusion Certificate for Erdős Problem 647", "id": "https://arxiv.org/abs/2608.17880", "authors": [ "Ibrahim Mian", "Shayaan Siddique" ], "type": "paper", "relationship": "formalizes", "endorsement": "" }, { "title": "Erdős Problem 647", "id": "https://www.erdosproblems.com/647", "authors": [], "type": "website", "relationship": "formalizes", "endorsement": "" }, { "title": "Formal Conjectures: ErdosProblems/647", "id": "https://github.com/google-deepmind/formal-conjectures", "authors": [ "The Formal Conjectures Authors" ], "type": "repository", "relationship": "background", "endorsement": "" }, { "title": "Erdős problem 647: certificate data and verification artifacts", "id": "https://doi.org/10.5281/zenodo.21996019", "authors": [ "Ibrahim Mian", "Shayaan Siddique" ], "type": "dataset", "relationship": "background", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "cs.LO", "math.NT" ], "msc2020": [ "11N56", "68V20" ] }, "automation": { "methods": [ { "method": "manual", "models": [], "framework": "" }, { "method": "other", "models": [], "framework": "Python generator and independent replayer" } ], "strongest": "manual", "models": [], "spend": "", "notes": "No proof step is closed by an automated prover. Nothing in the development uses native_decide: the whole-library audit in Erdos647/AxiomAudit.lean fails the build if any theorem under Erdos647.* depends on anything beyond propext, Classical.choice and Quot.sound, which excludes both sorryAx and the native_decide axioms." }, "scope": "", "sorry_count": null, "sorry_in_definitions": null, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "", "alignment": false, "original": false, "review": { "status": "internal", "bucket": "other", "reviewers": [], "notes": "Verified from source on two independent toolchains, gcc and clang, with the raw results of both legs archived at doi:10.5281/zenodo.21996019. Axioms are gated twice: a curated manifest in Erdos647/AxiomCheck.lean and a mechanical whole-library audit in Erdos647/AxiomAudit.lean, both required to pass in CI on every push. No external peer review of the formalization has taken place." }, "canonical": { "repo": "ibrahimmian36/palomar-decanus", "directory": "" }, "nodes": [ { "id": "erdos:647", "join": "names", "page": "s/erdos-647.html", "label": "Erdős Problem 647" } ], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-04-000003", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-04-000003", "commit": "776a5817bffcd9c30dd96259ccb7cfc822ec8ec4", "trust": "high", "theorems": 1, "date": "2026-09-04" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "itpplasma/navier-formal/formalization.yaml", "repo": "itpplasma/navier-formal", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/itpplasma/navier-formal/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Navier–Stokes critical-norm reduction on R3 (checkpoint CP1)", "description": "Lean 4 / Mathlib formalization of the checkpoint block of a research programme on the three-dimensional incompressible Navier–Stokes Cauchy problem: the energy identity, critical scaling, cubic enstrophy inequality, exact signed critical pressure balance, low-frequency pressure bound, cubic gradient-quotient functional, and the conditional theorem that a finite-horizon a priori L3 bound implies Clay alternative A. The arbitrary-data critical bound itself is an explicit hypothesis. No solution of the Millennium problem is claimed.", "authors": [ "Christopher Albert" ], "maintainers": [ "Christopher Albert" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Internal CP1 proof development for the Navier–Stokes critical-norm route", "id": "", "authors": [ "Christopher Albert" ], "type": "other", "relationship": "other", "endorsement": "" }, { "title": "Existence and smoothness of the Navier–Stokes equation", "id": "https://www.claymath.org/library/monographs/MPPc.pdf", "authors": [ "Charles L. Fefferman" ], "type": "book", "relationship": "background", "endorsement": "" }, { "title": "Localisation and compactness properties of the Navier–Stokes global regularity problem", "id": "10.2140/apde.2013.6.25", "authors": [ "Terence Tao" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "A profile decomposition approach to the L∞_t(L³_x) Navier–Stokes regularity criterion", "id": "arXiv:1012.0145", "authors": [ "Isabelle Gallagher", "Gabriel S. Koch", "Fabrice Planchon" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "L_{3,∞}-solutions of the Navier–Stokes equations and backward uniqueness", "id": "10.1070/RM2003v058n02ABEH000609", "authors": [ "Luis Escauriaza", "Gregory Seregin", "Vladimír Šverák" ], "type": "paper", "relationship": "background", "endorsement": "" } ], "related": [ { "id": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Millenium/NavierStokes.lean", "relationship": "builds-on", "note": "Reference statements of Fefferman's alternatives; alternative (A) is inlined in NavierFormal/ClayReference.lean with the Apache-2.0 header." }, { "id": "https://github.com/openai/NavierStokesAndEuler", "relationship": "builds-on", "note": "Solution-only pinned dependency at commit 8937a8f4 (forced alternatives C/D, Apache-2.0); only NavierFormal/External/ imports it." } ], "classification": { "arxiv": [ "math.AP" ], "msc2020": [ "35Q30", "76D05", "76D03", "35B65" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "claude-fable-5-1", "claude-opus-5", "claude-sonnet-5" ], "framework": "Claude Code" } ], "strongest": "agent", "models": [ "claude-fable-5-1", "claude-opus-5", "claude-sonnet-5" ], "spend": "subscription-based", "notes": "Human controller directs the programme and owns integration and promotion; Claude models draft proofs, Lean modules, and independent audits." }, "scope": "Partial. The advertised statement surface is written and proved: nine CP1 statements are declared with Mathlib-only definitions in Challenge.lean and re-declared and proved in Solution.lean, with type identity checked mechanically at pp.all. They are the Phase II complete results only: the norm-scaling half of prop:scaling including critical L3 invariance, the two scalar remarks of its proof, and prop:ode. Everything else in CP1 remains paper only and is deliberately not advertised; see the CP1 stage gates and the Phase II programme in itpplasma/navier PLAN.md.", "sorry_count": 9, "sorry_in_definitions": 0, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "none known; statement audit pending", "alignment": true, "original": false, "review": { "status": "self-assessed; within-family independent audits recorded in itpplasma/navier", "bucket": "self-assessed", "reviewers": [ "none" ], "notes": "Statement faithfulness audit is a stage gate before any registration." }, "canonical": { "repo": "itpplasma/navier-formal", "directory": "" }, "nodes": [ { "id": "millennium:NavierStokes", "join": "anchor", "page": "s/millennium-navierstokes.html", "label": "Existence And Smoothness Of The Navier–Stokes Equation" } ], "container": false, "anchors": { "millennium:NavierStokes": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Millenium/NavierStokes.lean" }, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "jaumededios/Schiffer/formalization.yaml", "repo": "jaumededios/Schiffer", "path": "formalization.yaml", "directory": "", "origins": [ "cited" ], "url": "https://github.com/jaumededios/Schiffer/blob/HEAD/formalization.yaml", "version": "v0.3", "missing": [], "name": "schiffer_counterexample", "description": "", "authors": [ "Jaume de Dios Pont", "OpenAI Codex" ], "maintainers": [], "license": "UNLICENSED", "role": "", "substantive": "", "sources": [ { "title": "Modular proof draft for a planar Schiffer counterexample", "id": "https://github.com/caolab97/Schiffer/blob/85235fb037cbfd336e3ae9801f61951c5e2d1c4f/Schiffer_GPT_V3.tex", "authors": [ "Jaume de Dios Pont", "OpenAI Codex" ], "type": "original proof manuscript", "relationship": "", "endorsement": "" } ], "related": [], "classification": { "arxiv": [], "msc2020": [] }, "automation": { "methods": [ { "method": "agent", "models": [ "GPT-5" ], "framework": "OpenAI Codex" }, { "method": "agent", "models": [ "GPT-5" ], "framework": "OpenAI Codex" } ], "strongest": "agent", "models": [ "GPT-5" ], "spend": "not available (subscription-hosted Codex usage)", "notes": "Work proceeded through frequent source checks, cached artifact builds, and coherent milestone commits. Specialist agents handled self-contained Bessel, origin-series, final-assembly, comparator, cleanup-audit, and referee tasks. The final branch and all local development branches were copied to the private jaumededios/schiffer backup repository." }, "scope": "A complete large-N construction of a bounded, star-shaped and simply connected planar Schiffer domain that is not a Euclidean disk. The project formalizes the weighted Fourier and anisotropic Banach-space foundations, the Bessel-function and zero-density spine, a reusable quantitative Lyapunov--Schmidt/parametric-contraction reduction, uniform nonlinear cone branches, the negative quadratic radius drift, near-integer crossings, integer landing, direct Cartesian regularity at the cone tip, one-sided C2 gluing, and graph/field nontriviality. It also proves a construction-free comparator statemen", "sorry_count": 3, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "Schiffer.Analysis.SchifferCounterexample.PlanarSchifferCounterexample", "file": "Schiffer/Analysis/SchifferCounterexample.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "Schiffer.exists_schifferCounterexample", "file": "Schiffer/Main.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "Schiffer.Bifurcation.exists_uniformConeBranch", "file": "Schiffer/Bifurcation/UniformBranch.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "Schiffer.Density.exists_nearIntegerCrossing", "file": "Schiffer/Density/NearInteger.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "Schiffer.Landing.exists_landedConeSolution", "file": "Schiffer/Landing/IntegerRadius.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": false, "literature": [] } ], "comparator": false, "literature_dependencies": 0, "divergences": "This is not a line-by-line translation of a pre-existing published proof: the large-N proof manuscript, its corrected density/landing argument, and the Lean formalization were co-developed. The manuscript often uses anisotropic Holder spaces and states C2,alpha boundary regularity plus interior smoothness; the public Lean comparator currently claims the classical counterexample at C2 regularity and does not claim C2,alpha or C-infinity interior regularity. The construction-native SchifferCandidate separately records star-convexity and simple connectedness, while the plain comparator states star-shapedness, which implies contractibility but does not repeat the simple-connectedness field. The comparator normalizes the positive eigenvalue to 1 by dilation and strengthens the Neumann boundary condition to vanishing of the full Frechet derivative. The formal proof uses weighted l1 Fourier sequences, continuous radial jets, quantitative parametric contraction, and reusable Lyapunov--Schmidt splitting rather than a black-box Nash--Moser theorem. It proves tip and interface regularity directly from integer-order Bessel power series and one-sided Cartesian jets, without invoking general ell", "alignment": true, "original": false, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [ "OpenAI Codex mathematical referee agent", "OpenAI Codex Lean/API referee agent", "OpenAI Codex OriginSeries and interface-sign referee agents" ], "notes": "Multiple agents independently audited the full-radius closed-ball regularity, modewise Bessel ODE argument, x=N-r radial sign, zero-mode normalization, independent inner/outer Hessian traces, non-circular one-sided gluing, nontriviality, public API/blueprint correspondence, placeholders, default builds, and axiom usage. All reported PASS. The result has not yet received independent human peer review, publication review, or author verification in the sense of this metadata standard." }, "canonical": { "repo": "jaumededios/schiffer", "directory": "" }, "nodes": [ { "id": "wikipedia:PompeiuProblem", "join": "artifact", "page": "s/wikipedia-pompeiuproblem.html", "label": "The Pompeiu problem" } ], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "jbaelaw/erdos190-lean/formalization.yaml", "repo": "jbaelaw/erdos190-lean", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/jbaelaw/erdos190-lean/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Erdős Problem 190: every canonical N exceeds (Ck)^k for large k, and H(k)^{1/k}/k → ∞ under the existence hypothesis — a Lean 4 formalization of the qualitative statement of arXiv:2604.20588", "description": "A Lean 4 formalization, against Mathlib, of the qualitative statement of J. H. Bae, \"A resolution of Erdős Problem #190: the canonical van der Waerden number satisfies H(k)^{1/k}/k → ∞\" (arXiv:2604.20588, v2), Corollary 1.2, via the elementary argument of its Section 4.3. Let H(k) be the least N such that every finite colouring of {1,…,N} contains a monochromatic or a rainbow k-term arithmetic progression; Erdős and Graham (1979) asked whether H(k)^{1/k}/k → ∞ (Problem 190 in Bloom's database). The compared theorems state that for every C there is K such that for all k ≥ K every N for which [N", "authors": [ "Ji Ho Bae" ], "maintainers": [ "Ji Ho Bae" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "A resolution of Erdős Problem #190: the canonical van der Waerden number satisfies H(k)^{1/k}/k → ∞", "id": "https://arxiv.org/abs/2604.20588", "authors": [ "Ji Ho Bae" ], "type": "paper", "relationship": "formalizes", "endorsement": "participated" }, { "title": "Erdős Problem #190 (erdosproblems.com)", "id": "https://www.erdosproblems.com/190", "authors": [ "Thomas F. Bloom (site)" ], "type": "web discussion", "relationship": "background", "endorsement": "" }, { "title": "Old and new problems and results in combinatorial number theory: van der Waerden's theorem and related topics", "id": "Enseign. Math. (2) 25 (1979), 325–344", "authors": [ "Paul Erdős", "Ronald L. Graham" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Three-color van der Waerden numbers grow super-exponentially", "id": "https://arxiv.org/abs/2606.02541", "authors": [ "Jacob Fox", "Zach Hunter" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "A new lower bound for van der Waerden numbers", "id": "https://doi.org/10.1016/j.ejc.2017.10.007", "authors": [ "Thomas Blankenship", "Jay Cummings", "Vladislav Taranchuk" ], "type": "paper", "relationship": "adapts", "endorsement": "" }, { "title": "Problems and results on 3-chromatic hypergraphs and some related questions", "id": "Colloq. Math. Soc. János Bolyai 10 (1975), 609–627", "authors": [ "Paul Erdős", "László Lovász" ], "type": "paper", "relationship": "adapts", "endorsement": "" } ], "related": [ { "id": "https://github.com/yidiq7/ProbMethodCombinatorics", "relationship": "builds-on", "note": "Lean dependency, pinned at commit 10f2997bf965df06cfbb0129651a1af6c7367618. Its theorem lovasz_local_lemma_symmetric (the symmetric Lovász local lemma, standard axioms only) is used in Erdos190/EL.lean. It is a dependency of the Solution only; the Challenge imports Mathlib alone." } ], "classification": { "arxiv": [ "math.CO", "math.NT" ], "msc2020": [ "05D10", "11B25", "05C65" ] }, "automation": { "methods": [ { "method": "agent", "models": [], "framework": "" } ], "strongest": "agent", "models": [], "spend": "", "notes": "" }, "scope": "COMPLETE for the qualitative statement. All four compared theorems are fully proved. The only sorries in the repository are the four deliberate holes in Challenge.lean (the Comparator convention); Solution.lean and the Erdos190 development library are sorry-free, and every compared declaration depends on exactly propext, Classical.choice and Quot.sound. Not formalized: the existence of H(k) (Erdős–Graham, via Szemerédi's theorem) and the paper's explicit rate with the constant 1/e (Baker–Harman–Pintz).", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "Erdos190.Palomar.divergence", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Erdos190.Palomar.divergence_eventually", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Erdos190.Palomar.H_divergence", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Erdos190.Palomar.explicit_bound", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The formal statements are the qualitative statement of the paper (Corollary 1.2), phrased for every canonical N so as not to assume the existence of H(k); divergence alone does not entail a statement about H(k), and the conclusion H(k) > (Ck)^k for all large k is formalized only under the explicit existence hypothesis (H_divergence). The paper's Proposition 4.1 uses r₀ = ⌊k/log k⌋ and gives the rate √k/log k; the formalization uses r₀ = ⌊k/3⌋, which avoids real analysis in the asymptotic step and gives the rate k^{1/6−o(1)}. The Erdős–Lovász base is formalized with the cruder constant r^(k−1)/(16k²) (the paper has r^(k−1)/(16k)), obtained with the dependency-degree bound k²N in the local lemma; only the shape r^(k−1)/poly(k) matters. [N] is modelled as Fin N (0-indexed), which does not affect any statement. The paper's main theorem (the explicit constant 1/e and ε(k), via the Baker–Harman–Pintz prime-gap theorem) is not formalized.", "alignment": true, "original": false, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [ "Ji Ho Bae (self-assessment)", "Palomar registry automated editorial review (codex:gpt-5.6-sol)" ], "notes": "The author reviewed the statements of record in Challenge.lean against the paper. The proofs are checked by Lean (and by #print axioms over the compared declarations). Commit be43a3ead08a9d9af352cf296d284dd3468ca805 of this repository was mechanically verified (Comparator and NanoDa kernel) and passed the automated editorial review of the Palomar registry of Lean-verified results with no problems identified; it is registered as PALOMAR-2026-09-15-000003, version 1 (https://palomar-registry.org/entry?id=PALOMAR-2026-09-15-000003&version=1), with an immutable source-preservation fork at PalomarArchive/jbaelaw--erdos190-lean--ffa29cd1b3a4. No independent human review of the Lean development has been performed." }, "canonical": { "repo": "jbaelaw/erdos190-lean", "directory": "" }, "nodes": [ { "id": "erdos:190", "join": "names", "page": "s/erdos-190.html", "label": "Erdős 190" } ], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-15-000003", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-15-000003", "commit": "be43a3ead08a9d9af352cf296d284dd3468ca805", "trust": "high", "theorems": 4, "date": "2026-09-15" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "JD-Jones-ASES/Hadamard-formal/formalization.yaml", "repo": "JD-Jones-ASES/Hadamard-formal", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/JD-Jones-ASES/Hadamard-formal/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "hadamard-formal: eight Hadamard orders unrecorded in the Cati-Pasechnik database, and the Miyamoto-erratum obstruction", "description": "A Lean 4 formalization of the periodic Cooper-Wallis construction and of Hadamard existence at eight orders: from kernel-checked periodic T-matrix quadruples of orders 103 and 163 and seven symmetric-circulant Williamson quadruples, the development proves that Hadamard matrices of orders 2884, 4532, 7004, 7172, 7828, 8476, 9476, and 11948 exist. Each of the eight orders lands on an entry of Table 4 of Cati and Pasechnik's maintained construction database (arXiv:2411.18897v2, 2025-08-30) recording no Hadamard matrix of that order as known, and the source repositories' firsthand audit of the con", "authors": [ "JD Jones" ], "maintainers": [ "JD Jones" ], "license": "MIT", "role": "", "substantive": "", "sources": [ { "title": "Hadamard-T: T-matrix witnesses and the Hadamard orders they close", "id": "https://github.com/JD-Jones-ASES/Hadamard-T/tree/eec1bed6216c44675d946fe80d47695a7ae2ab40", "authors": [ "JD Jones" ], "type": "other", "relationship": "formalizes", "endorsement": "participated" }, { "title": "Hadamard-M: an erratum at order 515, and an explicit Hadamard matrix of order 7796", "id": "https://github.com/JD-Jones-ASES/Hadamard-M/tree/0208cf661fcd6b1e7d69eb52078d873eefc4d7cf", "authors": [ "JD Jones" ], "type": "other", "relationship": "formalizes", "endorsement": "participated" }, { "title": "A construction of Hadamard matrices", "id": "J. Combin. Theory Ser. A 57 (1991) 86-108", "authors": [ "M. Miyamoto" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "A database of constructions of Hadamard matrices", "id": "arXiv:2411.18897", "authors": [ "M. Cati", "D. V. Pasechnik" ], "type": "paper", "relationship": "background", "endorsement": "" } ], "related": [ { "id": "https://github.com/leanprover-community/mathlib4/blob/v4.33.0/Mathlib/LinearAlgebra/Matrix/HadamardMatrix.lean", "relationship": "builds-on", "note": "Mathlib's Hadamard-matrix module (added 2026-06-16, mathlib4 PR #38582) supplies the Matrix.IsHadamard predicate the compared statements use, closure lemmas including the Kronecker product, and the 4-divides-n obstruction. It exhibits no Hadamard matrix of any order and contains no existence theorem, and Mathlib at this revision has no content on Williamson quadruples, T-matrices, or the Goethals-" }, { "id": "https://github.com/google-deepmind/formal-conjectures/blob/abe6ef73129989fb8cf94edb72b1ac21d30a411b/FormalConjectures/Wikipedia/Hadamard.lean", "relationship": "other", "note": "States the Hadamard conjecture and its known-for-4k-with-k-at-most-166 variant in Lean as unproved benchmark targets; as of that commit, every Hadamard statement there, including an explicit order-12 matrix, is left as sorry. No proved content overlaps this development." }, { "id": "Lu-Ming Zhang, Formal Verification of Constructions and Theorems on Hadamard Matrices, MSc dissertation, University of Oxford, 2021", "relationship": "other", "note": "The one prior proof-assistant effort on Hadamard constructions our search located, known to us only through Cati and Pasechnik's citation describing it as work towards verifying constructions covering orders up to 112. We could not locate its text, artifact, or even the proof assistant used, so it cannot be independently confirmed; its stated order range is disjoint from the eight orders proved he" } ], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05B20" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Claude Code (Fable 5)", "Codex (GPT 5.6 Sol)", "GPT 5.6 (external source review)", "Grok (external review; exact model not recorded)" ], "framework": "mixed AI-agent workflow" } ], "strongest": "agent", "models": [ "Claude Code (Fable 5)", "Codex (GPT 5.6 Sol)", "GPT 5.6 (external source review)", "Grok (external review; exact model not recorded)" ], "spend": "not tracked", "notes": "AI systems generated and audited the mathematics and code; JD Jones directed the work and is the responsible author and maintainer. AI assistance is disclosed at the project level and in both source repositories." }, "scope": "The compared surface consists of the handshake parity theorem, its empty-class corollary, the nonexistence theorem for the displayed C2-matrix of Miyamoto's Corollary 4, a generic periodic Cooper-Wallis existence theorem, and eight existence theorems at orders 2884, 4532, 7004, 7172, 7828, 8476, 9476, and 11948. The order-103 and order-163 predicates are periodic autocorrelation; no aperiodic T-sequence statement is made. The development proves existence without materializing the eight large matrices, and it formalizes neither the source repositories' Python certificates and search history nor", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "HadamardFormal.handshake_mod_four", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "HadamardFormal.no_handshake_matrix", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "HadamardFormal.no_c2_matrix", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "HadamardFormal.cooperWallis", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "HadamardFormal.hadamard_2884", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "HadamardFormal.hadamard_4532", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "HadamardFormal.hadamard_7004", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "HadamardFormal.hadamard_7172", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "HadamardFormal.hadamard_7828", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "HadamardFormal.hadamard_8476", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "HadamardFormal.hadamard_9476", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "HadamardFormal.hadamard_11948", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The formalization uses the periodic T-matrix predicate required by the Cooper-Wallis construction, not the stronger aperiodic T-sequence notion, and it proves existence without materializing the resulting large matrices. On the erratum, the displayed C2 form, the pairing consequence of the paper's hypothesis (4.1), and the emptiness of the class at every positive block order not congruent to 1 mod 4 are formalized, which closes both printed readings of the corollary's output order; what remains informal is the transcription of the printed text, human-audited in Hadamard-M, and the bibliographic propagation from the unsupported order-103 entry to the order-515 list entry and its order-2060 claim.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "JD Jones" ], "notes": "The Lean kernel checks all solution proofs and concrete certificates, and the build-time axiom audit permits only propext, Classical.choice, and Quot.sound. The source mathematics received external AI review; no independent human expert review is recorded. Kernel replay, Comparator statement matching, and the axiom audit establish proof replay and statement fidelity, not literature status; the dated status facts above are documentary claims resting on the cited public sources." }, "canonical": { "repo": "jd-jones-ases/hadamard-formal", "directory": "" }, "nodes": [ { "id": "wikipedia:Hadamard", "join": "anchor", "page": "s/wikipedia-hadamard.html", "label": "Hadamard's conjecture" } ], "container": false, "anchors": { "wikipedia:Hadamard": "https://github.com/google-deepmind/formal-conjectures/blob/abe6ef73129989fb8cf94edb72b1ac21d30a411b/FormalConjectures/Wikipedia/Hadamard.lean" }, "palomar": [ { "id": "PALOMAR-2026-08-31-000001", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-31-000001", "commit": "46544fab662e04db8b6fd18b47bd994585c4685c", "trust": "high", "theorems": 12, "date": "2026-08-31" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "jidodat/erdos684-lean/formalization.yaml", "repo": "jidodat/erdos684-lean", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/jidodat/erdos684-lean/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Erdős Problem 684: f(n)/log n is unbounded — a Lean 4 formalization of arXiv:2604.23784", "description": "A Lean 4 formalization, against Mathlib, of the main theorem of J. H. Bae, \"Unbounded logarithmic limsup in Erdős Problem 684 via shifted carry scheduling\" (arXiv:2604.23784, v3). For 0 ≤ k ≤ n write C(n,k) = u·v with u supported on the primes ≤ k and v on the primes in (k,n], and let f(n) be the least k with u > n²; Erdős asked for bounds on f(n) (Problem 684 in Bloom's database). The compared theorems state that for every ε > 0 there are infinitely many n with f(n) > (1/2 − ε)·log n·log log n / log log log n, equivalently limsup f(n)·log log log n /(log n·log log n) ≥ 1/2 (stated in the exte", "authors": [ "Ji Ho Bae" ], "maintainers": [ "Ji Ho Bae" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Unbounded logarithmic limsup in Erdős Problem 684 via shifted carry scheduling", "id": "https://arxiv.org/abs/2604.23784", "authors": [ "Ji Ho Bae" ], "type": "paper", "relationship": "formalizes", "endorsement": "participated" }, { "title": "Erdős Problem #684 (erdosproblems.com)", "id": "https://www.erdosproblems.com/684", "authors": [ "Thomas F. Bloom (site)" ], "type": "web discussion", "relationship": "background", "endorsement": "" }, { "title": "Some unconventional problems in number theory", "id": "https://doi.org/10.1007/BF01903382", "authors": [ "Paul Erdős" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Short proofs in combinatorics and number theory", "id": "https://arxiv.org/abs/2603.29961", "authors": [ "Boris Alexeev", "Max Putterman", "Mehtaab Sawhney", "Mark Sellke", "Gregory Valiant" ], "type": "paper", "relationship": "background", "endorsement": "" } ], "related": [ { "id": "https://github.com/AlexKontorovich/PrimeNumberTheoremAnd", "relationship": "builds-on", "note": "Lean dependency, pinned at commit a5154676af9aa3095150ee410cdda80555aa0642 (2026-08-30). Its theorem MediumPNT (ψ(x) = x + O(x·exp(−c(log x)^{1/10})), standard axioms only) is used in PNT.lean to discharge the prime-number-theorem hypothesis of the development. It is a dependency of the Solution only; the Challenge imports Mathlib alone." } ], "classification": { "arxiv": [ "math.NT", "math.CO" ], "msc2020": [ "11B65", "11N05", "11A51" ] }, "automation": { "methods": [ { "method": "agent", "models": [], "framework": "" } ], "strongest": "agent", "models": [], "spend": "", "notes": "" }, "scope": "COMPLETE. All five compared theorems are fully proved. The only sorries in the repository are the five deliberate holes in Challenge.lean (the Comparator convention); Solution.lean and the entire Erdos684Lean development library are sorry-free, and every one of its 88 public theorems depends on exactly propext, Classical.choice and Quot.sound (checked with #print axioms over all of them).", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "Erdos684.Palomar.main_theorem", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Erdos684.Palomar.main_theorem_limsup", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Erdos684.Palomar.unbounded", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The formal statements are those of the paper (Theorem 1.2, displays (4) and (5)), with f(n) valued in ℕ∞ so that the paper's convention f(n) = +∞ for an empty defining set is literal. The proof differs from the paper's text in simplifications only, all recorded in the README: in Lemma 3.1 only the upper bound is needed and it is obtained from θ(K) ≤ (1+o(1))K and the monotonicity of x ↦ 1 − log h/log x instead of partial summation, so that the Mertens-type sum (10) of the paper is not used; ψ(x) − θ(x) = O(√x) is replaced by Mathlib's ψ(x) − θ(x) ≤ 2√x log x; the prefix extraction in Step 3 of Lemma 4.1 is an abstract lemma proved by induction on the total depth; the decomposition into the ranges (I)–(IV) carries the hypothesis M ≤ K, which the paper uses implicitly; and the prime number theorem enters as θ(x) = x + O(x/log² x), weaker than the remainder (8) quoted in the paper, and is derived from PrimeNumberTheoremAnd's MediumPNT rather than cited.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Ji Ho Bae" ], "notes": "The author reviewed the statements of record in Challenge.lean against the paper. The proofs are checked by Lean (and, in the repository, by an axiom sweep over all public theorems). No external or independent human review of the Lean development has been performed." }, "canonical": { "repo": "jidodat/erdos684-lean", "directory": "" }, "nodes": [ { "id": "erdos:684", "join": "names", "page": "s/erdos-684.html", "label": "Erdős 684" } ], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-03-000006", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-03-000006", "commit": "4543ff7764f9e8f1b2732a8464d5c19c945bfc52", "trust": "high", "theorems": 5, "date": "2026-09-03" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "linrock/math-proofs/erdos-689/formalization.yaml", "repo": "linrock/math-proofs", "path": "erdos-689/formalization.yaml", "directory": "erdos-689", "origins": [ "palomar" ], "url": "https://github.com/linrock/math-proofs/blob/HEAD/erdos-689/formalization.yaml", "version": "v0.4", "missing": [], "name": "Erdős 689: eventual double covering by prime residue classes", "description": "A complete Lean 4 proof of Erdős problem 689: for all sufficiently large n, one residue class per prime p ≤ n can cover every integer from 1 through n at least twice. 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Erdős" ], "type": "paper", "relationship": "other", "endorsement": "" }, { "title": "A greedy matching proof of Erdős’s two-fold residue-class problem (working manuscript, 27 April 2026)", "id": "https://www.ulam.ai/research/erdos689.pdf", "authors": [ "Przemek Chojecki" ], "type": "paper", "relationship": "formalizes", "endorsement": "" }, { "title": "Erdős problem 689 discussion", "id": "https://www.erdosproblems.com/forum/thread/689", "authors": [], "type": "web discussion", "relationship": "background", "endorsement": "" } ], "related": [ { "id": "https://github.com/google-deepmind/formal-conjectures/blob/f19cf7f60d9bc650ff58462f540e236caf3a6a67/FormalConjectures/ErdosProblems/689.lean", "relationship": "other", "note": "Challenge and Solution restate the right-hand covering proposition of erdos_689 at this pinned revision in Lean 4.33.1 syntax. Solution proves that proposition. The upstream declaration is an unproved formalized conjecture. 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For each fixed C > 1 and all sufficiently large n, any n such numbers have some order 2 ≤ k ≤ n+1 with |∑_i z_i^k| > C^(-n), even without the original condition z₁ = 1. The proof follows the residual-polynomial method in Tan, Wang, Huang, and Chen’s August 2026 paper “Residual bounds for Schur-stable polynomials,” using a fixed-degree polynomial test. 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Also proves that, for each fixed C > 1 and all sufficiently large n, any n complex numbers with |z_i| ≥ 1 have some 2 ≤ k ≤ n+1 with |∑_i z_i^k| > C^(-n). The threshold depends only on C. Repeated points and points on the unit circle are allowed. The condition z₁ = 1 is not required. 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"origins": [ "search" ], "url": "https://github.com/plby/lean-proofs/blob/HEAD/src/latest/ErdosProblems/Erdos730/formalization.yaml", "version": "v0.4", "missing": [], "name": "lean-proofs: formal Lean 4 proofs of solved Erdős problems", "description": "Machine-checked Lean 4 proofs of solved research problems from the Erdős problems collection, hosted as stable formal_proof targets for Google DeepMind's Formal Conjectures. The principal result is an affirmative answer to Erdős Problem #730 (Erdős, Graham, Ruzsa and Straus, 1975): the set of pairs n < m for which binom(2n, n) and binom(2m, m) have the same set of prime divisors is infinite, and in fact contains infinitely many consecutive pairs (n, n + 1). The proof formalises the argument posted informally by Liam Price on the problem's discussion page on 24 June 2026: Kummer's theorem conve", "authors": [ "Will Blair" ], "maintainers": [ "Will Blair" ], "license": "MIT", "role": "", "substantive": "", "sources": [ { "title": "Erdős Problem #730 (erdosproblems.com), after P. Erdős, R. L. Graham, I. Z. Ruzsa and E. G. Straus, 'On the prime factors of C(2n, n)', Math. Comp. 29 (1975)", "id": "https://www.erdosproblems.com/730", "authors": [ "Paul Erdős", "Ronald L. Graham", "Imre Z. Ruzsa", "Ernst G. Straus" ], "type": "web discussion", "relationship": "background", "endorsement": "n/a" }, { "title": "Comment on Erdős Problem #730 asserting, with a one-paragraph gist, that GPT Pro proves infinitely many consecutive pairs (n, n+1)", "id": "https://www.erdosproblems.com/forum/thread/730", "authors": [ "Liam Price" ], "type": "web discussion", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "Closing derivation ('Proof Route Mapping') linked as a follow-up, reproducing the algebraic skeleton of the argument; the analytic sections exist only in a private document and are reconstructed in this formalisation", "id": "https://chatgpt.com/share/6a3c7e81-cab8-83eb-a247-cd10b62e80fb", "authors": [ "Tomodovodoo (forum handle)" ], "type": "web discussion", "relationship": "adapts", "endorsement": "not-contacted" }, { "title": "Bernt Lindström, 'Well distribution of Sidon sets in residue classes', J. Number Theory 69 (1998), 197–200", "id": "https://doi.org/10.1006/jnth.1997.2212", "authors": [ "Bernt Lindström" ], "type": "paper", "relationship": "adapts", "endorsement": "n/a" }, { "title": "Erdős Problem #154 discussion (erdosproblems.com), including the post presenting Lindström's proof", "id": "https://www.erdosproblems.com/forum/thread/154#post-4218", "authors": [ "Thomas F. Bloom (site)" ], "type": "web discussion", "relationship": "background", "endorsement": "n/a" }, { "title": "Formal statements of Erdős Problems #730, #154, #94, #399 and #1074 in Formal Conjectures (the objects each Challenge restates)", "id": "https://github.com/google-deepmind/formal-conjectures", "authors": [ "The Formal Conjectures Authors (Google DeepMind)" ], "type": "other", "relationship": "background", "endorsement": "n/a" } ], "related": [ { "id": "https://github.com/Woett/Lean-files/blob/main/ErdosProblem154.lean", "relationship": "builds-on", "note": "Lindström's residue-distribution theorem for the Sidon set itself, formalised by Aristotle (Harmonic) and Wouter van Doorn; reproduced as ErdosProblems/Erdos154/Lindstrom.lean with its header, and extended to the sumset statement in ErdosProblems/Erdos154/Sumset.lean. No licence is published upstream; see NOTICE." }, { "id": "https://github.com/AlexKontorovich/PrimeNumberTheoremAnd", "relationship": "builds-on", "note": "Pinned proof dependency supplying the prime number theorem in arithmetic progressions used by the #730 proof; not in any Challenge's import closure." }, { "id": "https://github.com/google-deepmind/formal-conjectures/pull/664", "relationship": "other", "note": "AlphaProof's formal witness for the non-consecutive #730 pair (10003, 10005), recorded in Formal Conjectures as erdos_730.variants.delta_ne_one." } ], "classification": { "arxiv": [ "math.NT", "math.CO" ], "msc2020": [ "11B65", "11N13", "11A41", "11B13" ] }, "automation": { "methods": [ { "method": "agent", "models": [], "framework": "Codex and Claude Code agent sessions" } ], "strongest": "agent", "models": [], "spend": "not reconstructed", "notes": "Every tracked theorem is rebuilt and axiom-audited on every push (.github/workflows/verify.yml), and each Palomar configuration is replayed through Comparator in CI with a non-adversarial sandbox stand-in; the registry's own run is the trusted one." }, "scope": "Complete, kernel-checked proofs of the results listed under main_results, together with the component lemmas of the #730 development (indexed in proofs.yaml and audited in Audit.lean). Each registry-shaped statement under Palomar// is one Comparator comparison: a Challenge importing only Mathlib, a Solution delegating to the proof here, and the configuration naming the compared declaration.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "Palomar.Erdos730.erdos_730_infinite", "file": "PalomarSolutions/Erdos730.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Palomar.Erdos154.erdos_154_sumset", "file": "PalomarSolutions/Erdos154.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Palomar.Erdos94.sum_multiplicity", "file": "PalomarSolutions/Erdos94.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "#730: none in the statement — the Challenge inlines the Formal Conjectures set verbatim; the proof establishes the stronger consecutive-pair statement, available in the development as Erdos730.FullDensityCore.GoodParameter and the density theorems around it. #154: the Challenge states the sumset form Formal Conjectures records, which is the consequence proved here of Lindström's theorem for A itself, with IsSidon in the Formal Conjectures shape (two representations agree up to order). #94: an elementary bounded identity only; it does not prove the cubic distance-multiplicity theorem or the regular-polygon conjecture of that problem.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "Kernel-checked and axiom-audited in this repository's CI and reproduced from source by the author; no independent human review of the mathematics has been performed, and the informal #730 source argument is itself unadjudicated on erdosproblems.com. The #730 statement is submitted to Palomar to obtain the registry's independent dual-kernel Comparator check; #154 waits on the upstream licence of the formalisation it builds on; #94 is below a registry's research-interest floor and is not submitted." }, "canonical": { "repo": "plby/lean-proofs", "directory": "src/latest/ErdosProblems/Erdos730" }, "nodes": [ { "id": "erdos:730", "join": "names", "page": "s/erdos-730.html", "label": "Erdős Problem 730" } ], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "Rogerhu12/Erdos1045/formalization.yaml", "repo": "Rogerhu12/Erdos1045", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/Rogerhu12/Erdos1045/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Eventual maximizers of the planar distance product", "description": "Characterizes planar configurations maximizing the product of squared pairwise distances under diameter or convex-hull perimeter constraints. The diameter result holds for odd orders at least 2^100000000 and even orders at least 2^(10^120), with attainment, direct-rigid uniqueness, the exact odd formula and the canonical even graph and symmetries. The full perimeter characterization holds above 2^100000000. Above the even cutoff the selected stationary polynomial system has a unique nonsingular algebraic root in its specified window. The development also proves both normalized parity limits an", "authors": [ "Boyang Hu" ], "maintainers": [ "Boyang Hu" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Eventual maximizers of the planar distance product", "id": "", "authors": [ "Boyang Hu" ], "type": "paper", "relationship": "formalizes", "endorsement": "" }, { "title": "Erdős problem 1045", "id": "", "authors": [], "type": "", "relationship": "background", "endorsement": "" } ], "related": [ { "id": "https://github.com/coleski/erdos1045-n6/tree/59ede5a12403d62d00fa21b59597dbb6c01fb0a2", "relationship": "independent", "note": "A Lean-verified proof of the six-point case of Erdos #1045, GitHub user coleski, September 11, 2026. Cited as related work for the exact six-point maximum; this development has no dependency on that project." } ], "classification": { "arxiv": [ "math.MG", "math.CV" ], "msc2020": [ "52A40", "30C85", "05C62" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "GPT-6 Astra", "GPT-5.6 Sol", "GPT-5.6 Luna" ], "framework": "OpenAI Codex" } ], "strongest": "agent", "models": [ "GPT-5.6 Luna", "GPT-5.6 Sol", "GPT-6 Astra" ], "spend": "", "notes": "The initial mathematical draft was generated using GPT-6 Pro. The Lean formalization was developed using OpenAI Codex agents, with human direction of scope and proof-route choices. The resulting formal proofs were checked by Lean 4. The author is responsible for the manuscript and for the correspondence between its mathematical statements and the formal statements." }, "scope": "Erdos1045.main is the conjunction of five fully specified propositions in Challenge.lean. The diameter characterization uses the cutoff 2^100000000 for odd orders and 2^(10^120) for even orders, including attainment, direct-rigid uniqueness, the odd exact formula, and the even graph and symmetry data. The perimeter characterization uses 2^100000000. The algebraic certificate uses 2^(10^120) and includes uniqueness and nonsingularity within its selected polynomial window. The remaining propositions are the normalized parity limits and unique positive KKT certificate at the same even-order cutof", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "Global analytic localization uses an exterior-map/Faber route in place of the manuscript's capacity-based localization argument. Activity of constraints is proved using feasible lens-coordinate variations before KKT. Finite-kernel monotonicity uses exact difference identities and function-value convergence. The diameter graph is expressed by the explicit cyclic adjacency predicate CanonicalEvenDiameterAdj. The algebraic certificate specifies the reference construction, polynomial system and window explicitly, and proves existence and uniqueness of the selected root. The explicit even-order extension uses quantitative Fourier, localization, pressure and Schur estimates to prove the displayed integer cutoff. Explicit localization and uniform local rigidity give the smaller common threshold for odd diameter and perimeter maximizers. The quantitative pressure margin, radial derivative and chart estimates also establish unique strictly positive KKT multipliers at the same even-order cutoff.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "See docs/VALIDATION.md for the verification record." }, "canonical": { "repo": "rogerhu12/erdos1045", "directory": "" }, "nodes": [ { "id": "erdos:1045", "join": "names", "page": "s/erdos-1045.html", "label": "Erdős 1045" } ], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "rwst/Pisot-Cantor-61/formalization.yaml", "repo": "rwst/Pisot-Cantor-61", "path": "formalization.yaml", "directory": "", "origins": [ "cited", "palomar" ], "url": "https://github.com/rwst/Pisot-Cantor-61/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Lean formalization toward Bugeaud Problem 10.61", "description": "A Lean 4 / Mathlib development of criteria for Bugeaud's Problem 10.61 ([Bug12, p. 222], due to Mendès France [MF67]): for a Pisot number α > 2 and the Cantor set C(α) = {(α−1) Σ_{k≥1} ε_k α^{−k} : ε_k ∈ {0,1}}, no ξ ∈ C(α) has (ξαⁿ)_{n≥1} uniformly distributed modulo one. The problem is open; what is proved here are criteria for it and two instances. Every compared statement that concludes Problem 10.61 does so for a *quadratic* setup — a real root α > 1 of X² − aX − b whose conjugate has modulus below one — and the two instances are quadratic; the arbitrary-degree material is conditional ing", "authors": [ "Ralf Stephan" ], "maintainers": [ "Ralf Stephan" ], "license": "CC0-1.0", "role": "", "substantive": "", "sources": [ { "title": "Machine-written notes on Bugeaud Problem 61", "id": "", "authors": [ "Ralf Stephan" ], "type": "original-proof", "relationship": "other", "endorsement": "participated" }, { "title": "Distribution Modulo One and Diophantine Approximation", "id": "", "authors": [ "Yann Bugeaud" ], "type": "book", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Nombres normaux. Applications aux fonctions pseudo-aléatoires", "id": "", "authors": [ "Michel Mendès France" ], "type": "paper", "relationship": "background", "endorsement": "n/a" }, { "title": "Dimension, entropy and Lyapunov exponents", "id": "", "authors": [ "Lai-Sang Young" ], "type": "paper", "relationship": "background", "endorsement": "n/a" }, { "title": "An Introduction to Ergodic Theory", "id": "", "authors": [ "Peter Walters" ], "type": "book", "relationship": "background", "endorsement": "n/a" }, { "title": "Uniform Distribution of Sequences", "id": "", "authors": [ "Lauwerens Kuipers", "Harald Niederreiter" ], "type": "book", "relationship": "background", "endorsement": "n/a" }, { "title": "Pisot and Salem Numbers", "id": "", "authors": [ "Marie-José Bertin", "Annette Decomps-Guilloux", "Marthe Grandet-Hugot", "Martine Pathiaux-Delefosse", "Jean-Pierre Schreiber" ], "type": "book", "relationship": "background", "endorsement": "n/a" } ], "related": [], "classification": { "arxiv": [ "math.NT", "math.DS" ], "msc2020": [ "11K06", "11J71", "11R06", "37B10", "28A80", "37D35", "37A35" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "claude-opus-5" ], "framework": "Claude Code" }, { "method": "agent", "models": [ "claude-fable-5" ], "framework": "Claude Code" }, { "method": "manual", "models": [], "framework": "" } ], "strongest": "agent", "models": [ "claude-fable-5", "claude-opus-5" ], "spend": "subscription-based Claude Code usage; per-project spend not recorded separately", "notes": "lake test runs leanprover/comparator over four configurations: two axiom lanes, std3 and cited-ly, against each of two challenge shapes. The Challenge tree is one module per module of the development that owns a compared declaration, which is what a human edits; ChallengeFlat.lean is generated from it by scripts/flatten-challenge.py as a single module importing only Mathlib, which is the shape this registry requires, and comparator/ std3-flat.json is the configuration submitted. All four pass. Two negative controls were confirmed to bite and should be re-run after any change: adding a cited-ax" }, "scope": "Problem 10.61 is not solved. The seventeen declarations named in comparator/std3-flat.json are the formalized clauses of the notes' Theorems A, B(ii), B(iv), C and D that need nothing beyond Lean's three standard axioms. What they establish about Problem 10.61 is confined to degree two. Every compared statement whose conclusion is non-equidistribution is quantified over a BB61.QuadSetup, and the two instances, α = 2+√5 and α = 2+√3, are quadratic. Theorem C(ii) characterizes when the Route A covering criterion applies to a quadratic setup — when A(α) < 1 — and not when Problem 10.61 holds for ", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "BB61.QuadSetup.equidistributed_iff_exists_invariant_measure", "file": "BB61/Saturation.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "BB61.QuadSetup.forall_not_equidistributed_iff_exists_trigCertificate", "file": "BB61/Saturation.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "MeasureTheory.le_integral_of_partitionPressure_le", "file": "ForMathlib/Dynamics/TopologicalPressure.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "BB61.forall_not_equidistributed_of_partitionPressure_lt", "file": "BB61/PressureCriterion.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "BB61.forall_not_equidistributed_of_transferBound", "file": "BB61/PressureCriterion.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "BB61.QuadSetup.routeAExponent_mul_hMin", "file": "BB61/EntropyBudget.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "BB61.slope_neg_iff", "file": "BB61/RouteADepth.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "BB61.logb_coverTotal_balancedDepth_le", "file": "BB61/RouteADepth.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "BB61.QuadSetup.exists_avoided_interval", "file": "BB61/Covering.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "BB61.QuadSetup.not_equidistributed_of_routeAExponent_lt_one", "file": "BB61/Criterion.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "BB61.QuadSetup.routeAExponent_lt_one_iff_quadratic", "file": "BB61/RouteANormalForm.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "BB61.isPisot_family", "file": "BB61/RouteAFamily.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "BB61.norm_le_familyConjBound", "file": "BB61/RouteAFamily.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "BB61.routeA_family_lt_one", "file": "BB61/RouteAFamily.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "BB61.routeA_two_add_sqrt5", "file": "BB61/RouteA.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "BB61.two_add_sqrt5_not_equidistributed", "file": "BB61/RouteA.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "BB61.GapThree.problem_10_61_two_add_sqrt3_axiom_free", "file": "BB61/GapSqrtThree.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "Theorem A(i) is compared in the unbundled form, with Measure.map and an explicit IsProbabilityMeasure hypothesis, rather than through the bundled ProbabilityMeasure statement of the development: the bundled type mentions a measurability proof and would drag some fifteen further proofs into the compared closure. The Σ₁ wrapper that the notes put around Corollary 3.2 is prose; what is compared is the pressure bracket it runs on. The most material divergence concerns degree. The notes state Theorem C(i) for a Pisot number of degree d ≥ 2, and state Theorem C(iii) as \"an explicit infinite family in every degree on which 10.61 holds in the strong form\". The Lean form of C(i), BB61.QuadSetup.not_equidistributed_of_routeAExponent_lt_one, is quantified over a BB61.QuadSetup and so covers degree two only, and C(iii) is formalized as three ingredients — the Pisot property, the conjugate bound, and a conditional numerical inequality giving A < 1 — with no formalized step from them to non-equidistribution above degree two. The registry account follows the development, not the notes. At degree two the family's members are quadratic setups and the development does package the conclusion, as BB61", "alignment": true, "original": true, "review": { "status": "unchecked", "bucket": "unchecked", "reviewers": [], "notes": "No review was performed in the sense this field asks about. The releaser directed the work and selected which outputs to keep, but states on the first page of paper.pdf that he cannot reconstruct or endorse the mathematical arguments and has not independently verified informal-to-formal fidelity. No external referee has read the development or the notes, and the agent that wrote the proofs is the one that ran the hygiene passes. What is established is mechanical: comparator compares each named statement against the challenge file constant by constant, checks the axiom list, and replays the export through the Lean kernel." }, "canonical": { "repo": "rwst/pisot-cantor-61", "directory": "" }, "nodes": [ { "id": "books:BugeaudDistributionModuloOne/Problem10_61", "join": "artifact", "page": "s/books-bugeauddistributionmoduloone-problem10-61.html", "label": "Bugeaud Collection of Conjectures and Open Questions: Pisot orbits on the Cantor set" } ], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-31-000013", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-31-000013", "commit": "d61132ffcdb748b641f9de59b7acfffc860796b2", "trust": "high", "theorems": 17, "date": "2026-09-06" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "SamPetkov/Erdos625-formalization/formalization.yaml", "repo": "SamPetkov/Erdos625-formalization", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/SamPetkov/Erdos625-formalization/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Erdős Problem 625: chromatic versus cochromatic number of a random graph", "description": "A Lean 4 formalization of the full-sequence quantitative theorem for G(n,1/2): with probability tending to one, the chromatic number minus the cochromatic number is at least ((log 2)^2/4) log(200/153) times n/(log n)^3. The formal statement uses the finite binomial random-graph measure, natural-valued chromatic and cochromatic numbers, real subtraction, and convergence along all natural n.", "authors": [], "maintainers": [], "license": "CC-BY-4.0", "role": "", "substantive": "", "sources": [ { "title": "Erdős Problem 625: manuscript and Lean formalization", "id": "https://github.com/SamPetkov/Erdos/tree/d02c8b6c541f916170fc9d0cd2fef7d727074629/625", "authors": [ "Samuil Petkov" ], "type": "original-proof", "relationship": "other", "endorsement": "participated" }, { "title": "Some Problems and Results in Cochromatic Theory", "id": "https://doi.org/10.1016/S0167-5060(08)70393-5", "authors": [ "Paul Erdős", "John Gimbel" ], "type": "article", "relationship": "background", "endorsement": "n/a" }, { "title": "The Difference Between the Chromatic and the Cochromatic Number of a Random Graph", "id": "https://doi.org/10.48550/arXiv.2409.17614", "authors": [ "Annika Heckel" ], "type": "article", "relationship": "background", "endorsement": "not-contacted" } ], "related": [], "classification": { "arxiv": [ "math.CO", "math.PR" ], "msc2020": [ "05C15", "05C80", "60C05" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "OpenAI GPT-5 family" ], "framework": "Codex and Aristotle" } ], "strongest": "agent", "models": [ "OpenAI GPT-5 family" ], "spend": "subscription-based and not separately tracked", "notes": "The proof was developed interactively with AI assistance and repeatedly audited by the human author. Automation does not establish peer review." }, "scope": "The principal compared theorem is Erdos625.erdos625. It proves the explicit uniform coefficient ((log 2)^2/4) log(200/153) and full-sequence convergence in probability. Comparator also treats Erdos625.halfProbability as an auxiliary definition boundary and checks Erdos625.coe_halfProbability, which pins its underlying real value to 1/2. These auxiliary declarations specify the random-graph parameter; they are not additional main results. The manuscript's nonconstant phase-resolved A_4(delta_n) refinement is not part of the formal principal theorem.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The formal theorem matches the manuscript's uniform quantitative result. It omits only the stronger nonconstant phase-resolved refinement, as stated explicitly in the scope field.", "alignment": true, "original": true, "review": { "status": "self-assessed with independent kernel replay; not peer-reviewed", "bucket": "self-assessed", "reviewers": [], "notes": "The statement, constants, source closure, prohibited-token scan, and axiom boundary were audited. A clean Aristotle workspace compiled the pinned standalone source and preserved the transcript and checksums. No external mathematical referee or community acceptance is claimed." }, "canonical": { "repo": "sampetkov/erdos625-formalization", "directory": "" }, "nodes": [ { "id": "erdos:625", "join": "names", "page": "s/erdos-625.html", "label": "Erdős 625" } ], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-02-000006", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-02-000006", "commit": "9702b5e734627e3fdaef8da0d65ad7394016f0a7", "trust": "high", "theorems": 2, "date": "2026-09-02" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "tadamcz/erdos1/formalization.yaml", "repo": "tadamcz/erdos1", "path": "formalization.yaml", "directory": "", "origins": [ "cited", "search" ], "url": "https://github.com/tadamcz/erdos1/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Erdős problem #1: disproof", "description": "Disproof of Erdős problem #1: there is no constant C > 0 such that N > C · 2^n for every sum-distinct set A ⊆ {1, …, N} of size n. Found autonomously by GPT-6 Astra in the FrontierMath Erdős benchmark (Adamczewski and Bloom, 2026).", "authors": [], "maintainers": [], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Disproof of Erdős problem #1: A sum-distinct set A ⊆ {1,…,N} of size n does not force N ≫ 2^n.", "id": "", "authors": [], "type": "original-proof", "relationship": "other", "endorsement": "" }, { "title": "Erdős problem #1 (erdosproblems.com)", "id": "https://www.erdosproblems.com/1", "authors": [], "type": "web discussion", "relationship": "background", "endorsement": "" }, { "title": "Some of my new and almost new problems and results in combinatorial number theory", "id": "MR1628841", "authors": [ "Paul Erdős" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Problems and results in additive number theory", "id": "MR0079027", "authors": [ "Paul Erdős" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "A construction for sets of integers with distinct subset sums", "id": "https://doi.org/10.37236/1341", "authors": [ "Tom Bohman" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "FrontierMath Erdős", "id": "", "authors": [ "Tom Adamczewski", "Thomas F. Bloom" ], "type": "paper", "relationship": "background", "endorsement": "" } ], "related": [ { "id": "https://github.com/google-deepmind/formal-conjectures/blob/488aade228ec37880b8fec178c173c07d279bb53/FormalConjectures/ErdosProblems/1.lean", "relationship": "builds-on", "note": "Formal Conjectures states the conjecture (and its definitions) with `sorry` as an open problem; the definitions and statement here are copied verbatim from this file at this commit, and the compared theorem is its negation, proved." }, { "id": "https://github.com/epoch-research/LeanOpenProblems/blob/77882c437ca1dfefab3b27fa00f1d29788100311/apn/data/erdos/Isolated/Erdos1.erdos_1.lean", "relationship": "builds-on", "note": "The isolated statement file given to the AI system in the FrontierMath Erdős benchmark (derived from the Formal Conjectures file above)." } ], "classification": { "arxiv": [ "math.NT", "math.CO" ], "msc2020": [ "11B13", "11B75" ] }, "automation": { "methods": [ { "method": "autonomous", "models": [ "GPT-6 Astra (pre-release version, OpenAI)" ], "framework": "Inspect AI (UK AISI) with the LeanOpenProblems task harness; deepagent-based agent for default-configuration attempts, basic ReAct agent for the larger-budget attempts" } ], "strongest": "autonomous", "models": [ "GPT-6 Astra (pre-release version, OpenAI)" ], "spend": "1790 USD across the 2 resolution attempt(s) included in this repository, at GPT-6 Astra's standard rates; the paper reports over 220,000 USD of compute across all attempts on all 68 benchmark problems", "notes": "The Lean development was written entirely by the AI system inside the benchmark harness. Humans (Thomas F. Bloom) selected the problem and reviewed the formal statement for faithfulness before the run; humans (Tom Adamczewski) ran the harness and, after the run, packaged the verified output into this repository. The only edits to the AI-written files are mechanical and are listed in the README: removed the sorry'd stub of the original conjecture `erdos_1` (lines 22–31 of the original) together with its docstring; restated `Erdos1.erdos_1.disproof` explicitly (the original used `¬ (type_of% @Er" }, "scope": "Fully formalized: the compared theorem `Erdos1.erdos_1.disproof` in `Challenge.lean` is proved in `Solution.lean` via the imported module `Erdos1.Resolutions.Erdos1_219usd_38h` with no `sorry` and no axioms beyond propext, Quot.sound and Classical.choice. The 1 further module(s) in `Erdos1/Resolutions/` are independent complete proofs of the same statement, built as the separate Lake library `Erdos1Alternates` but not compared by Comparator. Only the negation of the formal conjecture is advertised; the stronger facts visible in the proofs (for example explicit budgets or 3-colourability) are n", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "None known between the compared theorem and the conjecture as stated on erdosproblems.com. The formal conjecture `Erdos1.erdos_1` (Formal Conjectures) reads: there is `C > 0` such that `C · 2^|A| < N` for every `N ≠ 0` and every sum-distinct `A ⊆ {1, …, N}`; the compared theorem is exactly its negation. The hypothesis `N ≠ 0` only excludes the degenerate empty interval. The disproof is ineffective (no explicit `n(ε)`).", "alignment": true, "original": true, "review": { "status": "mechanically verified (comparator: lean kernel replay, standard axioms only) in the benchmark harness and again in this repository's ci; preliminary informal reading of the argument by thomas f. bloom; no independent refereeing", "bucket": "other", "reviewers": [ "Thomas F. Bloom" ], "notes": "Statement audit: the formal statement was selected and reviewed for faithfulness to the informal problem by Thomas F. Bloom before the benchmark run (for Formal Conjectures statements he chose the representative statement; autoformalized statements he checked individually). Mathematical review: the resolution was verified by Comparator against the trusted statement; Thomas F. Bloom made an initial, informal examination of the argument and wrote a short exposition (FrontierMath Erdős appendix and erdosproblems.com). The paper states that the proofs have not yet been properly digested by human experts." }, "canonical": { "repo": "tadamcz/erdos1", "directory": "" }, "nodes": [ { "id": "erdos:1", "join": "anchor", "page": "s/erdos-1.html", "label": "Erdős Problem 1" } ], "container": false, "anchors": { "erdos:1": "https://github.com/google-deepmind/formal-conjectures/blob/488aade228ec37880b8fec178c173c07d279bb53/FormalConjectures/ErdosProblems/1.lean" }, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "tadamcz/erdos1207/formalization.yaml", "repo": "tadamcz/erdos1207", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/tadamcz/erdos1207/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Erdős problem #1207 (isosceles-free subsets of planar point sets): proof", "description": "Proof of Erdős problem #1207: there is a constant c > 0 such that every sufficiently large n admits a set of n points in the plane whose largest isosceles-free subset has fewer than n^(1-c) points, i.e. P_2(n) < n^(1-c). Found autonomously by GPT-6 Astra in the FrontierMath Erdős benchmark harness (Adamczewski and Bloom, 2026).", "authors": [], "maintainers": [], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Proof of Erdős problem #1207: there is c > 0 with P_2(n) < n^(1-c) for all sufficiently large n, where P_2(n) is the least, over n-point planar sets, of the largest isosceles-free subset.", "id": "", "authors": [], "type": "original-proof", "relationship": "other", "endorsement": "" }, { "title": "Erdős problem #1207 (erdosproblems.com)", "id": "https://www.erdosproblems.com/1207", "authors": [], "type": "web discussion", "relationship": "background", "endorsement": "" }, { "title": "A survey of problems in combinatorial number theory", "id": "", "authors": [ "Paul Erdős" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Isosceles triangles determined by a planar point set", "id": "", "authors": [ "János Pach", "Gábor Tardos" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "The Minkowski grid has robustly many repeated distances", "id": "https://arxiv.org/abs/2607.05374", "authors": [ "Sungchul Lee", "Cosmin Pohoata", "Daniel G. Zhu" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "FrontierMath Erdős", "id": "", "authors": [ "Tom Adamczewski", "Thomas F. Bloom" ], "type": "paper", "relationship": "background", "endorsement": "" } ], "related": [ { "id": "https://github.com/epoch-research/LeanOpenProblems/blob/8fa3b4c6ee5e6f0ee25d027b6f110f20787db977/apn/data/erdos_autoformalized/Isolated/Erdos1207.erdos_1207.lean", "relationship": "builds-on", "note": "The statement file given to the AI system. The problem had no statement in Formal Conjectures; this statement was produced by the benchmark authors' autoformalization pipeline. The compared theorem is this statement, proved. Its auxiliary notions Set.Triplewise and IsIsosceles come from Formal Conjectures (FormalConjecturesForMathlib at commit 9cbe1d3c12998c786b7c2cd99ce28a21b6631f66) and are copi" } ], "classification": { "arxiv": [ "math.CO", "math.MG", "math.NT" ], "msc2020": [ "52C10", "11R29" ] }, "automation": { "methods": [ { "method": "autonomous", "models": [ "GPT-6 Astra (pre-release version, OpenAI)" ], "framework": "Inspect AI (UK AISI) with the LeanOpenProblems task harness; basic ReAct agent (the harness's larger-budget configuration, without subagents, memory or a literature snapshot)" } ], "strongest": "autonomous", "models": [ "GPT-6 Astra (pre-release version, OpenAI)" ], "spend": "549 USD for the single resolution attempt included in this repository, at GPT-6 Astra's standard rates; two further attempts on the same statement in the same eval set did not resolve it (one hit the 96-hour working-time limit at about 800 USD, one was lost to an infrastructure error at about 367 USD)", "notes": "The Lean development was written entirely by the AI system inside the benchmark harness. Humans (Tom Adamczewski) ran the harness and, after the run, packaged the output into this repository. The only edits to the AI-written file are mechanical and there are only two, both listed in the README: import FormalConjecturesUtil became import Mathlib, and the six declarations that import provided which are not in Mathlib were copied in verbatim from Formal Conjectures (two used by the statement, four by the proof). No mathematics was touched. Reversing the diff in provenance/ reproduces the harness'" }, "scope": "Fully formalized: the compared theorem Erdos1207.erdos_1207 in Challenge.lean is proved in Solution.lean via the Solution.lean itself with no sorry and no axioms beyond propext, Quot.sound and Classical.choice. The proof module is an ordinary readable Lean development of 19,687 lines, 664 theorems and about 1,975 declarations. Only the conjectured statement is advertised; the explicit value of c inside the proof, and an unconditional (log n)^2 bound also proved there, are not part of the compared statement. Nothing about the informal problem was omitted or weakened relative to the benchmark's ", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "Relative to the erdosproblems.com statement there is no known divergence. P d n is the infimum over n-element finsets of EuclideanSpace ℝ (Fin d) of the supremum of cardinalities of isosceles-free subsets of that finset, which is exactly P_d(n): for every n such a finset exists, and the inner set of cardinalities is nonempty and bounded by n, so neither sInf nor sSup takes its junk value 0. A set is isosceles-free when no three distinct points have two equal distances among them (IsIsosceles p q r is dist p q = dist q r or dist q r = dist r p or dist r p = dist p q, so degenerate collinear isosceles triples count, matching the problem's remark that the case d = 1 is the three-term-progression problem). The conclusion reads \"for some constant c > 0 and all sufficiently large n\" with the real power n^(1-c); the informal question does not specify the range of n, and any c at most 1 forces P_2(n) at least 2 for n at least 2, so the eventual quantifier is the natural reading. The value of c is existential and not advertised.", "alignment": true, "original": true, "review": { "status": "kernel-checked by lean 4 v4.27.0 when this repository builds, with an in-repo audit (audit.lean and scripts/audit.sh) confirming that challenge.lean and solution.lean give the theorem the same fully explicit type and that it depends only on propext, quot.sound and classical.choice. the external comparator and its independent nanoda replay could not be run, because no comparator or lean4export release supports this repository's pinned lean v4.27.0. no human has refereed the argument; no informal review and no independent refereeing.", "bucket": "other", "reviewers": [], "notes": "Statement audit: the formal statement was produced by the benchmark authors' autoformalization pipeline and given to the model unchanged; the packager's own reading of it is recorded under fidelity.divergences. Mathematical review: none. The proof is readable and can be audited by a mathematician, but no one has done so." }, "canonical": { "repo": "tadamcz/erdos1207", "directory": "" }, "nodes": [ { "id": "erdos:1207", "join": "names", "page": "s/erdos-1207.html", "label": "Erdős Problem 1207" } ], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "tadamcz/erdos126/formalization.yaml", "repo": "tadamcz/erdos126", "path": "formalization.yaml", "directory": "", "origins": [ "cited", "search" ], "url": "https://github.com/tadamcz/erdos126/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Erdős problem #126: proof", "description": "Proof of Erdős problem #126: if f(n) is the least number of distinct primes dividing ∏_{a≠b∈A} (a+b) over n-element sets A ⊆ ℕ, then f(n)/log n → ∞. Found autonomously by GPT-6 Astra in the FrontierMath Erdős benchmark (Adamczewski and Bloom, 2026).", "authors": [], "maintainers": [], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Proof of Erdős problem #126: If f(n) is the least number of distinct primes dividing ∏_{a≠b∈A}(a+b) over n-element sets A ⊆ ℕ, then f(n)/log n → ∞.", "id": "", "authors": [], "type": "original-proof", "relationship": "other", "endorsement": "" }, { "title": "Erdős problem #126 (erdosproblems.com)", "id": "https://www.erdosproblems.com/126", "authors": [], "type": "web discussion", "relationship": "background", "endorsement": "" }, { "title": "On a Problem in the Elementary Theory of Numbers", "id": "https://doi.org/10.2307/2301909", "authors": [ "Paul Erdős", "Paul Turán" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "FrontierMath Erdős", "id": "", "authors": [ "Tom Adamczewski", "Thomas F. Bloom" ], "type": "paper", "relationship": "background", "endorsement": "" } ], "related": [ { "id": "https://github.com/google-deepmind/formal-conjectures/blob/488aade228ec37880b8fec178c173c07d279bb53/FormalConjectures/ErdosProblems/126.lean", "relationship": "builds-on", "note": "Formal Conjectures states the conjecture (and its definitions) with `sorry` as an open problem; the definitions and statement here are copied verbatim from this file at this commit, and the compared theorem is that statement, proved." }, { "id": "https://github.com/epoch-research/LeanOpenProblems/blob/77882c437ca1dfefab3b27fa00f1d29788100311/apn/data/erdos/Isolated/Erdos126.erdos_126.lean", "relationship": "builds-on", "note": "The isolated statement file given to the AI system in the FrontierMath Erdős benchmark (derived from the Formal Conjectures file above)." } ], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11B75", "11A41" ] }, "automation": { "methods": [ { "method": "autonomous", "models": [ "GPT-6 Astra (pre-release version, OpenAI)" ], "framework": "Inspect AI (UK AISI) with the LeanOpenProblems task harness; deepagent-based agent for default-configuration attempts, basic ReAct agent for the larger-budget attempts" } ], "strongest": "autonomous", "models": [ "GPT-6 Astra (pre-release version, OpenAI)" ], "spend": "843 USD across the 4 resolution attempt(s) included in this repository, at GPT-6 Astra's standard rates; the paper reports over 220,000 USD of compute across all attempts on all 68 benchmark problems", "notes": "The Lean development was written entirely by the AI system inside the benchmark harness. Humans (Thomas F. Bloom) selected the problem and reviewed the formal statement for faithfulness before the run; humans (Tom Adamczewski) ran the harness and, after the run, packaged the verified output into this repository. The only edits to the AI-written files are mechanical and are listed in the README: removed trailing unused stub `theorem Erdos126.erdos_126.disproof : ¬ (type_of% @Erdos126.erdos_126) := sorry`." }, "scope": "Fully formalized: the compared theorem `Erdos126.erdos_126` in `Challenge.lean` is proved in `Solution.lean` via the imported module `Erdos126.Resolutions.Erdos126_132usd_25h` with no `sorry` and no axioms beyond propext, Quot.sound and Classical.choice. The 3 further module(s) in `Erdos126/Resolutions/` are independent complete proofs of the same statement, built as the separate Lake library `Erdos126Alternates` but not compared by Comparator. Only the conjectured statement is advertised; stronger quantitative bounds visible inside the proofs are not part of the compared statement. Nothing ab", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The compared theorem is exactly the Formal Conjectures statement. The definition `IsMaximalAddFactorsCard f` says that for every `n`, `f n` is the greatest `m` such that every `n`-element `A : Finset ℕ` has at least `m` distinct prime factors in `∏_{(a,b) ∈ A.offDiag} (a + b)` (product over ordered pairs `a ≠ b`; the same prime set as over unordered pairs). Such an `f` exists and is unique (the primary resolution verifies existence, uniqueness and monotonicity in its `Independent126` section), so the hypothesis is not vacuous; `A` may contain `0`, which changes `|S(A)|` by at most one relative to positive sets. The conclusion `Tendsto (fun n => f n / Real.log n) atTop atTop` coerces `f n` to `ℝ`; `Real.log n` is positive for `n ≥ 2`. The stronger polynomial bounds proved internally are not part of the compared statement.", "alignment": true, "original": true, "review": { "status": "mechanically verified (comparator: lean kernel replay, standard axioms only) in the benchmark harness and again in this repository's ci; preliminary informal reading of the argument by thomas f. bloom; no independent refereeing", "bucket": "other", "reviewers": [ "Thomas F. Bloom" ], "notes": "Statement audit: the formal statement was selected and reviewed for faithfulness to the informal problem by Thomas F. Bloom before the benchmark run (for Formal Conjectures statements he chose the representative statement; autoformalized statements he checked individually). Mathematical review: the resolution was verified by Comparator against the trusted statement; Thomas F. Bloom made an initial, informal examination of the argument and wrote a short exposition (FrontierMath Erdős appendix and erdosproblems.com). The paper states that the proofs have not yet been properly digested by human experts." }, "canonical": { "repo": "tadamcz/erdos126", "directory": "" }, "nodes": [ { "id": "erdos:126", "join": "anchor", "page": "s/erdos-126.html", "label": "Erdős Problem 126" } ], "container": false, "anchors": { "erdos:126": "https://github.com/google-deepmind/formal-conjectures/blob/488aade228ec37880b8fec178c173c07d279bb53/FormalConjectures/ErdosProblems/126.lean" }, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "tadamcz/erdos548/formalization.yaml", "repo": "tadamcz/erdos548", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/tadamcz/erdos548/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Erdős problem #548 (Erdős–Sós conjecture): proof", "description": "Proof of the Erdős–Sós conjecture (Erdős problem #548): every graph on n ≥ k+1 vertices with at least (k−1)n/2 + 1 edges contains every tree on k+1 vertices. Found autonomously by GPT-6 Astra in the FrontierMath Erdős benchmark (Adamczewski and Bloom, 2026).", "authors": [], "maintainers": [], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Proof of Erdős problem #548: Every graph on n ≥ k+1 vertices with at least (k−1)n/2 + 1 edges contains every tree on k+1 vertices.", "id": "", "authors": [], "type": "original-proof", "relationship": "other", "endorsement": "" }, { "title": "Erdős problem #548 (erdosproblems.com)", "id": "https://www.erdosproblems.com/548", "authors": [], "type": "web discussion", "relationship": "background", "endorsement": "" }, { "title": "Extremal problems in graph theory", "id": "", "authors": [ "Paul Erdős" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "FrontierMath Erdős", "id": "", "authors": [ "Tom Adamczewski", "Thomas F. Bloom" ], "type": "paper", "relationship": "background", "endorsement": "" } ], "related": [ { "id": "https://github.com/epoch-research/LeanOpenProblems/blob/77882c437ca1dfefab3b27fa00f1d29788100311/apn/data/erdos_autoformalized/Isolated/Erdos548.erdos_548.lean", "relationship": "builds-on", "note": "The statement file given to the AI system in the FrontierMath Erdős benchmark. The problem had no statement in Formal Conjectures; this statement was produced by the benchmark authors' autoformalization pipeline and reviewed for faithfulness by Thomas F. Bloom. The compared theorem is this statement, proved." } ], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05C35", "05C05" ] }, "automation": { "methods": [ { "method": "autonomous", "models": [ "GPT-6 Astra (pre-release version, OpenAI)" ], "framework": "Inspect AI (UK AISI) with the LeanOpenProblems task harness; deepagent-based agent for default-configuration attempts, basic ReAct agent for the larger-budget attempts" } ], "strongest": "autonomous", "models": [ "GPT-6 Astra (pre-release version, OpenAI)" ], "spend": "363 USD across the 1 resolution attempt(s) included in this repository, at GPT-6 Astra's standard rates; the paper reports over 220,000 USD of compute across all attempts on all 68 benchmark problems", "notes": "The Lean development was written entirely by the AI system inside the benchmark harness. Humans (Thomas F. Bloom) selected the problem and reviewed the formal statement for faithfulness before the run; humans (Tom Adamczewski) ran the harness and, after the run, packaged the verified output into this repository. The only edits to the AI-written files are mechanical and are listed in the README: port to Mathlib v4.28.0: Mathlib v4.28.0 changed the field `SimpleGraph.loopless` from `Irreflexive Adj` to the class `Std.Irrefl Adj`, so proofs of that field need a `constructor` step (or an anonymous" }, "scope": "Fully formalized: the compared theorem `Erdos548.erdos_548` in `Challenge.lean` is proved in `Solution.lean` via the imported module `Erdos548.Resolutions.Erdos548_192usd_21h` with no `sorry` and no axioms beyond propext, Quot.sound and Classical.choice. Only the conjectured statement is advertised; stronger quantitative bounds visible inside the proofs are not part of the compared statement. Nothing about the informal problem was omitted or weakened relative to the benchmark's formal statement; see fidelity.divergences for the relation between that formal statement and the informal problem.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "Relative to the cited source (the erdosproblems.com statement) there is no divergence: the compared theorem is its direct formalisation, with trees on `k+1` vertices and the hypothesis `|E| ≥ (k-1)n/2 + 1`. Relative to the classical phrasing of the Erdős–Sós conjecture (\"more than `(t-2)n/2` edges, trees on `t` vertices\"; here `t = k+1`) there is one marginal difference: when `(k-1)n` is odd, `|E| > (k-1)n/2` already holds with one edge fewer than `|E| ≥ (k-1)n/2 + 1` requires, so in that parity case the compared theorem assumes half an edge more and is very slightly weaker than the classical statement (the classical statement implies it, not conversely). The Lean proof's internal counting lemma derives `2|E(G)| ≤ (k-1)n` whenever the tree is absent, which is the sharp classical bound, but only the advertised statement is compared. The vertex type `Fin n` loses no generality; `G.edgeSet.ncard` is the number of edges; `T.IsTree` and `T.IsContained G` are Mathlib's notions (containment as a not necessarily induced subgraph); `k + 1 ≤ n` is `n ≥ k+1`.", "alignment": true, "original": true, "review": { "status": "mechanically verified (comparator: lean kernel replay, standard axioms only) in the benchmark harness and again in this repository's ci; preliminary informal reading of the argument by thomas f. bloom; no independent refereeing", "bucket": "other", "reviewers": [ "Thomas F. Bloom" ], "notes": "Statement audit: the formal statement was selected and reviewed for faithfulness to the informal problem by Thomas F. Bloom before the benchmark run (for Formal Conjectures statements he chose the representative statement; autoformalized statements he checked individually). Mathematical review: the resolution was verified by Comparator against the trusted statement; Thomas F. Bloom made an initial, informal examination of the argument and wrote a short exposition (FrontierMath Erdős appendix and erdosproblems.com). The paper states that the proofs have not yet been properly digested by human experts." }, "canonical": { "repo": "tadamcz/erdos548", "directory": "" }, "nodes": [ { "id": "erdos:548", "join": "names", "page": "s/erdos-548.html", "label": "Erdős Problem 548" } ], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "tadamcz/erdos571/formalization.yaml", "repo": "tadamcz/erdos571", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/tadamcz/erdos571/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Erdős problem #571 (rational exponents for Turán numbers of bipartite graphs): proof", "description": "Proof of Erdős problem #571 (Erdős–Simonovits rational exponents conjecture): for every rational α ∈ [1, 2) there is a bipartite graph G with ex(n; G) ≍ n^α. Found autonomously by GPT-6 Astra in the FrontierMath Erdős benchmark (Adamczewski and Bloom, 2026).", "authors": [], "maintainers": [], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Proof of Erdős problem #571: For every rational α ∈ [1,2) there is a bipartite graph G with ex(n; G) ≍ n^α.", "id": "", "authors": [], "type": "original-proof", "relationship": "other", "endorsement": "" }, { "title": "Erdős problem #571 (erdosproblems.com)", "id": "https://www.erdosproblems.com/571", "authors": [], "type": "web discussion", "relationship": "background", "endorsement": "" }, { "title": "Rational exponents in extremal graph theory", "id": "https://doi.org/10.4171/JEMS/798", "authors": [ "Boris Bukh", "David Conlon" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Cube-supersaturated graphs and related problems", "id": "", "authors": [ "Paul Erdős", "Miklós Simonovits" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "FrontierMath Erdős", "id": "", "authors": [ "Tom Adamczewski", "Thomas F. Bloom" ], "type": "paper", "relationship": "background", "endorsement": "" } ], "related": [ { "id": "https://github.com/epoch-research/LeanOpenProblems/blob/77882c437ca1dfefab3b27fa00f1d29788100311/apn/data/erdos_autoformalized/Isolated/Erdos571.erdos_571.lean", "relationship": "builds-on", "note": "The statement file given to the AI system in the FrontierMath Erdős benchmark. The problem had no statement in Formal Conjectures; this statement was produced by the benchmark authors' autoformalization pipeline and reviewed for faithfulness by Thomas F. Bloom. The compared theorem is this statement, proved." } ], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05C35" ] }, "automation": { "methods": [ { "method": "autonomous", "models": [ "GPT-6 Astra (pre-release version, OpenAI)" ], "framework": "Inspect AI (UK AISI) with the LeanOpenProblems task harness; deepagent-based agent for default-configuration attempts, basic ReAct agent for the larger-budget attempts" } ], "strongest": "autonomous", "models": [ "GPT-6 Astra (pre-release version, OpenAI)" ], "spend": "617 USD across the 1 resolution attempt(s) included in this repository, at GPT-6 Astra's standard rates; the paper reports over 220,000 USD of compute across all attempts on all 68 benchmark problems", "notes": "The Lean development was written entirely by the AI system inside the benchmark harness. Humans (Thomas F. Bloom) selected the problem and reviewed the formal statement for faithfulness before the run; humans (Tom Adamczewski) ran the harness and, after the run, packaged the verified output into this repository. The only edits to the AI-written files are mechanical and are listed in the README: port to Mathlib v4.28.0: Mathlib v4.28.0 changed the field `SimpleGraph.loopless` from `Irreflexive Adj` to the class `Std.Irrefl Adj`, so proofs of that field need a `constructor` step (or an anonymous" }, "scope": "Fully formalized: the compared theorem `Erdos571.erdos_571` in `Challenge.lean` is proved in `Solution.lean` via the imported module `Erdos571.Resolutions.Erdos571_325usd_42h` with no `sorry` and no axioms beyond propext, Quot.sound and Classical.choice. Only the conjectured statement is advertised; stronger quantitative bounds visible inside the proofs are not part of the compared statement. Nothing about the informal problem was omitted or weakened relative to the benchmark's formal statement; see fidelity.divergences for the relation between that formal statement and the informal problem.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The compared theorem is the benchmark's formalisation of the erdosproblems.com statement. `extremalNumber n G` is Mathlib's Turán number (maximum edge count of a `G`-free simple graph on `Fin n`); `G.IsBipartite` is Mathlib's bipartiteness; `Asymptotics.IsTheta atTop` encodes `≍` (two-sided `≪`) for `n → ∞`, with both sides cast to `ℝ` and `n^α` read as the real power `(n : ℝ) ^ (α : ℝ)`. The graph lives on `Fin q`, which loses no generality. No divergence from the informal statement is known. The relation between this proof and the substantial existing literature (Bukh–Conlon and later work) has not yet been worked out.", "alignment": true, "original": true, "review": { "status": "mechanically verified (comparator: lean kernel replay, standard axioms only) in the benchmark harness and again in this repository's ci; preliminary informal reading of the argument by thomas f. bloom; no independent refereeing", "bucket": "other", "reviewers": [ "Thomas F. Bloom" ], "notes": "Statement audit: the formal statement was selected and reviewed for faithfulness to the informal problem by Thomas F. Bloom before the benchmark run (for Formal Conjectures statements he chose the representative statement; autoformalized statements he checked individually). Mathematical review: the resolution was verified by Comparator against the trusted statement; Thomas F. Bloom made an initial, informal examination of the argument and wrote a short exposition (FrontierMath Erdős appendix and erdosproblems.com). The paper states that the proofs have not yet been properly digested by human experts." }, "canonical": { "repo": "tadamcz/erdos571", "directory": "" }, "nodes": [ { "id": "erdos:571", "join": "names", "page": "s/erdos-571.html", "label": "Erdős Problem 571" } ], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "tadamcz/erdos74/formalization.yaml", "repo": "tadamcz/erdos74", "path": "formalization.yaml", "directory": "", "origins": [ "cited", "search" ], "url": "https://github.com/tadamcz/erdos74/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Erdős problem #74: disproof", "description": "Disproof of Erdős problem #74 (Erdős–Hajnal–Szemerédi): there is a function f(n) → ∞ such that every graph whose n-vertex subgraphs can all be made bipartite by deleting at most f(n) edges has finite chromatic number. Found autonomously by GPT-6 Astra in the FrontierMath Erdős benchmark (Adamczewski and Bloom, 2026).", "authors": [], "maintainers": [], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Disproof of Erdős problem #74: Some f(n) → ∞ forces every graph whose n-vertex subgraphs are within f(n) edges of bipartite to have finite chromatic number.", "id": "", "authors": [], "type": "original-proof", "relationship": "other", "endorsement": "" }, { "title": "Erdős problem #74 (erdosproblems.com)", "id": "https://www.erdosproblems.com/74", "authors": [], "type": "web discussion", "relationship": "background", "endorsement": "" }, { "title": "On almost bipartite large chromatic graphs", "id": "MR806975", "authors": [ "Paul Erdős", "András Hajnal", "Endre Szemerédi" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "FrontierMath Erdős", "id": "", "authors": [ "Tom Adamczewski", "Thomas F. Bloom" ], "type": "paper", "relationship": "background", "endorsement": "" } ], "related": [ { "id": "https://github.com/google-deepmind/formal-conjectures/blob/488aade228ec37880b8fec178c173c07d279bb53/FormalConjectures/ErdosProblems/74.lean", "relationship": "builds-on", "note": "Formal Conjectures states the conjecture (and its definitions) with `sorry` as an open problem; the definitions and statement here are copied verbatim from this file at this commit, and the compared theorem is its negation, proved." }, { "id": "https://github.com/epoch-research/LeanOpenProblems/blob/77882c437ca1dfefab3b27fa00f1d29788100311/apn/data/erdos/Isolated/Erdos74.erdos_74.lean", "relationship": "builds-on", "note": "The isolated statement file given to the AI system in the FrontierMath Erdős benchmark (derived from the Formal Conjectures file above)." } ], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05C15", "05C63" ] }, "automation": { "methods": [ { "method": "autonomous", "models": [ "GPT-6 Astra (pre-release version, OpenAI)" ], "framework": "Inspect AI (UK AISI) with the LeanOpenProblems task harness; deepagent-based agent for default-configuration attempts, basic ReAct agent for the larger-budget attempts" } ], "strongest": "autonomous", "models": [ "GPT-6 Astra (pre-release version, OpenAI)" ], "spend": "955 USD across the 6 resolution attempt(s) included in this repository, at GPT-6 Astra's standard rates; the paper reports over 220,000 USD of compute across all attempts on all 68 benchmark problems", "notes": "The Lean development was written entirely by the AI system inside the benchmark harness. Humans (Thomas F. Bloom) selected the problem and reviewed the formal statement for faithfulness before the run; humans (Tom Adamczewski) ran the harness and, after the run, packaged the verified output into this repository. The only edits to the AI-written files are mechanical and are listed in the README: port to Mathlib v4.28.0: Mathlib v4.28.0 changed the field `SimpleGraph.loopless` from `Irreflexive Adj` to the class `Std.Irrefl Adj`, so proofs of that field need a `constructor` step (or an anonymous" }, "scope": "Fully formalized: the compared theorem `Erdos74.erdos_74.disproof` in `Challenge.lean` is proved in `Solution.lean` via the imported module `Erdos74.Resolutions.Erdos74_118usd_22h` with no `sorry` and no axioms beyond propext, Quot.sound and Classical.choice. The 5 further module(s) in `Erdos74/Resolutions/` are independent complete proofs of the same statement, built as the separate Lake library `Erdos74Alternates` but not compared by Comparator. Only the negation of the formal conjecture is advertised; the stronger facts visible in the proofs (for example explicit budgets or 3-colourability)", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The compared theorem is exactly the negation of the Formal Conjectures statement. Points a reader should be aware of: (1) `f : ℕ → ℕ` is integer-valued and `Tendsto f atTop atTop` is the formal sense of `f(n) → ∞`; (2) the graph ranges over `SimpleGraph V` for a vertex type `V` in an arbitrary universe `u`, and the theorem is universe-polymorphic; (3) `maxSubgraphEdgeDistToBipartite G n` is the supremum over *all* (not only induced) `n`-vertex subgraphs of the minimum number of edge deletions making the subgraph bipartite — since edge deletion distance is monotone in the edge set, this agrees with the maximum over induced subgraphs; it uses `sSup` on `ℕ`, which is `0` for the empty set (graphs with fewer than `n` vertices), and `sInf`, whose argument set is always nonempty; (4) `G.chromaticNumber = ⊤` is Mathlib's formalisation of infinite chromatic number. The formal statement asserts only existence of some `f → ∞`; the Lean proofs construct explicit budgets but no growth rate is advertised.", "alignment": true, "original": true, "review": { "status": "mechanically verified (comparator: lean kernel replay, standard axioms only) in the benchmark harness and again in this repository's ci; preliminary informal reading of the argument by thomas f. bloom; no independent refereeing", "bucket": "other", "reviewers": [ "Thomas F. Bloom" ], "notes": "Statement audit: the formal statement was selected and reviewed for faithfulness to the informal problem by Thomas F. Bloom before the benchmark run (for Formal Conjectures statements he chose the representative statement; autoformalized statements he checked individually). Mathematical review: the resolution was verified by Comparator against the trusted statement; Thomas F. Bloom made an initial, informal examination of the argument and wrote a short exposition (FrontierMath Erdős appendix and erdosproblems.com). The paper states that the proofs have not yet been properly digested by human experts." }, "canonical": { "repo": "tadamcz/erdos74", "directory": "" }, "nodes": [ { "id": "erdos:74", "join": "anchor", "page": "s/erdos-74.html", "label": "Erdős Problem 74" } ], "container": false, "anchors": { "erdos:74": "https://github.com/google-deepmind/formal-conjectures/blob/488aade228ec37880b8fec178c173c07d279bb53/FormalConjectures/ErdosProblems/74.lean" }, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "tadamcz/koethe/formalization.yaml", "repo": "tadamcz/koethe", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/tadamcz/koethe/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Köthe conjecture: disproof (Krempa's matrix form)", "description": "Disproof of the Köthe conjecture in Krempa's matrix form: there is a ring R with a nil two-sided ideal I such that the matrix ideal M_2(I) is not nil. Found autonomously by GPT-6 Astra in an Epoch AI evaluation run over the open problems of Formal Conjectures' Wikipedia collection.", "authors": [], "maintainers": [], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Köthe conjecture: disproof (Krempa's matrix form)", "id": "", "authors": [], "type": "original-proof", "relationship": "other", "endorsement": "" }, { "title": "Köthe conjecture (Wikipedia)", "id": "https://en.wikipedia.org/wiki/K%C3%B6the_conjecture", "authors": [], "type": "web discussion", "relationship": "background", "endorsement": "" }, { "title": "Die Struktur der Ringe, deren Restklassenring nach dem Radikal vollständig reduzibel ist", "id": "https://doi.org/10.1007/BF01194626", "authors": [ "Gottfried Köthe" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Logical connections between some open problems concerning nil rings", "id": "", "authors": [ "Jan Krempa" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Polynomial rings over nil rings need not be nil", "id": "https://doi.org/10.1006/jabr.2000.8451", "authors": [ "Agata Smoktunowicz" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "A counterexample to Köthe's conjecture and a question of Rowen", "id": "https://arxiv.org/abs/2609.07996", "authors": [ "Tom Adamczewski", "Bernhard Böhmler", "René Marczinzik" ], "type": "paper", "relationship": "other", "endorsement": "" } ], "related": [ { "id": "https://github.com/google-deepmind/formal-conjectures/blob/9cbe1d3c12998c786b7c2cd99ce28a21b6631f66/FormalConjectures/Wikipedia/Koethe.lean", "relationship": "builds-on", "note": "Formal Conjectures states the conjecture with `sorry` as an open problem; the definitions and statement here are copied verbatim from this file at this commit, and the compared theorem is its negation, proved." }, { "id": "https://github.com/epoch-research/LeanOpenProblems/blob/0ef96d7b12cfa96a93761b4bba1c635f4546c5ca/apn/data/wikipedia/Isolated/Koethe.KotherConjecture.variants.general_matrix.lean", "relationship": "builds-on", "note": "The isolated statement file given to the AI system (the Formal Conjectures statement with a `.disproof` negation appended)." } ], "classification": { "arxiv": [ "math.RA" ], "msc2020": [ "16N40", "16S50" ] }, "automation": { "methods": [ { "method": "autonomous", "models": [ "GPT-6 Astra (pre-release version, OpenAI)" ], "framework": "Inspect AI (UK AISI) with the LeanOpenProblems task harness; deepagent-based agent" } ], "strongest": "autonomous", "models": [ "GPT-6 Astra (pre-release version, OpenAI)" ], "spend": "", "notes": "The Lean development was written entirely by the AI system inside the harness, in the evaluation run `wikipedia-vega-1000usd` of Epoch AI's LeanOpenProblems harness (2026-09), in which a pre-release version of GPT-6 Astra attempted, autonomously and once each, all 222 research-open statements of the `Wikipedia` collection of Formal Conjectures under a budget of $1,000 and 96 hours of working time per statement. Tom Adamczewski ran the harness and packaged the verified output into this repository. The only edits to the AI-written file are mechanical and are listed in the README: removed the sor" }, "scope": "Fully formalized: the compared theorem `Koethe.KotherConjecture.variants.general_matrix.disproof` in `Challenge.lean` is proved in `Solution.lean` via the imported module `Koethe.Resolutions.Koethe` with no `sorry` and no axioms beyond propext, Quot.sound and Classical.choice. Only the negation of the formal conjecture is advertised; stronger or more specific facts visible in the proof are not part of the compared statement.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The compared theorem is exactly the negation of the Formal Conjectures statement `general_matrix`, restated explicitly: for all `R : Type u_1` with `[Ring R]`, all two-sided ideals `I` with `IsNil I` (every element nilpotent), and all finite index types `n : Type u_2`, `IsNil (TwoSidedIdeal.matrix n I)`. `TwoSidedIdeal.matrix n I` is Mathlib's entrywise matrix ideal `M_n(I)` and requires a `DecidableEq n` instance, supplied classically (`open scoped Classical in`) exactly as in the Formal Conjectures file. The definition `Koethe.IsNil` is copied verbatim from that file. The statement is universe-polymorphic in `R` and `n`; the witness is built in every universe (`AlgebraicClosure (ULift (ZMod 2))`, index type `ULift (Fin 2)`). The relation to Köthe's original formulation (sums of nil left ideals) is a standard argument stated in the README but not formalized.", "alignment": true, "original": true, "review": { "status": "mechanically verified (comparator: lean kernel replay, standard axioms only) in the harness and again in this repository's ci; no human mathematical review yet", "bucket": "other", "reviewers": [ "none" ], "notes": "Statement audit: the statement is Formal Conjectures' formalization of the problem, taken as is; its correspondence to the informal problem is discussed in fidelity.divergences and the README. Mathematical review: none beyond mechanical verification; the informal proof account in the README was machine-generated from the Lean proof and has not been checked by a human expert." }, "canonical": { "repo": "tadamcz/koethe", "directory": "" }, "nodes": [ { "id": "wikipedia:Koethe", "join": "anchor", "page": "s/wikipedia-koethe.html", "label": "Köthe conjecture" } ], "container": false, "anchors": { "wikipedia:Koethe": "https://github.com/google-deepmind/formal-conjectures/blob/9cbe1d3c12998c786b7c2cd99ce28a21b6631f66/FormalConjectures/Wikipedia/Koethe.lean" }, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "tadamcz/mean-value-problem/formalization.yaml", "repo": "tadamcz/mean-value-problem", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/tadamcz/mean-value-problem/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Smale's mean value conjecture (K = 1): disproof", "description": "Disproof of Smale's mean value conjecture with constant K = 1: there is a complex polynomial p of degree at least 2 and a point z such that every critical point c of p has |p(z) − p(c)|/|z − c| > |p'(z)|. Found autonomously by GPT-6 Astra in an Epoch AI evaluation run over the open problems of Formal Conjectures' Wikipedia collection.", "authors": [], "maintainers": [], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Smale's mean value conjecture (K = 1): disproof", "id": "", "authors": [], "type": "original-proof", "relationship": "other", "endorsement": "" }, { "title": "Smale's mean value conjecture (K = 1) (Wikipedia)", "id": "https://en.wikipedia.org/wiki/Mean_value_problem", "authors": [], "type": "web discussion", "relationship": "background", "endorsement": "" }, { "title": "The fundamental theorem of algebra and complexity theory", "id": "https://doi.org/10.1090/S0273-0979-1981-14858-8", "authors": [ "Steve Smale" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Critical points and values of complex polynomials", "id": "https://doi.org/10.1016/0885-064X(89)90019-8", "authors": [ "David Tischler" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Smale's mean value conjecture and the hyperbolic metric", "id": "https://doi.org/10.1007/s002080100276", "authors": [ "Alan F. Beardon", "David Minda", "T. W. Ng" ], "type": "paper", "relationship": "background", "endorsement": "" } ], "related": [ { "id": "https://github.com/google-deepmind/formal-conjectures/blob/9cbe1d3c12998c786b7c2cd99ce28a21b6631f66/FormalConjectures/Wikipedia/MeanValueProblem.lean", "relationship": "builds-on", "note": "Formal Conjectures states the conjecture with `sorry` as an open problem; the definitions and statement here are copied verbatim from this file at this commit, and the compared theorem is its negation, proved." }, { "id": "https://github.com/epoch-research/LeanOpenProblems/blob/0ef96d7b12cfa96a93761b4bba1c635f4546c5ca/apn/data/wikipedia/Isolated/MeanValueProblem.mean_value_problem.lean", "relationship": "builds-on", "note": "The isolated statement file given to the AI system (the Formal Conjectures statement with a `.disproof` negation appended)." } ], "classification": { "arxiv": [ "math.CV" ], "msc2020": [ "30C10", "30C15" ] }, "automation": { "methods": [ { "method": "autonomous", "models": [ "GPT-6 Astra (pre-release version, OpenAI)" ], "framework": "Inspect AI (UK AISI) with the LeanOpenProblems task harness; deepagent-based agent" } ], "strongest": "autonomous", "models": [ "GPT-6 Astra (pre-release version, OpenAI)" ], "spend": "", "notes": "The Lean development was written entirely by the AI system inside the harness, in the evaluation run `wikipedia-vega-1000usd` of Epoch AI's LeanOpenProblems harness (2026-09), in which a pre-release version of GPT-6 Astra attempted, autonomously and once each, all 222 research-open statements of the `Wikipedia` collection of Formal Conjectures under a budget of $1,000 and 96 hours of working time per statement. Tom Adamczewski ran the harness and packaged the verified output into this repository. The only edits to the AI-written file are mechanical and are listed in the README: removed the sor" }, "scope": "Fully formalized: the compared theorem `MeanValueProblem.mean_value_problem.disproof` in `Challenge.lean` is proved in `Solution.lean` via the imported module `MeanValueProblem.Resolutions.MeanValueProblem` with no `sorry` and no axioms beyond propext, Quot.sound and Classical.choice. Only the negation of the formal conjecture is advertised; stronger or more specific facts visible in the proof are not part of the compared statement.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The compared theorem is exactly the negation of the Formal Conjectures statement, restated explicitly: for every `p : Polynomial ℂ` with `2 ≤ p.degree`, every `z : ℂ` and every `K : ℝ`, there is `c` with `p.derivative.eval c = 0` and `‖p.eval z - p.eval c‖ / ‖z - c‖ ≤ ‖p.derivative.eval z‖`. The parameter `K` is unused in the Formal Conjectures statement (the constant is fixed at 1). Lean's convention `x / 0 = 0` makes the case `p'(z) = 0` hold trivially with `c = z`, so the formal conjecture is if anything weaker than the informal one; the counterexample uses `z = 0` with `p'(0) = 1`, so it refutes the informal conjecture as stated in the literature.", "alignment": true, "original": true, "review": { "status": "mechanically verified (comparator: lean kernel replay, standard axioms only) in the harness and again in this repository's ci; no human mathematical review yet", "bucket": "other", "reviewers": [ "none" ], "notes": "Statement audit: the statement is Formal Conjectures' formalization of the problem, taken as is; its correspondence to the informal problem is discussed in fidelity.divergences and the README. Mathematical review: none beyond mechanical verification; the informal proof account in the README was machine-generated from the Lean proof and has not been checked by a human expert." }, "canonical": { "repo": "tadamcz/mean-value-problem", "directory": "" }, "nodes": [ { "id": "wikipedia:MeanValueProblem", "join": "anchor", "page": "s/wikipedia-meanvalueproblem.html", "label": "Mean value problem" } ], "container": false, "anchors": { "wikipedia:MeanValueProblem": "https://github.com/google-deepmind/formal-conjectures/blob/9cbe1d3c12998c786b7c2cd99ce28a21b6631f66/FormalConjectures/Wikipedia/MeanValueProblem.lean" }, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "teorth/sendov/formalization.yaml", "repo": "teorth/sendov", "path": "formalization.yaml", "directory": "", "origins": [ "cited", "palomar", "search" ], "url": "https://github.com/teorth/sendov/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Sendov's conjecture and the Phelps-Rodriguez conjecture", "description": "A Lean formalization of Sendov's conjecture, and of its Phelps-Rodriguez strengthening, in full generality: if every zero of a complex polynomial p of degree n >= 2 lies in the closed unit disk, then every zero a of p has a critical point of p within distance 1 of it, and within distance strictly less than 1 unless |a| = 1 and p is a scalar multiple of z^n - a^n. The proof is by contradiction. Degrees 2 to 4 fall to an elementary integral computation; degree 5 upward goes through two estimates on the same integral -- a polar channel and an origin channel -- which are shown to be incompatible, ", "authors": [ "Terence Tao" ], "maintainers": [ "Terence Tao" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "A digestion of the proof of Sendov's conjecture", "id": "https://terrytao.wordpress.com/2026/08/12/a-digestion-of-the-proof-of-sendovs-conjecture/", "authors": [ "Terence Tao" ], "type": "web post", "relationship": "formalizes", "endorsement": "" }, { "title": "Research Problems in Function Theory (statement of Sendov's conjecture)", "id": "https://zbmath.org/3254142", "authors": [ "W. K. Hayman" ], "type": "book", "relationship": "background", "endorsement": "" }, { "title": "Some properties of extremal polynomials for the Ilieff conjecture", "id": "https://zbmath.org/3386011", "authors": [ "D. Phelps", "R. S. Rodriguez" ], "type": "article", "relationship": "formalizes", "endorsement": "" }, { "title": "On a problem of Ilyeff", "id": "", "authors": [ "Z. Rubinstein" ], "type": "article", "relationship": "background", "endorsement": "" }, { "title": "Sharp Schoenberg type inequalities and the de Bruin-Sharma problem", "id": "arXiv:2508.10341", "authors": [ "Quanyu Tang", "Teng Zhang" ], "type": "preprint", "relationship": "background", "endorsement": "" }, { "title": "On a conjecture of Ilieff", "id": "", "authors": [ "D. A. Brannan" ], "type": "article", "relationship": "background", "endorsement": "" }, { "title": "On Ilyeff's conjecture", "id": "", "authors": [ "A. Meir", "A. Sharma" ], "type": "article", "relationship": "background", "endorsement": "" }, { "title": "Sendov's conjecture for sufficiently high degree polynomials", "id": "https://zbmath.org/7681950", "authors": [ "Terence Tao" ], "type": "article", "relationship": "background", "endorsement": "" }, { "title": "Sendov conjecture proof (ProofAtlas manuscript)", "id": "https://www.proofatlas.ai/papers/sendov-conjecture/", "authors": [ "Lech Mazur" ], "type": "other", "relationship": "adapts", "endorsement": "" } ], "related": [ { "id": "https://www.proofatlas.ai/formalizations/sendov-conjecture/", "relationship": "builds-on", "note": "The earlier Lean formalization of Sendov's conjecture for all n >= 2, by Lech Mazur, reported there as 1160 files and 92816 lines with a single main theorem. Priority for the first machine-checked proof of Sendov's conjecture belongs to that work, and this repository is the second formalization of it. \"builds-on\" rather than \"independent\": no code is shared and that development was never read, but" } ], "classification": { "arxiv": [ "math.CV" ], "msc2020": [ "30C15", "30C10", "12D10" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Claude Opus 5 (Anthropic)" ], "framework": "Claude Code" }, { "method": "other", "models": [], "framework": "" }, { "method": "other", "models": [], "framework": "Python scripts in scripts/" } ], "strongest": "agent", "models": [ "Claude Opus 5 (Anthropic)" ], "spend": "", "notes": "" }, "scope": "Complete, with no unproved step. Both conjectures in full generality: every complex polynomial of degree at least two with all zeroes in the closed unit disk, and every zero of it. No restriction on the degree beyond n >= 2, on the modulus or argument of the distinguished zero, or on multiplicities. Nothing is assumed from the literature; the two compared theorems are proved from Mathlib alone.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "SendovConjecture.sendov", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SendovConjecture.phelps_rodriguez", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "Faithful to the source, and in two respects stronger than it. The blog post argues Sendov's conjecture for n >= 5; this development proves all n >= 2, adding degrees 2 to 4 in Sendov/Analytic/LowDegree.lean. The blog post states the non-strict conclusion; this development additionally extracts the Phelps-Rodriguez equality classification, strengthening the distance bound to a strict inequality except for p = c(z^n - a^n) with |a| = 1. Both are generalizations rather than weakenings: no hypothesis of the source was strengthened, and no part of its statement was dropped. One step is proved by a different route than the informal account: for n >= 101 the write-up in docs/proof-large-degree.md splits 0 <= alpha <= 17 at alpha = 16 and estimates the two pieces by exact rational endpoint bounds, whereas Sendov/LargeDegree/Endgame.lean settles the whole interval with one generated Bernstein certificate, Sendov.F_pos, which the sharp Beta constant leaves enough margin for. The conclusion is the same. Lemma names differ from the informal text throughout; the roadmap in README.md records the correspondence.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Terence Tao" ], "notes": "The author directed and reviewed the development as it was produced. No external or independent review has been performed, and no mathematician other than the author has read the Lean source. Every step is checked by Lean, and scripts/audit.sh checks the trust conditions below." }, "canonical": { "repo": "teorth/sendov", "directory": "" }, "nodes": [ { "id": "wikipedia:Sendov", "join": "artifact", "page": "s/wikipedia-sendov.html", "label": "Sendov's conjecture" } ], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-13-000001", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-13-000001", "commit": "1ddea92d89f951a0a7cbbffa6c267cf7e6640b1d", "trust": "high", "theorems": 2, "date": "2026-08-20" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "wcook04/plectis-erdos/formalization.yaml", "repo": "wcook04/plectis-erdos", "path": "formalization.yaml", "directory": "", "origins": [ "cited" ], "url": "https://github.com/wcook04/plectis-erdos/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Eight Erdős problem programmes: checked mathematics", "description": "Lean 4 formalization across eight Erdős problem programmes, including unconditional theorems, exact conditional endpoint reductions, arithmetic classifications, counterexamples, and method barriers. The precise release boundary is recorded below.", "authors": [ "Will Cook" ], "maintainers": [ "Will Cook" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Erdős Problems 68, 243, 249, 251, 257, 269, 1041, and 1049", "id": "https://www.erdosproblems.com/", "authors": [ "Paul Erdős and collaborators" ], "type": "web discussion", "relationship": "background", "endorsement": "n/a" }, { "title": "Repository problem notes and cited-source registry", "id": "docs/papers/corpus.json", "authors": [ "Will Cook" ], "type": "other", "relationship": "adapts", "endorsement": "n/a" } ], "related": [], "classification": { "arxiv": [ "math.NT", "math.CO" ], "msc2020": [ "11J72" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "OpenAI Codex and other disclosed agent systems" ], "framework": "agent-assisted Lean formalisation and proof engineering" }, { "method": "other", "models": [], "framework": "Pinned Lean kernel and statement-isolated Comparator CI" } ], "strongest": "agent", "models": [ "OpenAI Codex and other disclosed agent systems" ], "spend": "", "notes": "" }, "scope": "Using the degree-seven polynomial constructed by the erdosproblems.com contributor ani, Lean proves that every preconnected strict-lemniscate set containing two distinct roots has one-dimensional Hausdorff measure greater than two. This refutes the exact Formal Conjectures path-image-length statement; the separate total-variation bound is also checked. The other seven targets remain open. Independent human review of correspondence with the 1958 wording has not been recorded. Comparator checks only selected exact statements, axioms and kernel acceptance; it does not assess novelty or historical", "sorry_count": 8, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "ErdosProblems.Erdos68.irrational_factorialGapSeries_iff_cofinal_strictFacTopRat_misses", "file": "ErdosProblems/Erdos68/FactorialGapPlateauCore.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "ErdosProblems.Erdos68.irrational_factorialGapSeries_iff_cofinal_nonunit_carries", "file": "ErdosProblems/Erdos68/FactorialGapPlateauCore.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "ErdosProblems.Erdos243.no_cofinallyBoundedNegative_of_normalizedVanishes", "file": 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Each registry-shaped statement under Palomar// is one Comparator comparison: a Challenge importing only Mathlib, a Solution delegating to the proof here, and the configuration naming the compared declaration.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "Palomar.Erdos730.erdos_730_infinite", "file": "PalomarSolutions/Erdos730.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Palomar.Erdos154.erdos_154_sumset", "file": "PalomarSolutions/Erdos154.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Palomar.Erdos94.sum_multiplicity", "file": "PalomarSolutions/Erdos94.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "#730: none in the statement — the Challenge inlines the Formal Conjectures set verbatim; the proof establishes the stronger consecutive-pair statement, available in the development as Erdos730.FullDensityCore.GoodParameter and the density theorems around it. #154: the Challenge states the sumset form Formal Conjectures records, which is the consequence proved here of Lindström's theorem for A itself, with IsSidon in the Formal Conjectures shape (two representations agree up to order). #94: an elementary bounded identity only; it does not prove the cubic distance-multiplicity theorem or the regular-polygon conjecture of that problem.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "Kernel-checked and axiom-audited in this repository's CI and reproduced from source by the author; no independent human review of the mathematics has been performed, and the informal #730 source argument is itself unadjudicated on erdosproblems.com. The #730 statement is submitted to Palomar to obtain the registry's independent dual-kernel Comparator check; #154 waits on the upstream licence of the formalisation it builds on; #94 is below a registry's research-interest floor and is not submitted." }, "canonical": { "repo": "williamjblair/lean-proofs", "directory": "" }, "nodes": [ { "id": "erdos:1188", "join": "artifact", "page": "s/erdos-1188.html", "label": "Erdős Problem 1188" }, { "id": "erdos:130", "join": "artifact", "page": "s/erdos-130.html", "label": "Erdős Problem 130" }, { "id": "erdos:254", "join": "artifact", "page": "s/erdos-254.html", "label": "Erdős Problem 254" }, { "id": "erdos:267", "join": "artifact", "page": "s/erdos-267.html", "label": "Erdős Problem 267" }, { "id": "erdos:394", "join": "artifact", "page": "s/erdos-394.html", "label": "Erdős Problem 394" }, { "id": "erdos:450", "join": "artifact", "page": "s/erdos-450.html", "label": "Erdős Problem 450" }, { "id": "erdos:489", "join": "artifact", "page": "s/erdos-489.html", "label": "Erdős Problem 489" }, { "id": "erdos:521", "join": "artifact", "page": "s/erdos-521.html", "label": "Erdős Problem 521" }, { "id": "erdos:538", "join": "artifact", "page": "s/erdos-538.html", "label": "Erdős Problem 538" }, { "id": "erdos:730", "join": "artifact", "page": "s/erdos-730.html", "label": "Erdős Problem 730" }, { "id": "erdos:769", "join": "artifact", "page": "s/erdos-769.html", "label": "Erdős Problem 769" }, { "id": "erdos:796", "join": "artifact", "page": "s/erdos-796.html", "label": "Erdős Problem 796" }, { "id": "erdos:959", "join": "artifact", "page": "s/erdos-959.html", "label": "Erdős Problem 959" } ], "container": true, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-22-000001", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-22-000001", "commit": "03729c9cbb0b602f5a828bb850c85e84c5a6d460", "trust": "high", "theorems": 1, "date": "2026-08-22" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "zhangjun725/erdos557/formalization.yaml", "repo": "zhangjun725/erdos557", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/zhangjun725/erdos557/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Erdős problem #557 (multicolour Ramsey numbers of trees): proof", "description": "Proof that the k-colour Ramsey number of every tree on n vertices is at most k(n-2)+3, hence R_k(T) ≤ kn + O(1) (Erdős problem #557). Derived in Lean 4 / Mathlib from the machine-checked proof of the Erdős–Sós conjecture (Erdős problem #548) found by GPT-6 Astra and packaged by Tom Adamczewski.", "authors": [], "maintainers": [], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Erdős problem #557 (erdosproblems.com)", "id": "https://www.erdosproblems.com/557", "authors": [], "type": "web discussion", "relationship": "background", "endorsement": "" }, { "title": "On partition theorems for finite graphs", "id": "", "authors": [ "Paul Erdős", "Ronald L. Graham" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "The Erdős-Sós conjecture in dense graphs", "id": "", "authors": [ "Bruce Reed", "Maya Stein" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Erdős problem #548 (Erdős–Sós conjecture): proof", "id": "", "authors": [ "Tom Adamczewski" ], "type": "other", "relationship": "builds-on", "endorsement": "" } ], "related": [ { "id": "https://github.com/tadamcz/erdos548/blob/82ffb751f3d37768927df9239ed08439bbe0dd09/Challenge.lean", "relationship": "builds-on", "note": "The compared theorem `Erdos548.erdos_548` of that repository is the only external result used." } ], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05C55", "05D10", "05C05" ] }, "automation": { "methods": [ { "method": "assisted", "models": [ "Claude (Anthropic)" ], "framework": "Claude Code" } ], "strongest": "other", "models": [ "Claude (Anthropic)" ], "spend": "", "notes": "Human involvement: problem selection, statement review and final responsibility for the development by Jun Zhang. The mathematical reduction (Erdős–Sós ⇒ #557 by pigeonhole) is folklore and is recorded on the erdosproblems.com page for #557." }, "scope": "Fully formalized: `Erdos557.erdos_557_tight` and `Erdos557.erdos_557_tight_odd` (the best-possible bounds R_k(T) ≤ k(n-2)+2, and k(n-2)+1 for odd n and even k, via the sharp lemma `Erdos548.tree_free_edge_bound`), `Erdos557.erdos_547` (R(T) ≤ 2n-2, Erdős problem #547, same reduction), `Erdos557.erdos_557_explicit` (every k-edge-labelling of the complete graph on k(n-2)+3 vertices has a colour class containing every tree on n vertices, for all k and n) and `Erdos557.erdos_557` (∃ c, ∀ k n T, T.IsTree → R_k(T) ≤ k n + c, with `R_k` defined as `Erdos557.multicolourRamsey`), with no `sorry` and no", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The informal problem asks whether R_k(T) ≤ kn + O(1) for every tree T on n vertices. `erdos_557` states this with an absolute constant c (c = 3 is exhibited), which is the strongest reading of O(1) (it does not depend on k). A k-colouring of the edges of K_m is modelled by Mathlib's `TopEdgeLabeling (Fin m) (Fin k)`, a function from the edge set of the complete graph to `Fin k`; a monochromatic copy is `T.IsContained (C.labelGraph i)` (not necessarily induced). `multicolourRamsey k G` is the infimum of the set of m for which every such labelling has a monochromatic copy of G; by `Nat.sInf_le` the explicit theorem gives the bound. Trees with n ≥ 1 vertices are covered; for n = 0 there are no trees (`IsTree` requires a vertex). The value k = 0 is allowed and degenerate: no 0-labelling of a graph with an edge exists.", "alignment": true, "original": false, "review": { "status": "mechanically verified (lean kernel, standard axioms only; see ci); no independent mathematical refereeing", "bucket": "other", "reviewers": [], "notes": "Statement audit: the author compared the Lean statement with the erdosproblems.com phrasing. External review by the erdosproblems.com curators has not yet taken place." }, "canonical": { "repo": "zhangjun725/erdos557", "directory": "" }, "nodes": [ { "id": "erdos:557", "join": "names", "page": "s/erdos-557.html", "label": "Erdős 557" } ], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "32805433/Adian-Rabin/formalization.yaml", "repo": "32805433/Adian-Rabin", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/32805433/Adian-Rabin/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Small undecidable groups and fixed-size Adian–Rabin families", "description": "Lean 4 proofs of the algebraic results in Small undecidable groups and unrecognizable 4-manifolds (arXiv:2609.10461v1): existence of a group with a three-generator, nine-relator presentation and unsolvable word problem; an effective family of four-generator, eleven-relator presentations with undecidable triviality; and an effective family of two-generator, ten-relator presentations with undecidable triviality. These are Theorems 1.1, 1.3, and 1.4 of the paper. The four-manifold consequences in the paper are outside the formalized scope.", "authors": [ "Marc Kegel", "Shana Yunsheng Li", "Qiuyu Ren" ], "maintainers": [ "Qiuyu Ren" ], "license": "MIT", "role": "", "substantive": "", "sources": [ { "title": "Small undecidable groups and unrecognizable 4-manifolds", "id": "https://arxiv.org/abs/2609.10461v1", "authors": [ "Marc Kegel", "Shana Yunsheng Li", "Qiuyu Ren" ], "type": "paper", "relationship": "formalizes", "endorsement": "participated" }, { "title": "Simple examples of groups with unsolvable word problem", "id": "https://doi.org/10.1007/BF01101402", "authors": [ "V. V. Borisov" ], "type": "paper", "relationship": "adapts", "endorsement": "" }, { "title": "Word problem for Thue systems with a few relations", "id": "https://doi.org/10.1007/3-540-59340-3_4", "authors": [ "Yuri Matiyasevich" ], "type": "paper", "relationship": "adapts", "endorsement": "" }, { "title": "Recursive unsolvability of a problem of Thue", "id": "https://doi.org/10.2307/2267170", "authors": [ "Emil L. Post" ], "type": "paper", "relationship": "adapts", "endorsement": "" }, { "title": "Decision problems for groups—survey and reflections", "id": "https://doi.org/10.1007/978-1-4613-9730-4_1", "authors": [ "Charles F. Miller III" ], "type": "paper", "relationship": "adapts", "endorsement": "" }, { "title": "Simpler algorithmically unrecognizable 4-manifolds", "id": "https://arxiv.org/abs/2310.07421v2", "authors": [ "Martin Tancer" ], "type": "paper", "relationship": "adapts", "endorsement": "" }, { "title": "On the homeomorphism problem for 4-manifolds", "id": "https://doi.org/10.53733/205", "authors": [ "Cameron McA. Gordon" ], "type": "paper", "relationship": "adapts", "endorsement": "" }, { "title": "The Word Problem for Groups", "id": "https://doi.org/10.1112/plms/s3-8.4.493", "authors": [ "J. L. Britton" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Formalizing Computability Theory via Partial Recursive Functions", "id": "https://doi.org/10.4230/LIPIcs.ITP.2019.12", "authors": [ "Mario Carneiro" ], "type": "paper", "relationship": "background", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.GR", "math.LO" ], "msc2020": [ "20F10", "20F05", "03D35" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "GPT-5.6 Sol Pro" ], "framework": "" }, { "method": "agent", "models": [ "GPT-5.6 Sol" ], "framework": "Codex" }, { "method": "agent", "models": [ "GPT-6" ], "framework": "Codex" } ], "strongest": "agent", "models": [ "GPT-5.6 Sol", "GPT-5.6 Sol Pro", "GPT-6" ], "spend": "", "notes": "Essentially all Lean code was generated with AI assistance under human direction. The paper, Section 1.6, reports manual checking of the theorem statements. Detailed cost and runtime measurements are not available." }, "scope": "Complete proofs of the three formalized algebraic existence theorems and their required dependencies. The statement interface counts generator and relator slots. No optimality or minimality claim, four-manifold theorem, or general theorem about all Markov properties is within scope.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "Undecidability.exists_three_generator_nine_relator_group_with_unsolvable_word_problem", "file": "GroupUndecidability.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Undecidability.exists_four_generator_eleven_relator_adian_rabin_family", "file": "GroupUndecidability.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Undecidability.exists_two_generator_ten_relator_adian_rabin_family", "file": "GroupUndecidability.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The Lean interface represents a presentation by exactly n generator slots and m relator slots, without claiming these counts are minimal or that relators are distinct. An Adian–Rabin family is a total computable sequence of relator tuples with an undecidable set of trivial-group indices; the paper phrases this as a recursively enumerable family with undecidable triviality. This effective indexing gives such a family: a decision procedure on its presentations would decide its indices by composition. The Thue-system input is proved using Post's machine simulation and Matiyasevich's 1995 compression. Only the algebraic Theorems 1.1, 1.3, and 1.4 are represented here. The topological and smooth conclusions of Corollary 1.5 are not checked.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "Section 1.6 of the paper reports manual checking of the formal statements by the authors. The current revision passed a clean build and transitive axiom audit with Lean and Mathlib 4.33.0; AXIOM_AUDIT.md records these checks and their scope. Separate statement comparison and proof replay have not been rerun for this version. The review status does not claim independent human review of all proof code." }, "canonical": { "repo": "32805433/adian-rabin", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-22-000002", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-22-000002", "commit": "fb4a401e8fbdba36e7b35c11fde2baa15592e185", "trust": "high", "theorems": 3, "date": "2026-09-22" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "abhisaha-git/fourier-jacobi-core-evaluation/formalization.yaml", "repo": "abhisaha-git/fourier-jacobi-core-evaluation", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/abhisaha-git/fourier-jacobi-core-evaluation/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Explicit Fourier–Jacobi Haar-core evaluation and Abel limits", "description": "For a nonarchimedean local field with residue cardinality q > 2, this development evaluates a sum of three normalized Haar integrals of an explicit eight-term Weyl kernel defined from valuations of entries and two-by-two minors of a concrete 4 by 4 matrix. It proves measurability and Bochner integrability on the natural damped range, an explicit rational evaluation in the undamped interior, principal Abel convergence, and the whole signed special Abel limit at delta = plus or minus q^(-1/2). The Satake parameters are unitary and avoid the stated Weyl walls; the special quotient also excludes p", "authors": [ "Abhishek Saha" ], "maintainers": [ "Abhishek Saha" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "An explicit refined Gan–Gross–Prasad identity for Fourier–Jacobi periods of degree 2 Siegel cusp forms", "id": "https://arxiv.org/abs/2608.26007", "authors": [ "Biplab Paul", "Ameya Pitale", "Abhishek Saha", "Ralf Schmidt" ], "type": "paper", "relationship": "adapts", "endorsement": "" }, { "title": "Expanded Section 2 and corrected core-evaluation brief", "id": "PROVENANCE.md", "authors": [ "Abhishek Saha" ], "type": "other", "relationship": "adapts", "endorsement": "" } ], "related": [ { "id": "verification/preservation-recheck.json", "relationship": "builds-on", "note": "The earlier author-directed Fourier–Jacobi theorem-2-2-v2 development supplied thirty modules: finite Weyl and region algebra, four infinite I0 rows, and local-field valuation, Haar, collision and matrix lemmas. Those sources are included in this substantive repository; this is not a wrapper around a separately required unpublished Lean project. No public repository URL for that earlier snapshot i" } ], "classification": { "arxiv": [ "math.NT", "math.RT" ], "msc2020": [ "11F70", "11F85" ] }, "automation": { "methods": [ { "method": "agent", "models": [], "framework": "OpenAI Codex" }, { "method": "agent", "models": [ "GPT-6" ], "framework": "OpenAI Codex" } ], "strongest": "agent", "models": [ "GPT-6" ], "spend": "", "notes": "AI systems are credited as tools, not as human authors or maintainers. Four deliberate theorem placeholders occur only in Challenge.lean, with the author's explicit authorization. They are excluded from the unresolved proof counts; Solution.lean and the substantive proof development contain none. The solution never imports the challenge." }, "scope": "Complete explicit-kernel Haar-core evaluation, integrability, principal Abel limit and whole signed special Abel limit. The spherical-coefficient identification and full period theorem are outside the registered claims.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "FourierJacobiPalomar.measure_normalizations", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "FourierJacobiPalomar.integrability", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "FourierJacobiPalomar.principal_evaluation", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "FourierJacobiPalomar.special_abel_limit", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "Explicit-kernel result only; no spherical-coefficient or full-period bridge. Arbitrary characteristic with residue cardinality q > 2. Regular unitary alpha,beta with alpha,beta != 1, alpha != beta, alpha*beta != 1. With t=q^(-1/2), integrability requires 0 2; this formalization restricts n to 3 through 6.", "alignment": true, "original": false, "review": { "status": "self-assessed; no independent human expert review", "bucket": "self-assessed", "reviewers": [], "notes": "Complete local Lean 4.19 and Lean 4.32 builds passed, including the final theorem. Palomar Comparator and NanoDa acceptance are not yet claimed." }, "canonical": { "repo": "aivv73/kourovka-21-130-lean", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "ajt60gaibb/OpenProblemsInNLA/eigenvalues-and-inverse-problems/IS-02/lean/formalization.yaml", "repo": "ajt60gaibb/OpenProblemsInNLA", "path": "eigenvalues-and-inverse-problems/IS-02/lean/formalization.yaml", "directory": "eigenvalues-and-inverse-problems/IS-02/lean", "origins": [ "search" ], "url": "https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/HEAD/eigenvalues-and-inverse-problems/IS-02/lean/formalization.yaml", "version": "v0.4", "missing": [], "name": "IS-02: counterexample to the spectral-uniqueness locus condition", "description": "Complete exact order-four counterexample to the original spectral-uniqueness locus condition, independently reviewed and verified by the actual non-root Linux sandbox, Comparator, permitted-axiom audit and default kernel.", "authors": [ "George Stepaniants" ], "maintainers": [ "George Stepaniants" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "IS-02 canonical problem", "id": "https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/50838e37dd793830e2cecd1055cfc7e0349490f1/eigenvalues-and-inverse-problems/IS-02/README.md", "authors": [], "type": "web-post", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "IS-02 complete counterexample manuscript", "id": "https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/50838e37dd793830e2cecd1055cfc7e0349490f1/eigenvalues-and-inverse-problems/IS-02/solution.md", "authors": [ "Matthew J. 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Source mathematics and exact certificate remain credited to Sidney Holden. No external human peer review, source-author endorsement, official Tau Ceti service, priority certification or measured cost is claimed." }, "scope": "Complete original universal negation from the exact finite counterexample; all fifteen contracts, including attained full/subset minima and positive actual subset maximum. Stronger amplification is outside formalized scope.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "NLA.IE16.explicitL_card", "file": "NLA/IE16/Numeric.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "NLA.IE16.explicitL_admissible", "file": "NLA/IE16/Numeric.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "NLA.IE16.witness_feasible", "file": "NLA/IE16/Numeric.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "NLA.IE16.witness_objective", "file": "NLA/IE16/Numeric.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "NLA.IE16.full_minimum_exact", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "NLA.IE16.full_minimum_isLeast", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "NLA.IE16.full_lower_bound", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "NLA.IE16.every_five_point_subset_upper", "file": "NLA/IE16/SubsetGeometryDraft.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "NLA.IE16.every_five_point_subset_minimum_isLeast", "file": "NLA/IE16/SubsetGeometryDraft.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "NLA.IE16.subset_max_upper", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "NLA.IE16.subset_max_positive", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "NLA.IE16.ratio_lower_bound", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "NLA.IE16.ratio_exceeds_candidate", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "NLA.IE16.counterexample", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "NLA.IE16.not_IE16Conjecture", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The boundary retains every n >= 3, every n-point nonzero complex spectrum and every 1 <= k <= n-2. The finite counterexample fixes epsilon=1/1000, n=9 and k=4. Holden’s Section 6 positive weighted orthogonality supplies the full minimum and attainment. Generic complex Lagrange interpolation, injective nine-point labels, complete cluster occupancy and uniform product estimates 109 and 146 give the all-subset bound. The 126-row certificate is supplementary; it is not imported as a theorem or used to define M or subsetMax. The stronger amplification/unbounded-ratio theorem, asymptotic limit, operator-level GMRES construction, exact subsetMax value and minimality of dimension are outside scope.", "alignment": true, "original": false, "review": { "status": "two pre-proof statement approvals; two independent final mathematical and canonical linux-evidence approvals; coordinator accepted.", "bucket": "other", "reviewers": [ "/root/is02_cleanup_referee (independent AI agent)", "/root (AI agent; independent of statement authorship at that phase)", "/root/lean_iv01_next (independent AI agent)", "/root/nr03_independent_referee (independent AI agent)" ], "notes": "Statement acceptance precedes implementation. Historical source and compiled-development reports retain their exact scopes. The final addenda review the unchanged complete mathematics and actual canonical Linux evidence at 697a2a1d88337a6747aa5c82fb6e554d3ff1b356. Original compiler findings are resolved. Root and the source implementer do not count as independent final referees. Reviews follow the repository’s Tau Ceti adaptation; no official service or human peer review is claimed." }, "canonical": { "repo": "ajt60gaibb/openproblemsinnla", "directory": "linear-systems-and-elimination/IE-16/lean" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "ajt60gaibb/OpenProblemsInNLA/linear-systems-and-elimination/IE-17/lean/formalization.yaml", "repo": "ajt60gaibb/OpenProblemsInNLA", "path": "linear-systems-and-elimination/IE-17/lean/formalization.yaml", "directory": "linear-systems-and-elimination/IE-17/lean", "origins": [ "search" ], "url": "https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/HEAD/linear-systems-and-elimination/IE-17/lean/formalization.yaml", "version": "v0.4", "missing": [], "name": "IE-17: both LSMR backward errors can increase", "description": "Complete exact counterexample to both canonical monotonicity claims: attained matrix-only spectral minimum and the literal Moore–Penrose approximation along actual exact LSMR iterates.", "authors": [ "George Stepaniants" ], "maintainers": [ "George Stepaniants" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "IE-17 — Monotonic optimal backward error along LSMR", "id": "https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/8f04b905eb2e0827b6b84f37d9d080ae1f05b202/linear-systems-and-elimination/IE-17/README.md", "authors": [], "type": "web-post", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "IE-17: an exact counterexample for both LSMR backward errors", "id": "https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/8f04b905eb2e0827b6b84f37d9d080ae1f05b202/references/colbrook-recovered-2026-09-11/manuscripts/IE-17.tex", "authors": [ "Matthew J. 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Independent statement/final referees /root/reference_api_review and /root/existing_verification_audit contributed no proof code. Reviews follow the repository Tau Ceti adaptation; no official Tau Ceti endorsement, human peer review, priority certification or unmeasured costs claimed." }, "scope": "Complete original target verified: both separate monotonicity claims are false for the same actual exact LSMR run, with attained all-real spectral minima and unique literal Moore–Penrose approximation values. All eight reviewed declarations passed fresh sandboxed Linux Comparator and default-kernel replay in run 35016724809, attempt 1. Both independent source and operational reviews approved; canonical status is Lean verified. Challenge contains eight deliberate specification-only placeholders, excluded from proof-development sorry counts.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "NLA.IE17.spectralNorm_semantics", "file": "NLA/IE17/Proof.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "NLA.IE17.witness_full_column_rank", "file": "NLA/IE17/Proof.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "NLA.IE17.exact_lsmr_run", "file": "NLA/IE17/Proof.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "NLA.IE17.optimal_errors", "file": "NLA/IE17/Proof.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "NLA.IE17.approximation_values", "file": "NLA/IE17/Proof.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "NLA.IE17.terminal_errors_zero", "file": "NLA/IE17/Proof.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "NLA.IE17.both_errors_increase", "file": "NLA/IE17/Proof.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "NLA.IE17.canonical_counterexamples", "file": "NLA/IE17/Proof.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "", "alignment": false, "original": false, "review": { "status": "independent statements, complete proof source and authoritative linux evidence approved", "bucket": "other", "reviewers": [ "/root/reference_api_review (independent Codex AI agent)", "/root/existing_verification_audit (independent Codex AI agent)" ], "notes": "Exact pre-proof freeze, both statement approvals, both final source approvals and both operational audits are retained under reviews/. Authoritative run https://github.com/ajt60gaibb/OpenProblemsInNLA/actions/runs/35016724809/attempts/1 at proof revision 6e53192977d87097666c039f6d8134a800ff7501. Raw artifact, provenance, source hashes and all controls are retained under verification/linux-2026-09-15/. Artifact 10416087458 has SHA-256 b2b2a164604c4922ebd9e614e39aa9c8dcb1d184be07e845e71adf93272c03c3. Publication updates change documentation and evidence only; mathematical statements, proofs and dependency inputs match that run." }, "canonical": { "repo": "ajt60gaibb/openproblemsinnla", "directory": "linear-systems-and-elimination/IE-17/lean" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "ajt60gaibb/OpenProblemsInNLA/linear-systems-and-elimination/IE-18/lean/formalization.yaml", "repo": "ajt60gaibb/OpenProblemsInNLA", "path": "linear-systems-and-elimination/IE-18/lean/formalization.yaml", "directory": "linear-systems-and-elimination/IE-18/lean", "origins": [ "search" ], "url": "https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/HEAD/linear-systems-and-elimination/IE-18/lean/formalization.yaml", "version": "v0.4", "missing": [], "name": "IE-18: a counterexample to the four-step Anderson identity", "description": "Derives the actual two residual-map executions and the maximum over the genuine Hermitian eigenvalue list, then refutes the complete original universal maximum identity by an admissible strict counterexample.", "authors": [ "George Stepaniants" ], "maintainers": [ "George Stepaniants" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "IE-18 — The exact four-step amplification of restarted Anderson acceleration", "id": "https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/5adea969c17391693978ada2674d25bb5c3daeb1/linear-systems-and-elimination/IE-18/README.md", "authors": [], "type": "web-post", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "IE-18: counterexamples to the exact four-step identity", "id": "https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/5adea969c17391693978ada2674d25bb5c3daeb1/references/colbrook-recovered-2026-09-11/manuscripts/IE-18.tex", "authors": [ "Matthew J. 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No external human peer review, source-author endorsement, unrecorded exact model identity or measured monetary cost is claimed." }, "scope": "Complete negative answer to the canonical IE-18 universal identity. Local Lean compilation, standard-axiom checks and two independent final proof reviews passed. Authoritative Linux Comparator accepted all three exports on 12 September 2026 at commit 7b8512e21c50adc8597dcdbed32f2aec13c3b43e; run 34703657954. The canonical original negative target is Lean verified. 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No external human peer review, formalization-author endorsement by the informal source author, unrecorded model identity, or measured monetary cost is claimed." }, "scope": "Complete negative answer to the original IE-19 yes/no conjecture. Local Lean 4.33.1 compilation and transitive standard-axiom checks have passed. Two independent final proof reviews have passed. GitHub Actions run 34703188616 completed real Linux Comparator statement matching and Lean default-kernel replay for all three exports on Ubuntu 24.04 at immutable proof revision 531941ca0062049ccf03d3f2df418ea805ac4036. Its original artifacts, tool receipt, input hashes and independent operational audit are retained in verification/linux-2026-09-12/. 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The maximum row-sum norm is an explicit finite NNReal supremum coerced to the reals, rather than Mathlib's default entrywise matrix norm. No invertibility or strict-dominance premise is added to the universal target. The concrete inverses are proved genuine. 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No Schiffer mathematical theorem is imported." } ], "classification": { "arxiv": [], "msc2020": [] }, "automation": { "methods": [ { "method": "agent", "models": [], "framework": "OpenAI Codex" } ], "strongest": "agent", "models": [], "spend": "", "notes": "AI-assisted formalization requested by George Stepaniants. The implementing agent /root/leancert_examples is neither independent final referee. Coordinator /root is independent final referee 2; /root/solved_statement_inventory is independent final referee 1. Candidate documentation is prepared by /root/formal_review_standards and is not an additional mathematical review. No external human review, official Tau Ceti endorsement, model identity, measured cost or priority is claimed." }, "scope": "Complete negative answer to the entire canonical direct induced-norm uniqueness conjecture. The exact p=4 matrices are two distinct global minimizers over all complex right inverses; generic denominator/supremum semantics cover every original domain. All eight exports passed two independent statement reviews, two independent final proof reviews and actual Linux sandboxed Comparator/default-kernel verification in run 34725525250 on 2026-09-12 at unchanged proof revision a40e5608f61dd4086708cb2e03901ffd01e4c0a9. 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The same source matrices at m=2,n=3,p=4 suffice to refute its all-dimension/all-p statement. Generic denominator/supremum semantics are proved on every original domain. The actual complex inverse, rank, Euclidean norm, real powers, all-input suprema and all-right-inverse global minimality are retained. The stronger source all-p formulas and full minimizer classifications, higher-dimensional families, smallest-dimension classification, endpoints and separate product objective are outside these exports.", "alignment": true, "original": false, "review": { "status": "agent-reviewed; linux-comparator-and-default-kernel-verified", "bucket": "agent-reviewed", "reviewers": [ "OpenAI Codex agent /root/solved_statement_inventory: independent statement referee 1 and final proof referee 1", "OpenAI Codex agent /root: independent statement referee 2 and final proof referee 2" ], "notes": "Both statement approvals preceded implementation and include the unchanged Comparator supplement. Both independent final referees freshly elaborated ten commands in private prefixes excluding old IE-23 objects and inspected actual proof terms. Each traversed 93 project declarations; referee 1 checked 40 dependencies and referee 2 checked 28 plus 12 kernel assertions. All sixteen internal/public axiom reports use exactly propext, Classical.choice and Quot.sound. LeanCert supplies explicit kernel trust auditing of a pure exact proof; no numerical interval certificate. Local macOS reviews reused clean matching dependency objects read-only. 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Kim" ], "type": "article", "relationship": "background", "endorsement": "not-contacted" } ], "related": [ { "id": "https://github.com/sgstepaniants/Forsythe/tree/8d1b0c0545a77b40245e84705aa7d273e6c81e62/lean-proof", "relationship": "other", "note": "Statement/proof separation, kernel trust and Comparator organization studied; no mathematical theorem imported." }, { "id": "https://github.com/jaumededios/Schiffer/tree/2938e277969c329caf154e48a3d8823f3635c7f1", "relationship": "other", "note": "Independent Challenge/Solution organization studied; no mathematical theorem imported." }, { "id": "https://github.com/leanprover-community/mathlib4/blob/0df444a360eaa60ab8c11dca51a86af692955474/Mathlib/Analysis/SpecialFunctions/Trigonometric/Chebyshev/Extremal.lean", "relationship": "builds-on", "note": "Reuses Yuval Filmus's formal exterior Chebyshev comparison at derivative order zero, retaining library attribution and license." } ], "classification": { "arxiv": [], "msc2020": [] }, "automation": { "methods": [ { "method": "agent", "models": [], "framework": "OpenAI Codex" } ], "strongest": "agent", "models": [], "spend": "", "notes": "AI-assisted implementation by /root/next_matrix_functions. Statement referees /root/next_elimination and /root approved the frozen boundary before bodies. The author is not an independent final referee. No external human peer review, official Tau Ceti endorsement, source-author endorsement beyond participation, or novelty claim is asserted. Independent final proof referees and the actual accepted source-bound mechanical evidence are retained in reviews/final and verification/linux-2026-09-15." }, "scope": "Complete original canonical uniform asymptotic-order result, all thirteen frozen declarations, every real gap 0 < δ < 1 and every natural multiplication budget, including m=0,1. Actual non-root Linux verification run35016473743 at b873ead85b3c71cdba017a915d5ef6e73b1711ae passed LeanCert kernel assertions, Comparator, default-kernel replay, standard transitive axioms and rejection controls. Both independent final referees accepted the complete mathematical source and actual logs. The exact optimum beyond the two small budgets and stronger same-budget error comparison are not claimed.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "NLA.MF02.uniform_error_is_maximum", "file": "NLA/MF02/Errors.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "NLA.MF02.program_degree_bound", "file": "NLA/MF02/Model.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "NLA.MF02.cubic_degree_and_cost", "file": "NLA/MF02/Model.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "NLA.MF02.degree_error_lower_bound", "file": "NLA/MF02/Chebyshev.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "NLA.MF02.optimized_cubic_interval", "file": "NLA/MF02/Cubic.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "NLA.MF02.optimized_cubic_ratio", "file": "NLA/MF02/Cubic.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "NLA.MF02.cubic_error_bounds", "file": "NLA/MF02/Bounds.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "NLA.MF02.cubic_error_strictly_decreases", "file": "NLA/MF02/Bounds.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "NLA.MF02.unrestricted_error_small_budgets", "file": "NLA/MF02/Bounds.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "NLA.MF02.stage_minimum_attained", "file": "NLA/MF02/Minimum.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "NLA.MF02.stage_minimum_small_budgets", "file": "NLA/MF02/Minimum.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "NLA.MF02.stage_minimum_bounds", "file": "NLA/MF02/Minimum.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "NLA.MF02.uniform_asymptotic_order", "file": "NLA/MF02/Minimum.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "No target restriction. The formal proof reverses the affine interval map so that only the positive exterior Chebyshev comparison is needed. It uses an arbitrary strict positive tolerance to include zero actual error without a separate polynomial identity theorem. Derivative and hyperbolic-function arguments are replaced by exact polynomial factors and recurrence identities. The coefficient infima are never assumed attained. Neither exact optimal stage counts beyond m=0,1 nor an optimal leading constant is claimed.", "alignment": true, "original": false, "review": { "status": "accepted by two independent ai-agent final mathematical and evidence referees on exact tested source", "bucket": "other", "reviewers": [ "OpenAI Codex agent /root/next_elimination: independent statement and proof-source referee", "OpenAI Codex agent /root/next_inequalities: independent proof-source referee", "OpenAI Codex agent /root: independent original statement referee and coordinator; implemented the two final Nat.cast_one rewrites, so not a final proof referee" ], "notes": "The numerical obligations, definitions and thirteen Challenge signatures were frozen after two independent approvals and actual Linux statement elaboration. Final reports in reviews/final reconcile the full source and actual canonical Comparator/default-kernel/axiom/control logs at b873ead85b3c71cdba017a915d5ef6e73b1711ae. Historical draft labels remain in frozen provenance. The coordinator made two Nat.cast_one proof repairs and is not counted as an independent final mathematical referee. No external human peer review or official Tau Ceti service is asserted." }, "canonical": { "repo": "ajt60gaibb/openproblemsinnla", "directory": "matrix-functions-and-stability/MF-02/lean" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "ajt60gaibb/OpenProblemsInNLA/matrix-functions-and-stability/MF-05/lean/formalization.yaml", "repo": "ajt60gaibb/OpenProblemsInNLA", "path": "matrix-functions-and-stability/MF-05/lean/formalization.yaml", "directory": "matrix-functions-and-stability/MF-05/lean", "origins": [ "search" ], "url": "https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/HEAD/matrix-functions-and-stability/MF-05/lean/formalization.yaml", "version": "v0.4", "missing": [], "name": "MF-05: local two-family Holder continuity of the joint spectral radius", "description": "Complete Lean verification of the original local two-family Holder assertion for arbitrary nonempty compact complex matrix families. All 14 exact contracts passed local compilation, two independent final source reviews, and actual canonical non-root Linux verification at the immutable proof revision.", "authors": [ "George Stepaniants" ], "maintainers": [ "George Stepaniants" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "MF-05 — Local Hölder continuity of the joint spectral radius", "id": "https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/5308b2cfa30826e1314d6a8830415c14b5f1d7a3/matrix-functions-and-stability/MF-05/README.md", "authors": [], "type": "web-post", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "Uniform polynomial product bounds and sharp Hölder continuity of the joint spectral radius", "id": "https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/5308b2cfa30826e1314d6a8830415c14b5f1d7a3/references/colbrook-jsr-growth-2026-09-11/manuscripts/uniform_growth_and_holder.tex", "authors": [ "Matthew J. 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None of these changes adds a final hypothesis or changes the requested conclusion.", "alignment": true, "original": false, "review": { "status": "independent full source approvals and nonauthor repair continuations retained; canonical runtime evidence accepted by coordinator and independent nonauthor.", "bucket": "other", "reviewers": [ "OpenAI Codex agent /root (independent original statement referee)", "OpenAI Codex agent /root/next_elimination (independent statement and nonimplementing complete mathematical source referee)", "OpenAI Codex agent /root/next_matrix_functions (nonimplementing complete mathematical source referee)", "OpenAI Codex agent /root/ie13_continuation (nonauthor source continuation and canonical runtime evidence audit)", "OpenAI Codex agent /root/mf22_publication_referee (nonauthor canonical source/package continuation)" ], "notes": "See reviews/INDEX.json for exact scopes and role disclosures. Source approval does not replace canonical runtime evidence. 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No external human peer review, source-author endorsement, unrecorded exact model identity or measured monetary cost is claimed." }, "scope": "Complete negative answer to the canonical original constant-one MI-07 conjecture. Local Lean compilation, all 23 kernel trust/standard-axiom checks and two independent final proof reviews passed. Actual Linux run 34709291624 verified the seven exports at immutable revision f55777156432043de4201747a3759e0c6485e568 and replayed their exported declaration closure through Lean's default kernel. An independent operational audit passed; the canonical status is Lean verified on 2026-09-12. 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Implementation agent /root/leancert_examples is not an independent referee. No human review, source-author endorsement, historical priority, unrecorded model identity or measured cost is claimed." }, "scope": "Complete negative resolution of the full canonical complex positive-definite singular-value log-majorization conjecture. All eight exports have two independent statement approvals and two independent final proof approvals. Actual Linux run 34720684925 at immutable 26f526cf8b6232af9528b30616076dc7a2c66ac6 passed sandboxed Comparator, Lean default-kernel replay, standard-three axioms and both real control suites; a separate independent operational audit passed. The canonical publication package records Lean verified. Final repository publication review remains separate.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "NLA.MI22.singular_values_semantics", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "NLA.MI22.spectral_power_semantics", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "NLA.MI22.euclidean_norm_bounds", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "NLA.MI22.witness_rational_data", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "NLA.MI22.witness_principal_powers", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "NLA.MI22.witness_operator_gap", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "NLA.MI22.counterexample", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "NLA.MI22.not_weightedLogMajorizationConjecture", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "No narrowing of the original target. The witness B differs from Colbrook’s printed integer B and is explicitly defined by an exact eighth-power congruence; source thresholds change to 10500 and 11000. All principal-power, positivity, actual norm and ordered singular-value bridges are proved rather than assumed. Every original complex input quantifier, all real t in [0,1], every proper prefix and full-product equality remain in the conjecture. A first-prefix violation at n=3,t=1/8 refutes the entire conjunction. The original printed witness and root-error lemma are not advertised as formalized.", "alignment": true, "original": false, "review": { "status": "agent-reviewed; actual linux comparator/default-kernel verification and independent operational audit passed", "bucket": "agent-reviewed", "reviewers": [ "OpenAI Codex agent /root/solved_statement_inventory: independent statement referee 1 and final proof referee 1", "OpenAI Codex agent /root: independent statement referee 2 and final proof referee 2", "OpenAI Codex agent /root/formal_review_standards: independent operational referee for actual Linux run 34720684925; not an implementer or final mathematical referee" ], "notes": "Both statement reports precede implementation; all mathematical statements, targets, source correspondence and pins remain unchanged. Both final referees checked the complete actual proof and fresh compiled dependencies, with 17 internal/public standard-three reports and 195 project declarations inspected. Referee 1 independently parsed and reconstructed 48 actual Lean rational-table entries and printed the auxiliary kernel Boolean checker proof. The retained LeanCert task is only 10500<11000 on [0,0], explicitly kernel checked and consumed in the full negation; other matrix and analytic obligations use exact Lean proofs. The only authorized build change appended the Solution lean_lib, with the exact original lakefile archived and defaultTargets Challenge retained; complete proof command i" }, "canonical": { "repo": "ajt60gaibb/openproblemsinnla", "directory": "matrix-inequalities-and-norms/MI-22/lean" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "ajt60gaibb/OpenProblemsInNLA/matrix-inequalities-and-norms/MI-23/lean/formalization.yaml", "repo": "ajt60gaibb/OpenProblemsInNLA", "path": "matrix-inequalities-and-norms/MI-23/lean/formalization.yaml", "directory": "matrix-inequalities-and-norms/MI-23/lean", "origins": [ "search" ], "url": "https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/HEAD/matrix-inequalities-and-norms/MI-23/lean/formalization.yaml", "version": "v0.4", "missing": [], "name": "MI-23: generalized geometric-mean log-majorization counterexample", "description": "Formalizes the complete negative answer to the corrected canonical conjecture using complex positive-definite matrices, genuine CFC real powers, full characteristic-root lists and exact log-majorization. Proved spectral and Euclidean norm bridges connect a small exact rational certificate to the original target.", "authors": [ "George Stepaniants" ], "maintainers": [ "George Stepaniants" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "MI-23 — Corrected generalized geometric-mean product conjecture", "id": "https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/8b3d1157b73cd526bb4d0c2fd2dd1666d41812dc/matrix-inequalities-and-norms/MI-23/README.md", "authors": [], "type": "web-post", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "MI-23: Resolution", "id": "https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/8b3d1157b73cd526bb4d0c2fd2dd1666d41812dc/matrix-inequalities-and-norms/MI-23/solution.tex", "authors": [ "Matthew J. 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No Forsythe mathematical result is assumed." }, { "id": "https://github.com/jaumededios/Schiffer/tree/2938e277969c329caf154e48a3d8823f3635c7f1", "relationship": "other", "note": "Campaign reference for separating numerical targets from proof implementation; no Schiffer mathematical result or code is imported." } ], "classification": { "arxiv": [], "msc2020": [] }, "automation": { "methods": [ { "method": "agent", "models": [], "framework": "OpenAI Codex" } ], "strongest": "agent", "models": [], "spend": "", "notes": "AI-assisted formalization requested by George Stepaniants. Implementation agent /root/formal_review_standards is not counted as an independent referee. No human peer review, source-author endorsement, unrecorded model identity or measured monetary cost is claimed." }, "scope": "Complete negative answer to canonical MI-23, with every original complex PD and real-parameter quantifier preserved. All eight exports passed two independent statement reviews, two independent final proof reviews and actual Linux sandboxed Comparator/default-kernel verification in run 34716784038 on 2026-09-12 at unchanged proof revision 17194f9060609acae429e14d3dc3c4562b84f2bd. The independent operational audit binds all 133 input files, original artifact digests and both actual isolation/rejection-control suites. Canonical status is Lean verified.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "NLA.MI23.positive_powers_and_means", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "NLA.MI23.product_eigenvalue_semantics", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "NLA.MI23.squared_product_largest", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "NLA.MI23.operator_norm_bounds", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "NLA.MI23.witness_data", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "NLA.MI23.witness_squared_gap", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "NLA.MI23.counterexample", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "NLA.MI23.not_generalizedGeometricMeanConjecture", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "No narrowing of the original target. Fin-indexed complex matrices represent every original dimension. Real-part mapping in the actual characteristic-root list is supported by a proved reverse complex-root multiset equality, positivity and dimension theorem for all positive-definite products. Log-majorization retains every proper nonempty prefix inequality and full product equality; only its first necessary inequality is needed to refute it. CFC real powers retain both parameter regions, negative and zero exponents and t endpoints. The r=s=1,p=2,t=1/8 rational witness negates the full universal assertion. Exact algebra replaces approximate fractional-power or eigenvalue calculations.", "alignment": true, "original": false, "review": { "status": "agent-reviewed; linux-comparator-and-default-kernel-verified", "bucket": "agent-reviewed", "reviewers": [ "OpenAI Codex agent /root/leancert_examples: independent statement referee 1", "OpenAI Codex agent /root: independent statement referee 2 and final proof referee 2", "OpenAI Codex agent /root/solved_statement_inventory: independent final proof referee 1" ], "notes": "Both statement approvals precede implementation and bind unchanged definitions, Challenge and numerical targets. Both final referees freshly re-elaborated the actual proof in separate object prefixes and inspected complete matrix/CFC/spectral semantics and certificate consumption. The eight intentional Challenge placeholders remain isolated. All 64 distinct implementation/public axiom reports contain exactly propext, Classical.choice and Quot.sound; all kernel trust checks passed. The single kernel-mode LeanCert point certificate remains in the complete negation. Local macOS checks used exact clean dependency caches. 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Assi" ], "type": "article", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "On a conjecture of Bhatia, Lim and Yamazaki", "id": "https://doi.org/10.1016/j.laa.2017.06.040", "authors": [ "Trung Hoa Dinh", "Raluca Dumitru", "Jose A. 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Finite real p>=1 and infinity are separate conjuncts. CFC powers and genuine Euclidean/singular-value norms give concrete semantics. Positive-cone norm lemmas are sufficient because positivity is proved; they are not added assumptions. The infinity Heron endpoint uses a proved finite-power contradiction. The half-power Furuta slice is the full slice actually consumed; a more general Furuta theorem is not claimed. Frozen source comments describing preproof draft status are historical and retained unchanged; current execution status is in STATE.json.", "alignment": true, "original": false, "review": { "status": "two independent statement and two independent final source approvals", "bucket": "other", "reviewers": [ "/root/mi24_full_referee1", "/root/mi13_full_referee2" ], "notes": "Nonauthor agent source reviews under pinned Tau Ceti rubrics. Historical request-changes reports and their resolving continuations are retained. Reviewer integrity replays did not invoke Lean or Comparator. 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Frozen mathematical statements and dependency pins are unchanged. The build-only addition of the Solution library is recorded in verification/build-config-revision.json. Ten deliberate Challenge placeholders establish no result and are excluded from active-proof counts. Two additional independent nonimplementing campaign referees read the entire 23-file source and independently reconciled the authenticated historical execution. Their reports and source bindings are retained under reviews/campaign. Root prepares integration metadata and the default-target amendment without changing any mathematical file. 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Exact inspected bytes bound in REUSE-AUDIT.json." }, { "id": "https://github.com/sgstepaniants/Forsythe/tree/8d1b0c0545a77b40245e84705aa7d273e6c81e62/lean-proof", "relationship": "other", "note": "Numerical-first and independent Challenge structure inspected; no theorem imported." }, { "id": "https://github.com/leanprover-community/mathlib4/tree/0df444a360eaa60ab8c11dca51a86af692955474", "relationship": "builds-on", "note": "Actual complex PSD congruence, decreasing Hermitian spectrum, Euclidean Rayleigh infimum, finite sums/products and arbitrary-real-power asymptotics." }, { "id": "https://github.com/alerad/leancert/tree/621a43d7cf21f87872392a01e874f2f1dbddc926", "relationship": "builds-on", "note": "One kernel-mode point certificate exp(1)<=3, consumed by the full tail-factor and final counterexample chain." } ], "classification": { "arxiv": [], "msc2020": [] }, "automation": { "methods": [ { "method": "agent", "models": [], "framework": "OpenAI Codex" } ], "strongest": "agent", "models": [], "spend": "", "notes": "Substantial OpenAI Codex assistance. 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Every real dimension, rank, nonnegative error parameter, continuous nonnegative nondecreasing concave half-line function and selected ordered orthogonal eigenbasis is retained. Generic f(0)=0 is not assumed. Actual CFC, PSD order, Euclidean operator norm and every-basis truncation semantics are proved. 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A real extension of the half-line function is used, with proved independence of values below zero. The source residual theorem is connected to the stronger canonical premise by the actual PSD trace identity; no premise is silently substituted. The source negative-discriminant step is replaced by an exact sum of squares, and its pseudoinverse harmonic presentation by an actual diagonal quadratic test with zero-coordinate support proved first. 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No Forsythe mathematical theorem or proof module is imported." }, { "id": "https://github.com/jaumededios/Schiffer/tree/2938e277969c329caf154e48a3d8823f3635c7f1", "relationship": "other", "note": "Pinned exact mathematical statement/proof separation example. No Schiffer theorem or proof implementation is assumed." } ], "classification": { "arxiv": [], "msc2020": [] }, "automation": { "methods": [ { "method": "agent", "models": [], "framework": "OpenAI Codex" } ], "strongest": "agent", "models": [], "spend": "", "notes": "AI-assisted formalization requested by George Stepaniants. Implementers: /root/formal_review_standards (statements, Algebra, Tangent, Count, Proof, Solution); /root/leancert_examples (Smooth, Critical); /root/mf16_final_referee (Differential); /root (SmoothTransport, Generic), with Generic scoped inspection/sealing by /root/mf16_final_referee. None counts as an independent final mathematical referee. 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The informal Jacobian smoothness explanation is implemented through genuine localized algebraic smoothness, retractions and a square-zero lifting obstruction. The whole-ideal tangent and arbitrary generic-open intersection are proved. Supporting derivative/Hessian statements are generalized where stated. Only the full original conjunction is refuted; the remaining individual formulas and corrected values are not settled. The source abcLocalChart comment uses complete local ring for the ordinary localization: no adic completion or completeness theorem is claimed. Hessian nondegeneracy is exported, but no separate scheme-theoretic multiplicity theorem is formalized.", "alignment": true, "original": false, "review": { "status": "agent-reviewed; mathematical-packaging-operational-reviews-accepted; actual-linux-kernel-and-comparator-pass", "bucket": "agent-reviewed", "reviewers": [ "OpenAI Codex agent /root: independent historical statement referee 1, subsequently a proof contributor and ineligible as an independent final mathematical referee", "OpenAI Codex agent /root/leancert_examples: independent historical statement referee 2, subsequently a proof contributor and ineligible as an independent final mathematical referee", "OpenAI Codex agent /root/ra20_final_referee1: independent final mathematical referee 1, subsequently candidate-document author; no additional mathematical or packaging approval", "OpenAI Codex agent /root/ra20_final_referee2: independent final mathematical referee 2", "OpenAI Codex agent /root/ra20_final_referee2: independent candidate-packaging and actual Linux operational reviewer; these mechanical audits add no extra mathematical approval" ], "notes": "The accepted statement boundary binds 68 project inputs and 16 original source/policy files; the complete proof freeze binds 521 and those same 16 original identities at base 5830ed4fb06da0659414a3deb2a40ad327aca052. 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All twelve exact el" }, "canonical": { "repo": "ajt60gaibb/openproblemsinnla", "directory": "randomized-and-low-rank-approximation/RA-20/lean" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "ajt60gaibb/OpenProblemsInNLA/tensor-computations/TR-15/lean/formalization.yaml", "repo": "ajt60gaibb/OpenProblemsInNLA", "path": "tensor-computations/TR-15/lean/formalization.yaml", "directory": "tensor-computations/TR-15/lean", "origins": [ "search" ], "url": "https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/HEAD/tensor-computations/TR-15/lean/formalization.yaml", "version": "v0.4", "missing": [], "name": "TR-15: odd-order Hankel H-eigenvalue inheritance counterexample", "description": "Formalizes the full negative answer to the canonical inheritance conjecture with actual ordered tensor contractions, real nonzero H-eigenpairs, universal lower positivity, an explicit upper negative pair and unconditional lower-pair existence from the intermediate value theorem.", "authors": [ "George Stepaniants" ], "maintainers": [ "George Stepaniants" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "TR-15 — Nonnegative H-eigenvalue inheritance from odd-order Hankel tensors", "id": "https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/8b3d1157b73cd526bb4d0c2fd2dd1666d41812dc/tensor-computations/TR-15/README.md", "authors": [], "type": "web-post", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "An exact counterexample to odd-order Hankel H-eigenvalue inheritance", "id": "https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/8b3d1157b73cd526bb4d0c2fd2dd1666d41812dc/references/colbrook-unclaimed-2026-09-11/manuscripts/TR-15.md", "authors": [ "Matthew J. 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All seven exports passed two independent statement reviews, two independent final proof reviews and actual Linux sandboxed Comparator/default-kernel verification in run 34716902324 on 2026-09-12 at unchanged proof revision 6a2d0868e8b7905dbd5c04d4beaae8cf43288e1f. The independent operational audit binds all 135 inputs, original artifact digests and both actual isolation/rejection-control suites. 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Current README/metadata, completion addenda and canonical status are post-run publication documentation; the exact earlier input snapshot, including then-pending documentation, is retained in verification/linux/source. No mathematical, build or Comparator input changed. No official Tau Ceti endorsement or human peer review is claimed." }, "canonical": { "repo": "ajt60gaibb/openproblemsinnla", "directory": "tensor-computations/TR-27/lean" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "akopjan/Grunbaum-counterexample/formalization.yaml", "repo": "akopjan/Grunbaum-counterexample", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/akopjan/Grunbaum-counterexample/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "A convex body variant of Soberon's counterexample to the Grünbaum conjecture.", "description": "A Lean formalization of a geometric strengthening/variant of Pablo Soberón's counterexample to the Grünbaum four-hyperplane mass-partition conjecture. The formalized result constructs a convex body K in R^4 with nonempty interior such that no four affine hyperplanes divide K into sixteen sign cells of equal four-dimensional volume. Equivalently, the obstructed mass can be chosen to be uniform Lebesgue volume on a convex body. The proof adapts Soberón's explicit cubic/quartic tensor obstruction and replaces the smooth positive density used in the source by a small Wulff-body perturbation of the", "authors": [ "Arseniy Akopyan" ], "maintainers": [ "Arseniy Akopyan" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Four hyperplanes do not always equipartition a mass in R^4", "id": "arXiv:2608.23312v2", "authors": [ "Pablo Soberón" ], "type": "paper", "relationship": "adapts", "endorsement": "not-contacted" } ], "related": [], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "52C35", "52A37" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "OpenAI GPT-5.6 Sol" ], "framework": "ChatGPT" }, { "method": "manual", "models": [], "framework": "n/a" } ], "strongest": "agent", "models": [ "OpenAI GPT-5.6 Sol" ], "spend": "not tracked", "notes": "Prepared by Arseniy Akopyan with substantial assistance from ChatGPT. Automated assistance is disclosed as tooling rather than bibliographic authorship. The mathematical source and tensor construction are credited to Pablo Soberón." }, "scope": "The project formalizes the convex-body consequence using the Wulff body supportBody tau eps, the sixteen literal sign cells, the Walsh transform, the explicit Soberón tensor obstruction, localization near centered orthonormal frames, and quadratic stability estimates. The main statement removes the internal unit-normal parametrization: zero normals give an empty cell, while nonzero normals are normalized by positive rescaling. 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The proof uses the same explicit tensor obstruction but a Wulff-body/radial comparison in place of the source's Gaussian-density perturbation.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "The statement and provenance were manually reviewed during preparation. 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The Lean formalization proves the constant-Jacobian identity by `ring`, and proves F_e is not injective by exhibiting two distinct points, p = (0, 0, -3/4) and q = (1, -3/2, 39/2), with F_e p = F_e q for all e.", "authors": [], "maintainers": [], "license": "MIT", "role": "", "substantive": "", "sources": [ { "title": "Jacobian conjecture", "id": "", "authors": [], "type": "", "relationship": "independently-proves", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.AC" ], "msc2020": [ "14R15" ] }, "automation": { "methods": [ { "method": "manual", "models": [], "framework": "" } ], "strongest": "manual", "models": [], "spend": "", "notes": "" }, "scope": "", "sorry_count": null, "sorry_in_definitions": null, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "", "alignment": false, "original": true, "review": { "status": "pending", "bucket": "other", "reviewers": [], "notes": "" }, "canonical": { "repo": "alacosta2025/jacobian-conjecture-counterexample", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "anastasiosfragkosmath/quadratic-carleson-palomar-submission/formalization.yaml", "repo": "anastasiosfragkosmath/quadratic-carleson-palomar-submission", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/anastasiosfragkosmath/quadratic-carleson-palomar-submission/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Endpoint estimates for Stein's purely quadratic Carleson operator", "description": "A Lean 4 formalization of four principal endpoint assertions from Fragkos, Krause, and Lacey's Endpoint Estimates for Stein's Purely Quadratic Carleson Operator: two failures of modular estimates below t log₂ t, a full-modulation L log L endpoint estimate, and a lacunary L (log₂ L)² log₄ L endpoint estimate.", "authors": [ "Anastasios Fragkos" ], "maintainers": [ "Anastasios Fragkos" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Endpoint Estimates for Stein's Purely Quadratic Carleson Operator", "id": "https://arxiv.org/abs/2609.04101v1", "authors": [ "Anastasios Fragkos", "Ben Krause", "Michael Lacey" ], "type": "paper", "relationship": "formalizes", "endorsement": "" } ], "related": [ { "id": "https://github.com/anastasiosfragkosmath/lean-verification-quadratic-carleson/tree/3611a2ba0b28f135680156290f63ec72a3acd77e", "relationship": "other", "note": "This repository is a Palomar-prepared copy of the substantive Lean development at commit 3611a2ba0b28f135680156290f63ec72a3acd77e. 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Neither dominates the other over the whole parameter ran", "authors": [ "Andrew Fiori" ], "maintainers": [ "Andrew Fiori" ], "license": "CC-BY-4.0", "role": "", "substantive": "", "sources": [ { "title": "Zero Density Theorems for Short Intervals", "id": "arXiv:2609.22624", "authors": [ "Andrew Fiori" ], "type": "paper", "relationship": "formalizes", "endorsement": "participated" }, { "title": "New improved explicit estimates for zeta on the half line", "id": "", "authors": [ "Michael Revers" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Explicit bounds for zeta in the critical strip and a zero-free region", "id": "", "authors": [ "Andrew Yang" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Explicit bounds for zeta and a new zero-free region", "id": "", "authors": [ "Chiara Bellotti" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Explicit estimates for the logarithmic derivative of zeta", "id": "", "authors": [ "Nicol Leong" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "An improved explicit estimate for zeta on the half line", "id": "", "authors": [ "Ghaith Hiary", "Nicol Patel", "Andrew Yang" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "An explicit sub-Weyl bound for zeta", "id": "", "authors": [ "Nicol Patel", "Andrew Yang" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "On the accuracy of asymptotic approximations to the log-Gamma and Riemann-Siegel theta functions", "id": "", "authors": [ "Richard P. 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Every cited result was read in the original paper before being assumed; every numeric claim was isolated as its own statement and certified by an independent script in Code/, usually twice, once with arb ball arithmetic and once with mpmath; and every declaration carries a docstring recording its proof sketch, its correspondence with the source, and any divergence found." }, "scope": "Formalized and proved: both main theorems (Jensen and Littlewood mechanisms), both corollaries, the Littlewood short-interval theorem, the argument-integral machinery, the appendix of explicit zeta bounds, the Laurent expansion of zeta'/zeta, the reflection step, and the headline Corollary 1 in three forms. SEVEN statements are submitted and compared, not one. Both main theorems (zero_density_jensen for Theorem 2, zero_density_littlewood for Theorem 3), both corollaries (edge_proportion_jensen for Corollary 4, edge_proportion_littlewood for Corollary 5), and all three forms of Corollary 1 (pos", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "ZeroDensityShortIntervals.zero_density_jensen", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "ZeroDensityShortIntervals.zero_density_littlewood", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "ZeroDensityShortIntervals.edge_proportion_jensen", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "ZeroDensityShortIntervals.edge_proportion_littlewood", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "ZeroDensityShortIntervals.positive_proportion", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "ZeroDensityShortIntervals.positive_proportion_sixteenth", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "ZeroDensityShortIntervals.positive_proportion_in_middle_band", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [ { "statement": "Explicit sixth-power and subconvexity bounds for zeta on the critical line", "source": "New improved explicit estimates for zeta on the half line" }, { "statement": "Explicit bounds for zeta inside the critical strip", "source": "Explicit bounds for zeta in the critical strip and a zero-free region" }, { "statement": "Richert's bound in explicit form", "source": "Explicit bounds for zeta and a new zero-free region" }, { "statement": "A lower bound for |zeta(1+it)|", "source": "Explicit estimates for the logarithmic derivative of zeta" }, { "statement": "Stirling's expansion with an explicit remainder bound", "source": "On the accuracy of asymptotic approximations to the log-Gamma and Riemann-Siegel theta functions" }, { "statement": "A multi-function Phragmen-Lindelof interpolation principle", "source": "A note on the Phragmen-Lindelof theorem" }, { "statement": "The Riemann hypothesis verified below height 3 x 10^12", "source": "The Riemann hypothesis is true up to 3 x 10^12" }, { "statement": "The Riemann-von Mangoldt formula with an explicit error term", "source": "Counting zeros of Artin L-functions" } ] } ], "comparator": true, "literature_dependencies": 8, "divergences": "The k = 0 anchor of the piecewise-linear majorant has two admissible values in the source. 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No code from it is imported or ported here; it neither formalizes this Abel-function result nor supplies the complement theorem needed for it." } ], "classification": { "arxiv": [ "math.LO" ], "msc2020": [ "03C64" ] }, "automation": { "methods": [ { "method": "manual", "models": [], "framework": "n/a" }, { "method": "agent", "models": [ "ChatGPT 6 Astra" ], "framework": "ChatGPT" }, { "method": "agent", "models": [ "gpt-6-astra", "gpt-5.6-sol" ], "framework": "Codex" } ], "strongest": "agent", "models": [ "ChatGPT 6 Astra", "gpt-5.6-sol", "gpt-6-astra" ], "spend": "not tracked", "notes": "Artem Chernikov is the human author and responsible maintainer. Automated systems made material contributions to the research, formalization, and audits; they are credited here rather than in the human authorship fields." }, "scope": "The project proves both the universal conditional theorem AbelFormalization.mainTheorem and the unconditional existence theorem AbelFormalization.exists_abel_ominimal_expansion. For every total real function A whose restriction to (0,infinity) is analytic with positive derivative, is normalized by A(1)=0, and satisfies A(exp(x)-1)=A(x)+1 for x>0, the expansion of the real ordered field by C0(x)=A(1+x^2) is o-minimal. Definability is first-order definability with arbitrary real parameters. 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The invariant-density construction developed for this formalization normalizes a positive analytic invariant density by its positive primitive increment, rather than proving the precise Szekeres density asymptotic; it therefore makes no uniqueness or distinguished-solution claim. Total values of A outside (0,infinity) are arbitrary and play no role.", "alignment": true, "original": false, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [ "Codex gpt-6-astra agents", "Codex gpt-5.6-sol agents" ], "notes": "Agent-led statement-correspondence, dependency, reuse, compilation, and axiom audits are recorded in the repository. No completed independent human-expert review has been reported. 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The proof follows the classical decisive-coalition argument: field expansion (a coalition decisive for one pair is decisive for every pair) and group contraction (a decisive coalition with at least two voters contains a smaller decisive coalition), iterated down to a singleton dictator. This is a reusable formalization artifact, not a claim of new mathematics.", "authors": [ "Arthur Freitas Ramos" ], "maintainers": [ "Arthur Freitas Ramos" ], "license": "BSD-3-Clause", "role": "", "substantive": "", "sources": [ { "title": "A Difficulty in the Concept of Social Welfare", "id": "https://doi.org/10.1086/256963", "authors": [ "Kenneth J. Arrow" ], "type": "article", "relationship": "formalizes", "endorsement": "not-contacted" } ], "related": [ { "id": "https://palomar-registry.org/entry?id=PALOMAR-2026-09-01-000011&version=1", "relationship": "independent", "note": "Joris Roos's registered Palomar entry \"Arrow's theorem via Fourier analysis\" (PALOMAR-2026-09-01-000011, v1) proves the theorem for three-candidate elections via the Fourier-analytic route (Kalai's argument, following O'Donnell's \"Analysis of Boolean Functions\" Sec. 2.5), with a single voting rule applied to every pairwise election. This development is independent of it: it proves the theorem in f" } ], "classification": { "arxiv": [ "math.LO", "cs.GT" ], "msc2020": [ "91B14" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "gpt-6-luna (Codex CLI)" ], "framework": "Lean 4 and Mathlib" } ], "strongest": "agent", "models": [ "gpt-6-luna (Codex CLI)" ], "spend": "subscription-based", "notes": "AI assistance was used for proof engineering. The final definitions, statements, and proofs are checked by Lean's kernel." }, "scope": "The library formalizes ballots as injective rankings, profiles, social welfare functions, the unanimity and IIA axioms, decisive coalitions, and dictatorship; it proves the field expansion lemma, the group contraction lemma, and the main impossibility theorem. Challenge.lean repeats the compared statements with deliberate proof holes; those statement holes are excluded from the implementation sorry_count.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "Finite electorate and finite alternative set with at least three alternatives; ballots are strict linear orders represented as injective rankings into the naturals, which represent every strict linear order on a finite type.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "none" ], "notes": "The implementation library and Solution are proof-hole-free under the pinned local build. Challenge contains deliberate Palomar statement holes, excluded from the implementation sorry_count. Hosted Comparator verification, Palomar editorial review, and registration are separate states and are not inferred from local compilation." }, "canonical": { "repo": "arthur742ramos/arrow-impossibility-lean", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "Arthur742Ramos/blackwell-approachability-lean/formalization.yaml", "repo": "Arthur742Ramos/blackwell-approachability-lean", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/Arthur742Ramos/blackwell-approachability-lean/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Irreducible improper phi-regret in Lean", "description": "A Mathlib-only formalization of two explicit three-action improper phi-regret families in Section 4.4.1 of Dann, Mansour, Mohri, Schneider, and Sivan (COLT 2025). They are finite counterexample families for the paper's rate-preserving linear-reduction program. It proves valid-improper certificates, the affine-hyperplane invariant-transport core of Lemma 4, the extreme-point antipodal-displacement core of Lemma 5, a direct finite one-way affine action/loss-transfer-to-canonical bridge, a source-style two-direction affine linear-equivalence-to-canonical bridge in an explicit bijective-coordinate", "authors": [ "Arthur Freitas Ramos" ], "maintainers": [ "Arthur Freitas Ramos" ], "license": "BSD-3-Clause", "role": "", "substantive": "", "sources": [ { "title": "Rate-Preserving Reductions for Blackwell Approachability", "id": "https://proceedings.mlr.press/v291/dann25a.html", "authors": [ "Christoph Dann", "Yishay Mansour", "Mehryar Mohri", "Jon Schneider", "Balasubramanian Sivan" ], "type": "paper", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "An analog of the minimax theorem for vector payoffs", "id": "https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-6/issue-1/An-analog-of-the-minimax-theorem-for-vector-payoffs/pjm/1103044269.full", "authors": [ "David Blackwell" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" } ], "related": [], "classification": { "arxiv": [ "cs.GT", "cs.LG" ], "msc2020": [ "91A26", "68T05" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "GPT-5 Codex" ], "framework": "Lean 4 and Mathlib" } ], "strongest": "agent", "models": [ "GPT-5 Codex" ], "spend": "", "notes": "AI assistance was used for proof engineering. The final definitions, statements, and proofs are checked by Lean." }, "scope": "Thirty selected statements formalize source-backed finite-dimensional results. Both cited families receive fixed-point/impropriety certificates. A generic theorem proves that an exact one-way affine action/loss transfer with invertible linear parts, translated action and loss coordinates, properness on the transformed action image, and equality on Delta_3 times the unit loss cube yields an invertible canonical properizer; it cancels the loss translation at zero and does not assume Equation (16). For the skew simplex family, affine-hyperplane transport yields a nonzero common invariant from any", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The skew family follows the cited Section 4.4.1 construction after identifying q=(a,b,c) in Delta_3 and using phi_q=Id-M_q. The second family uses the published matrices A and B with coefficient square [-1,1]^2; its fixed point is proved from an explicit homogeneous kernel vector and square-completion positivity certificates, including the zero-parameter case. The generic selected lemmas formalize affine-normal transport (Lemma 4), the extreme-antipodal obstruction (Lemma 5), and finite homogeneous and affine one-way action/loss transfers with explicit inverse matrices, plus the source's two directional pairing identities in the explicit bijective-coordinate finite-image regime. The affine transfer theorem reconstructs the canonical correction from the zero loss and cube basis losses, rather than assuming Equation (16). A separate full-rank target-loss basis lemma derives M_psi = S M_phi from the source's Equation (15), and another derives Equation (17). The remaining selected layers prove the action-vertex half of Equation (18) and its exact nine-/twelve-constraint specialization for the two source families. This package intentionally does not formalize the source's general affine", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "No independent human or Palomar editorial approval is claimed. The thirty Challenge holes occur only in the independent statement surface and are excluded from status.sorry_count; the implementation and Solution contain no proof placeholders. Hosted Comparator verification and editorial review remain separate states." }, "canonical": { "repo": "arthur742ramos/blackwell-approachability-lean", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-10-000003", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-10-000003", "commit": "8a53170bbf91309d687d4f0131b2147457a3dbc3", "trust": "high", "theorems": 30, "date": "2026-09-10" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "Arthur742Ramos/blackwell-approachability-lean/rate-preserving-reduction/formalization.yaml", "repo": "Arthur742Ramos/blackwell-approachability-lean", "path": "rate-preserving-reduction/formalization.yaml", "directory": "rate-preserving-reduction", "origins": [ "palomar", "search" ], "url": "https://github.com/Arthur742Ramos/blackwell-approachability-lean/blob/HEAD/rate-preserving-reduction/formalization.yaml", "version": "v0.4", "missing": [], "name": "Finite tight Blackwell approachability-to-improper phi-regret reduction in Lean", "description": "A Mathlib-only nonempty finite-simplex, algorithmic specialization of Theorem 4 in Dann, Mansour, Mohri, Schneider, and Sivan (COLT 2025). It realizes the tensor action space as a joint simplex, defines the source improper comparator by adding a mixed-constraint tensor the canonical action marginal, proves that comparator escapes the action set, and proves exact finite-horizon equality between approachability loss and improper phi-regret under explicit causal strategy translations in both directions.", "authors": [ "Arthur Freitas Ramos", "Ruy Jose Guerra Barretto de Queiroz", "David Barros Hulak" ], "maintainers": [ "Arthur Freitas Ramos" ], "license": "BSD-3-Clause", "role": "", "substantive": "", "sources": [ { "title": "Rate-Preserving Reductions for Blackwell Approachability", "id": "https://proceedings.mlr.press/v291/dann25a.html", "authors": [ "Christoph Dann", "Yishay Mansour", "Mehryar Mohri", "Jon Schneider", "Balasubramanian Sivan" ], "type": "paper", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "An analog of the minimax theorem for vector payoffs", "id": "https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-6/issue-1/An-analog-of-the-minimax-theorem-for-vector-payoffs/pjm/1103044269.full", "authors": [ "David Blackwell" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" } ], "related": [ { "id": "PALOMAR-2026-09-10-000003", "relationship": "builds-on", "note": "A separate approved Palomar entry at the repository root formalizes finite irreducibility obstructions. This nested project has an independent Challenge, Solution, Comparator configuration, and intended blank existing-id field; it is not a version update of that entry." } ], "classification": { "arxiv": [ "cs.GT", "cs.LG" ], "msc2020": [ "91A26", "68T05" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "GPT-5 Codex" ], "framework": "Lean 4 and Mathlib" } ], "strongest": "agent", "models": [ "GPT-5 Codex" ], "spend": "", "notes": "AI assistance was used for proof engineering. The final definitions, statements, and proofs are checked by Lean." }, "scope": "Nine selected statements formalize the finite-simplex reduction. The action and constraint index types are explicitly nonempty and their spaces are probability simplexes; joint tensor actions are nonnegative tables of mass one; the action marginal and anchored rank-one lift are shown to preserve the required action spaces. The shift comparator adds a rank-one table with the same marginal, so its total mass is two and it is improper. For every coordinate loss, the source pairing difference B(x,l)-B(shift(w,x),l) equals the original mixed constraint score. The equality is summed over arbitrary f", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The cited theorem writes X = U tensor P. This project takes finite U=Delta_m and P=Delta_n and realizes their convex tensor hull as the joint simplex Delta_(m x n). The action marginal is consequently canonical. The source comparator phi_w(u' tensor p)=(u'+w) tensor p is implemented as x + w tensor marginal(x), which agrees on elementary tensors and is well-defined on every joint action. The source loss map M_B is represented by reducedLoss. The development proves the exact loss identities for deterministic history-based strategies and finite loss lists. It does not formalize the general continuous setting, randomized algorithms, or the source's rate-definition limits.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "No independent human or Palomar editorial approval is claimed. The nine Challenge holes occur only in the independent statement surface and are excluded from status.sorry_count; implementation, Solution, and examples have no proof placeholders. The prior automated review's empty-index finding is addressed by explicit nonemptiness assumptions on every central objective, improperness, trajectory-reduction, and strategy-reduction declaration. Hosted Comparator verification and editorial review remain separate states." }, "canonical": { "repo": "arthur742ramos/blackwell-approachability-lean", "directory": "rate-preserving-reduction" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-20-000001", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-20-000001", "commit": "42b9d7c77e77fc158d44cb31ef320798c3c73492", "trust": "high", "theorems": 12, "date": "2026-09-20" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "Arthur742Ramos/chess-lean/formalization.yaml", "repo": "Arthur742Ramos/chess-lean", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/Arthur742Ramos/chess-lean/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Chess in Lean 4", "description": "A standalone Lean 4 reimplementation of orthodox-chess move rules over generalized total finite position records. It formalizes a finite board model, geometric attacks, special moves, check and legal-move semantics, an exhaustive verified move generator, histories and draw policies, FEN/UCI/SAN interfaces, perft regressions, symmetry, and a recursively checked mate-in-two certificate. The selected reachability-invariant theorem proves that legal transitions from the initial position preserve the structural safety conditions; standard orthodox positions are checked concrete instances of this br", "authors": [ "Arthur Freitas Ramos", "David Barros Hulak", "Ruy J. G. B. de Queiroz" ], "maintainers": [ "Arthur Freitas Ramos", "David Barros Hulak", "Ruy J. G. B. de Queiroz" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "FIDE Laws of Chess", "id": "https://handbook.fide.com/chapter/e012023", "authors": [ "FIDE" ], "type": "other", "relationship": "adapts", "endorsement": "" } ], "related": [ { "id": "https://github.com/Arthur742Ramos/isabelle-afp-monorepo", "relationship": "adapts", "note": "Prior Isabelle/HOL formalization adapted by this Lean reimplementation. It originated in the private repository Arthur742Ramos/isabelle-afp-monorepo at immutable revision a81eecf7b7a77064380bdf1f8915d73ee9955fa3 and the commit-relative project path projects/chess-isabelle. A complete public snapshot is committed at the location above, so review does not depend on access to the upstream repository." } ], "classification": { "arxiv": [ "cs.LO" ], "msc2020": [ "03B35" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "GPT-5" ], "framework": "Lean 4 and Mathlib" }, { "method": "manual", "models": [], "framework": "Lean 4 and Mathlib" } ], "strongest": "agent", "models": [ "GPT-5" ], "spend": "subscription-based", "notes": "The Isabelle source entry was inspected before the Lean port. Definitions were implemented in small modules, each touched module was compiled, and concrete rule and perft examples were checked by Lean's native evaluator." }, "scope": "The proved Lean scope covers generalized total finite position records and transitions, including potentially malformed or unreachable records; attacks, check, pseudo-legal and legal moves, all standard special moves, exhaustive finite move generation, histories and reachability predicates, repetition and FIDE clock predicates, bounded forced mate, finite mate certificates, typed and strict textual FEN, UCI, canonical SAN, perft, rank-reflection symmetry, concrete standard perft regressions, and the Isabelle mate-in-two witness. The capstone theorem advertises `Chess.reachable_positionInvarian", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The Lean implementation preserves the modeled orthodox-rule state and rule scope but reorganizes the thirty Isabelle theories into cohesive Lean modules. It uses Boolean evaluators for executable checking and proposition wrappers for exact logical statements. The selected reachability theorem is stated over generalized Position records and concludes a structural invariant only under reachability; it does not claim that every represented record is an orthodox legal position or that every invariant record is reachable. The current port also proves the finite rule-level certificate soundness/completeness result and separately exposes the history-aware checker; it does not claim one-for-one Lean counterparts for every Isabelle invariant-preservation or symmetry transport lemma.", "alignment": true, "original": false, "review": { "status": "self-assessed with clean-build and independent-comparator checks", "bucket": "self-assessed", "reviewers": [ "Arthur Freitas Ramos", "David Barros Hulak", "Ruy J. G. B. de Queiroz" ], "notes": "Review covers source-to-module alignment, executable special-move cases, standard perft values, parser/printer boundaries, the legal-transition invariant proof, certificate soundness and constructive completeness, metadata, pinned manifests, and Palomar structural checks. The formalization should still receive Palomar's normal editorial and independent mathematical review after intake." }, "canonical": { "repo": "arthur742ramos/chess-lean", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "Arthur742Ramos/classical-svk-lean/formalization.yaml", "repo": "Arthur742Ramos/classical-svk-lean", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/Arthur742Ramos/classical-svk-lean/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Classical Seifert–van Kampen theorem for the fundamental groupoid", "description": "Proves that for every topological space X and open cover X = U ∪ V, the inclusion-induced square of Mathlib's ordinary continuous-path fundamental groupoids for U ∩ V, U, V, and X is a pushout in Cat. The statement retains all points as objects and assumes no basepoint, connectedness, or separation conditions. The proof uses continuous paths and endpoint-preserving homotopies, specializes an auxiliary preorder to the indiscrete relation, and proves a natural equivalence with Mathlib's ordinary fundamental groupoid. It constructs the categorical descent functor by path subdivision and proves ho", "authors": [ "Arthur Freitas Ramos" ], "maintainers": [ "Arthur Freitas Ramos" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Groupoids and Van Kampen's Theorem", "id": "10.1112/plms/s3-17.3.385", "authors": [ "Ronald Brown" ], "type": "article", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "The Directed Van Kampen Theorem in Lean", "id": "10.4230/LIPIcs.ITP.2024.8", "authors": [ "Henning Basold", "Peter Bruin", "Dominique Lawson" ], "type": "conference paper and formalization", "relationship": "adapts", "endorsement": "not-contacted" } ], "related": [ { "id": "https://github.com/Dominique-Lawson/Directed-Topology-Lean-4/tree/009529606c66d37ef93b4b81b8587f71ce4d2c56", "relationship": "builds-on", "note": "The source project is Dominique-Lawson/Directed-Topology-Lean-4 at full commit 009529606c66d37ef93b4b81b8587f71ce4d2c56. Its Lean4/ module tree is vendored and compatibility-ported into this repository; the exact upstream path Lean4/directed_van_kampen.lean contains the DirectedVanKampen.PushoutFunctor path-subdivision and homotopy-grid lemmas extracted as Lean4/path_descent_helpers.lean and reuse" }, { "id": "https://github.com/leanprover-community/mathlib4/tree/8f9d9cff6bd728b17a24e163c9402775d9e6a365", "relationship": "builds-on", "note": "Exact Mathlib commit pin for the ordinary path fundamental groupoid, induced functors, topological subspaces, and categorical pushout API. Challenge imports only Mathlib." }, { "id": "https://github.com/Arthur742Ramos/ComputationalPathsLean/tree/257c659b7973aeda900d86a5da73b208712c7523", "relationship": "independent", "note": "This immutable snapshot contains computational-path based SVK equivalences and presentation results. No definitions, source files, or proof terms are reused. The result selected here is instead the pushout for actual continuous-path groupoids of arbitrary open subsets, with all points as objects and without a basepoint or connectedness hypotheses." }, { "id": "https://isa-afp.org/entries/Seifert-Van-Kampen.html", "relationship": "independent", "note": "Related Isabelle/HOL formalization of a based fundamental-group theorem under path-connectedness hypotheses. No Isabelle source or proof is reused; the present selected result is an all-object groupoid pushout." } ], "classification": { "arxiv": [ "math.AT", "math.CT", "math.LO" ], "msc2020": [ "55Q05", "18B40", "03B35" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "gpt-6-astra" ], "framework": "Codex" }, { "method": "manual", "models": [], "framework": "Lean, Git, and mathematical review" } ], "strongest": "agent", "models": [ "gpt-6-astra" ], "spend": "subscription-based", "notes": "AI assistance is disclosed without treating the agent as an author. Mechanical checks establish proof/build and package properties; they do not establish novelty, independent mathematical review, or editorial acceptance." }, "scope": "An arbitrary topological space X, open subsets U and V with U ∪ V = X, and the four continuous-path fundamental groupoids of U ∩ V, U, V, and X. No basepoint, path-connectedness, connectedness of the overlap, separation axiom, or manifold structure is assumed. The conclusion is the categorical pushout property for the maps induced by the four subspace inclusions.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "ClassicalSVK.seifert_van_kampen_groupoid", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The statement is the full fundamental-groupoid case of the two-open-cover theorem: its groupoids contain every point of each space as an object, unlike a based fundamental-group statement. Mathlib represents paths on the unit interval and endpoint-preserving homotopy; the proof identifies these path classes with the directed path classes for the indiscrete preorder. Its interval-subdivision and homotopy-grid helpers are adapted from the directed-topology formalization by Basold, Bruin, and Lawson; the selected proof constructs the descent universal property and does not invoke their packaged directed theorem. It does not claim a new mathematical theorem, reproduce the earlier computational-path proof, or claim independent authorship of the helper infrastructure.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "The proof and statement boundary are checked by Lean, the package scripts, and the pinned Palomar workflows. No independent human line-by-line review, Palomar editorial acceptance, intake, or registration is claimed." }, "canonical": { "repo": "arthur742ramos/classical-svk-lean", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-25-000004", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-25-000004", "commit": "874680e16db88b4a41daadf56eafbac79fd2f748", "trust": "high", "theorems": 1, "date": "2026-09-25" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "Arthur742Ramos/ComputationalPathsLean/formalization.yaml", "repo": "Arthur742Ramos/ComputationalPathsLean", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/Arthur742Ramos/ComputationalPathsLean/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Proof-Relevant Associativity Rewriting for Free Magmas", "description": "A compact Lean development of directed associativity rewriting on Mathlib's free magmas. It proves strict termination, constructs irreducible canonical right-associated normal forms, derives global confluence, and establishes soundness and completeness of the induced symmetric rewrite equality against both Mathlib's free semigroup semantics and its standard associativity quotient. Two explicit routes around Mac Lane's pentagon have the same endpoints but retain distinct proof-relevant traces with two and three primitive rotations.", "authors": [ "Arthur Freitas Ramos", "Ruy J. G. B. de Queiroz", "Anjolina Grisi de Oliveira", "Tiago M. L. de Veras" ], "maintainers": [ "Arthur Freitas Ramos" ], "license": "MIT", "role": "", "substantive": "", "sources": [ { "title": "Computational Paths: A Calculus of Equality", "id": "https://github.com/Arthur742Ramos/ComputationalPathsLean/blob/2a2baa1f31c68f0e696021db91f8381dd2854652/paper/main.tex", "authors": [ "Arthur Freitas Ramos", "Ruy J. G. B. de Queiroz", "Anjolina Grisi de Oliveira" ], "type": "paper", "relationship": "adapts", "endorsement": "" }, { "title": "Natural associativity and commutativity", "id": "https://hdl.handle.net/1911/62865", "authors": [ "Saunders Mac Lane" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Confluent Reductions: Abstract Properties and Applications to Term Rewriting Systems", "id": "https://doi.org/10.1145/322217.322230", "authors": [ "Gerard Huet" ], "type": "paper", "relationship": "background", "endorsement": "" } ], "related": [ { "id": "https://github.com/leanprover-community/mathlib4/blob/db584cd6d46c92f209a44c0f1c829460d327499d/Mathlib/Algebra/Free.lean", "relationship": "builds-on", "note": "Mathlib supplies the standard free objects and quotient semantics. The present development adds a proof-relevant directed rewrite system, constructive normalization traces, confluence, and completeness." } ], "classification": { "arxiv": [ "math.LO", "math.CT", "cs.LO" ], "msc2020": [ "03B35", "18M05", "68Q42", "68V20" ] }, "automation": { "methods": [ { "method": "manual", "models": [], "framework": "" }, { "method": "agent", "models": [ "Codex" ], "framework": "Lean 4.33.0, Lake, and Mathlib v4.33.0" } ], "strongest": "agent", "models": [ "Codex" ], "spend": "not tracked", "notes": "Lean and Mathlib revisions are pinned by lean-toolchain and lake-manifest.json. Challenge and Solution are compiled independently. Challenge imports only Mathlib.Algebra.Free and exposes exactly eight theorem holes; Solution proves all eight declarations." }, "scope": "The selected declarations cover the associativity-only rewrite calculus on free magma trees: strict termination; canonical normalization and irreducibility; global confluence; soundness and completeness against FreeSemigroup equality; equivalence with Mathlib's associativity quotient; and concrete distinct routes around Mac Lane's pentagon. The result does not include units, inverses, general monoidal categories, higher associahedra, or a complete omega-groupoid. 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Palomar editorial review, registry acceptance, and mathematical endorsement are not claimed." }, "canonical": { "repo": "arthur742ramos/computationalpathslean", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "Arthur742Ramos/correlated-equilibrium-lean/formalization.yaml", "repo": "Arthur742Ramos/correlated-equilibrium-lean", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/Arthur742Ramos/correlated-equilibrium-lean/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Finite correlated equilibria and regret certificates", "description": "For finite games with player-specific action types, a valid independent product distribution is a correlated equilibrium exactly when its mixed profile is Nash. The correlated-equilibrium set is nonempty, compact, and convex when every player's action type is nonempty. For every positive finite horizon, cumulative external-regret and swap-regret bounds give corresponding approximate coarse-correlated and correlated equilibria of the empirical distribution. The development also proves equivalence with conditional obedience inequalities, arbitrary mixed-deviation sufficiency, and extension-by-ze", "authors": [ "Arthur Freitas Ramos", "David Barros Hulak", "Ruy Jose Guerra Barretto de Queiroz" ], "maintainers": [ "Arthur Freitas Ramos" ], "license": "BSD-3-Clause", "role": "", "substantive": "", "sources": [ { "title": "Subjectivity and correlation in randomized strategies", "id": "https://doi.org/10.1016/0304-4068(74)90037-8", "authors": [ "Robert J. 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The selected certificate compression and empirical certificate theorems reduce swap-rule bounds to per-recommendation inequalities.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "Payoffs are maximized, not costs minimized. Approximate CE bounds each entire swap rule's total expected gain, not each recommendation separately. Empirical results assume a supplied cumulative bound and positive horizon; no algorithm, stochastic process, or convergence rate is supplied. The product theorem assumes valid mixed weights. Existence requires all action types nonempty but permits no players. Legal equilibria exclude illegal deviations. All statements use finite real sums and products.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "Nine deliberate CEChallenge theorem holes are excluded from the implementation count. There are no definition holes. Local checks, CI, registry intake, automated review, and registration are distinct; no registry approval is claimed. 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The final definitions, statements, and proofs are checked by Lean." }, "scope": "Four selected statements cover the new base reduction, its RM/RM+ numeric specialization, the averaged-product gain identity, and the transport to approximate CE. The library additionally proves the dependent-action algorithmic theorem, rational refinement, exact executable certificates, and a coordination example.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The classical reduction is presented for finite nonempty action types, full-information unit-interval payoffs, simultaneous updates, and uniform averages. N is a finite Markov-averaging parameter and T is the positive output horizon. The approximation bound includes both the external learner term and T/N. No last-iterate, stochastic, floating-point, general-sum Nash, or practical runtime claim is supplied. Exact rational output is a verification example, not a performance benchmark.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "No independent human or Palomar editorial approval is claimed. The earlier CE and regret-matching packages are explicit immutable dependencies rather than copied source or repeated Comparator targets. The four Challenge theorem holes occur only in the independent statement surface and are excluded from `status.sorry_count`; the implementation and Solution contain no proof placeholders. 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The structural results are extensional over mathematical values; the IEEE-facing layer is a self-contained field-level semantic model and does not claim hardware-level bit identity.", "authors": [ "Arthur Freitas Ramos", "David Barros Hulak", "Ruy J. G. B. de Queiroz" ], "maintainers": [ "Arthur Freitas Ramos" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "Attention Is All You Need", "id": "https://arxiv.org/abs/1706.03762", "authors": [ "Ashish Vaswani", "Noam Shazeer", "Niki Parmar", "Jakob Uszkoreit", "Llion Jones", "Aidan N. 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No general algorithmic or complexity-theoretic claim is made.", "authors": [ "Arthur Freitas Ramos", "David Barros Hulak", "Ruy Jose Guerra Barretto de Queiroz" ], "maintainers": [ "Arthur Freitas Ramos" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "A Formal Counterexample to the Cost-Preserving Single-Source Unsplittable Flow Conjecture", "id": "https://isa-afp.org/entries/Dinitz_Garg_Goemans_Counterexample.html", "authors": [ "Arthur Freitas Ramos", "David Barros Hulak", "Ruy Jose Guerra Barretto de Queiroz" ], "type": "formalization", "relationship": "independently-proves", "endorsement": "" }, { "title": "On the Single-Source Unsplittable Flow Problem", "id": "https://doi.org/10.1007/s004930050043", "authors": [ "Yefim Dinitz", "Naveen Garg", "Michel X. 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It proves both weak and strict conjecture-negation corollaries. It does not formalize the general Dinitz--Garg--Goemans rounding theorem, an algorithm, a complexity classification, or the Isabelle implementation.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The Isabelle source uses a generic model with real-valued data, while this Lean certificate uses exact rational arithmetic. Since every displayed datum and inequality is rational, the finite certificate embeds into the real-valued model. The Lean project reimplements the certificate rather than importing Isabelle code, and it does not formalize the general rounding theorem or any algorithmic/complexity claims.", "alignment": true, "original": true, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "none" ], "notes": "The proof builds with Lean 4.33.0 and pinned Mathlib. The theorem surface, source attribution, exact numerical certificate, and rational scope were audited locally; no external peer review is claimed." }, "canonical": { "repo": "arthur742ramos/dinitz-garg-goemans-counterexample-lean", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-29-000003", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-29-000003", "commit": "cc7284cf415fc2a773f00d400319ef39663c277a", "trust": "high", "theorems": 7, "date": "2026-08-29" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "Arthur742Ramos/equality-saturation-checker-lean/formalization.yaml", "repo": "Arthur742Ramos/equality-saturation-checker-lean", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/Arthur742Ramos/equality-saturation-checker-lean/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Native Lean equality-saturation certificate checking", "description": "Formalizes a native Lean checker for certified finite acyclic e-graphs. The principal result proves that accepted class certificates connect a canonical e-class term to the executable extracted representative and establish its minimum additive cost among all represented terms. The development also covers chronological merge certificates, egg-style flat explanations, replay-backed extraction, and bounded certificate search.", "authors": [ "Arthur Freitas Ramos" ], "maintainers": [ "Arthur Freitas Ramos" ], "license": "BSD-3-Clause", "role": "", "substantive": "", "sources": [ { "title": "Checking Equality-Saturation Merge and Extraction Certificates", "id": "https://isa-afp.org/entries/Equality_Saturation_Checker.html", "authors": [ "Arthur Freitas Ramos", "David Barros Hulak", "Ruy Jose Guerra Barretto de Queiroz" ], "type": "website", "relationship": "adapts", "endorsement": "not-contacted" } ], "related": [ { "id": "https://isa-afp.org/entries/Equality_Saturation_Checker.html", "relationship": "adapts", "note": "Preserves the checked-certificate semantic boundary while changing the term, context, parser, extraction, and proof organization to Lean-native definitions." } ], "classification": { "arxiv": [ "cs.LO", "cs.PL", "math.LO" ], "msc2020": [ "03B35", "68Q42" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "GPT-5 Codex" ], "framework": "Lean 4 and Mathlib" } ], "strongest": "agent", "models": [ "GPT-5 Codex" ], "spend": "subscription-based", "notes": "AI assistance was used for proof engineering. The final definitions, statements, and proofs are checked by Lean." }, "scope": "The compared theorem proves that a checked finite acyclic e-graph has a canonical class representative, an executable extracted representative, a conversion between them under the input rewrite rules, and minimum additive cost among every term represented by the selected class. The merge-log theorem establishes chronological prefix reuse. The egg layer parses nested flat S-expressions, rejects malformed multiple annotations at the decoder boundary, recovers atomic-child positions, and infers substitutions. The replay-backed extraction layer checks source-to- candidate paths, selected membershi", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The Lean development uses its own recursive term and context datatypes, total list lookup, explicit Boolean replay, a chronological merge-record structure, and a parser AST that retains annotation positions. The executable memoization, certified DAG checker, replay-backed extraction certificate, and finite frontier producer are independent of any engine implementation. These choices preserve the certificate semantics while avoiding a translation-shaped file layout.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "none" ], "notes": "The implementation library and Solution are proof-hole-free under the pinned local build. Challenge contains one deliberate Palomar statement hole. Hosted Comparator verification, registry review, and registration are separate states and are not inferred from local compilation." }, "canonical": { "repo": "arthur742ramos/equality-saturation-checker-lean", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-01-000007", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-01-000007", "commit": "e74222fe2591b15837d1bc239b58a14170d4bccc", "trust": "high", "theorems": 1, "date": "2026-09-01" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "Arthur742Ramos/finite-graph-fundamental-group/formalization.yaml", "repo": "Arthur742Ramos/finite-graph-fundamental-group", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/Arthur742Ramos/finite-graph-fundamental-group/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Finite graph fundamental groups: combinatorial and geometric models", "description": "Formalizes the fundamental group of a finite connected graph, represented as a finite weakly connected directed multigraph with its standard interval-cell quotient realization. The main result identifies the topological fundamental group at a chosen vertex with `FreeGroup (Fin (E + 1 - V))`. The proof compares the topological group with the endomorphism group at the root in Mathlib's free groupoid, whose explicit spanning-tree basis supplies the non-tree-edge generators. The development also proves the cycle-rank identities, root-change equivalences, the tree/triviality criterion, and the corr", "authors": [ "Arthur Freitas Ramos", "David Hulak", "Ruy de Queiroz" ], "maintainers": [ "Arthur Freitas Ramos", "David Hulak", "Ruy de Queiroz" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Algebraic Topology", "id": "ISBN 978-0-521-79540-1", "authors": [ "Allen Hatcher" ], "type": "book", "relationship": "independently-proves", "endorsement": "not-contacted" }, { "title": "Mathlib GroupTheory.FreeGroup.NielsenSchreier", "id": "", "authors": [], "type": "formalization", "relationship": "adapts", "endorsement": "" }, { "title": "Mathlib CategoryTheory.Groupoid.FreeGroupoid", "id": "", "authors": [], "type": "formalization", "relationship": "adapts", "endorsement": "" }, { "title": "Mathlib AlgebraicTopology.FundamentalGroupoid.FundamentalGroup", "id": "", "authors": [], "type": "formalization", "relationship": "adapts", "endorsement": "" }, { "title": "Mathlib Combinatorics.Quiver.Arborescence", "id": "", "authors": [], "type": "formalization", "relationship": "adapts", "endorsement": "" }, { "title": "Mathlib Combinatorics.Quiver.Covering", "id": "", "authors": [], "type": "formalization", "relationship": "adapts", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.AT", "math.LO" ], "msc2020": [ "05C25", "18B40", "20F65" ] }, "automation": { "methods": [ { "method": "agent-assisted", "models": [ "GPT-5 Codex" ], "framework": "Lean 4 and Mathlib" } ], "strongest": "other", "models": [ "GPT-5 Codex" ], "spend": "subscription-based", "notes": "The formalization was developed interactively, then checked with a fresh pinned dependency manifest and the repository's reproducible verification scripts." }, "scope": "The proof covers finite directed multigraphs represented by quivers with finite vertex and edge types, assuming connectivity after symmetrizing edges. It defines the combinatorial fundamental group as the endomorphism group at a root in the free groupoid, constructs the non-tree-edge basis, proves its cardinality, and derives the explicit free-group, root-change, cycle-rank, triviality, and abelianization results. It also constructs a path-lifting quiver over the graph and proves the projection is a quiver covering by explicit star and costar equivalences. The geometric realization is the quot", "sorry_count": 1, "sorry_in_definitions": 0, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The formalization uses a finite quiver, hence permits directed parallel edges, and defines the graph group combinatorially through Mathlib's free groupoid before identifying it with the topological fundamental group of the quotient realization. The topological comparison is presented through an explicit path-lifting cover with contractible realization; the proof uses Mathlib's covering and monodromy interfaces rather than postulating a universal-cover object. The rank is written as E + 1 - V in Nat to avoid an ambiguous truncated subtraction. The one deliberate `sorry` is only in the Palomar statement surface `Challenge.lean`; the proved library and `Solution.lean` contain no proof holes or custom axioms.", "alignment": true, "original": true, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "none" ], "notes": "The Challenge statement was kept separate from the proof and checked for exact Comparator correspondence for the public headline statement. The proof was audited for sorries and custom axioms and built from the pinned Lean/Mathlib dependency closure. The repository records the standard quotient graph realization and its canonical map into Mathlib's topological fundamental groupoid, while recording the comparison proof and its path-lifting/monodromy argument in `TopologicalComparison.lean`. The arborescence realization is separately certified contractible and simply connected in `TreeContraction.lean`; the checked combinatorial covering construction is recorded in `Cover.lean`." }, "canonical": { "repo": "arthur742ramos/finite-graph-fundamental-group", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-27-000004", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-27-000004", "commit": "dd57e3ab8bc5a7042fcc5498e778b008d87ac032", "trust": "high", "theorems": 1, "date": "2026-08-27" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "Arthur742Ramos/GraphCoveringTheory/formalization.yaml", "repo": "Arthur742Ramos/GraphCoveringTheory", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/Arthur742Ramos/GraphCoveringTheory/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Schreier and Kurosh subgroup theorems via covering graphs", "description": "Formalizes the sharp Schreier index formula for finite-index subgroups of free groups and the Kurosh subgroup theorem for arbitrary subgroups of indexed free products. The Schreier proof realizes a subgroup as the vertex group of a finite action-groupoid covering, while the Kurosh proof constructs the Bass--Serre tree, a locally bijective graph-of-groups cover, and its path-lifting normal form. The Kurosh factor-only statement is a free product of the nontrivial quotient vertex stabilizers and the free group of the quotient graph; each vertex stabilizer is identified with an intersection with ", "authors": [ "Arthur Freitas Ramos", "David Barros Hulak", "Ruy J. G. B. de Queiroz" ], "maintainers": [ "Arthur Freitas Ramos" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Die Untergruppen der freien Produkte von beliebigen Gruppen", "id": "EUDML:159700", "authors": [ "A. G. Kurosh" ], "type": "article", "relationship": "independently-proves", "endorsement": "not-contacted" }, { "title": "Die Untergruppen der freien Gruppen", "id": "10.1007/BF02952517", "authors": [ "Otto Schreier" ], "type": "article", "relationship": "independently-proves", "endorsement": "not-contacted" }, { "title": "Mathlib GroupTheory.FreeGroup.NielsenSchreier", "id": "", "authors": [], "type": "formalization", "relationship": "adapts", "endorsement": "not-contacted" }, { "title": "Mathlib GroupTheory.Schreier", "id": "", "authors": [], "type": "formalization", "relationship": "extends", "endorsement": "not-contacted" }, { "title": "Mathlib Combinatorics.Quiver.Covering", "id": "", "authors": [], "type": "formalization", "relationship": "adapts", "endorsement": "not-contacted" }, { "title": "Mathlib Combinatorics.Quiver.Arborescence", "id": "", "authors": [], "type": "formalization", "relationship": "adapts", "endorsement": "not-contacted" }, { "title": "Mathlib CategoryTheory.Action.Concrete", "id": "", "authors": [], "type": "formalization", "relationship": "extends", "endorsement": "not-contacted" }, { "title": "Finite graph fundamental groups: combinatorial and geometric models", "id": "", "authors": [], "type": "formalization", "relationship": "extends", "endorsement": "not-contacted" } ], "related": [], "classification": { "arxiv": [ "math.GR", "math.AT", "math.LO" ], "msc2020": [ "20E07", "05C25", "18B40" ] }, "automation": { "methods": [ { "method": "agent-assisted", "models": [ "GPT-5 Codex" ], "framework": "Lean 4 and Mathlib" } ], "strongest": "other", "models": [ "GPT-5 Codex" ], "spend": "subscription-based", "notes": "The public statement is isolated from candidate-local implementation modules; the solution imports the reusable covering and spanning-tree development." }, "scope": "The Schreier theorem covers every finite nonempty generating type alpha and every subgroup H of FreeGroup alpha carrying Mathlib's finite-index typeclass. It proves an explicit group equivalence with FreeGroup (Fin (1 + H.index * (Fintype.card alpha - 1))) and, internally, a FreeGroupBasis indexed by that type. The Kurosh theorem covers every subgroup H of an arbitrary indexed free product. Its factor-only product is indexed by the nontrivial vertex stabilizers of the quotient Bass--Serre graph together with the quotient graph's free group, and each such stabilizer is proved equal to `H ∩ g G_", "sorry_count": 1, "sorry_in_definitions": 0, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The formalization states group equivalence rather than only equality of abstract ranks, and uses finite free groups indexed by Fin. The positive hypothesis is explicit because natural-number subtraction is truncated; the implementation also proves the equivalent all-n truncation-safe form. The finite covering is the action groupoid on the right coset quotient supplied by Mathlib, with a rose as base. This is the finite Schreier graph construction, not a claim to have formalized the entire topological subgroup-cover correspondence. The Kurosh statement uses the canonical quotient-vertex stabilizer index rather than choosing a separate double-coset representative for every factor; its accompanying factor theorem proves the equivalent intersection description without making those choices part of the public product type.", "alignment": true, "original": true, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "none" ], "notes": "The Challenge theorem is a small, closure-safe statement surface with one deliberate proof hole, while Solution and the implementation library are proof-hole-free. The Schreier covering maps, free-groupoid generators, vertex count, edge count, subgroup identification, and final reindexing equivalence are checked by Lean's kernel. The Kurosh development adds a checked Bass--Serre tree, local cover bijections, path lifting, trivial factor removal, and a kernel proof for the graph-of-groups normal form. No custom axioms are introduced." }, "canonical": { "repo": "arthur742ramos/graphcoveringtheory", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-31-000002", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-31-000002", "commit": "be06b655ae6d2cfee8fee8033b2ff0f5a4f99bf5", "trust": "high", "theorems": 1, "date": "2026-08-31" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "Arthur742Ramos/GrushkoNeumannTheorem/formalization.yaml", "repo": "Arthur742Ramos/GrushkoNeumannTheorem", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/Arthur742Ramos/GrushkoNeumannTheorem/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Grushko--Neumann rank additivity for free products", "description": "Formalizes the binary Grushko--Neumann theorem: the minimal number of generators of a free product of two finitely generated groups is the sum of the minimal numbers of generators of the factors. The central theorem is proved by a kernel-checked finite labelled-graph reduction. For a minimal null path, the implemented reduction uniformly performs a source unfold, a subsequent safe fold, and an explicit monochromatic-vertex contraction; this gives the strict decrease used by strong induction and the separated-generator lower bound. The repository also retains the finite-core and free-group rank", "authors": [ "Arthur Freitas Ramos", "David Barros Hulak", "Ruy J. G. B. de Queiroz" ], "maintainers": [ "Arthur Freitas Ramos" ], "license": "Apache-2.0", "role": "thin-wrapper", "substantive": "https://github.com/Arthur742Ramos/MarshallHallTheorem", "sources": [ { "title": "On the Number of Generators of a Free Product", "id": "10.1112/jlms/s1-18.1.12", "authors": [ "B. H. Neumann" ], "type": "article", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "Topology of finite graphs", "id": "10.1007/BF02095993", "authors": [ "John R. Stallings" ], "type": "article", "relationship": "background", "endorsement": "not-contacted" } ], "related": [ { "id": "https://github.com/Arthur742Ramos/HowsonIntersectionTheorem", "relationship": "related", "note": "Accepted companion development in the same subgroup/finite-core program; it is not imported by this repository and is not a source for the selected rank theorem." }, { "id": "https://github.com/leanprover-community/mathlib4/blob/v4.32.0/Mathlib/GroupTheory/Rank.lean", "relationship": "builds-on", "note": "Provides the `Group.rank` definition and the generator/rank interface used in the statement and lower-bound bridge." }, { "id": "https://github.com/leanprover-community/mathlib4/blob/v4.32.0/Mathlib/GroupTheory/Coprod/Basic.lean", "relationship": "builds-on", "note": "Provides the free-product type and canonical factor injections used by the finite labelled-graph and rank constructions." }, { "id": "https://github.com/Arthur742Ramos/GraphCoveringTheory", "relationship": "other", "note": "Earlier covering-graph and Schreier work that motivated the broader program; it is not imported by the proof modules in this focused submission." } ], "classification": { "arxiv": [ "math.GR" ], "msc2020": [ "20E07", "20F65", "05C25" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "GPT-5.6-sol" ], "framework": "Codex with Lean 4 and Mathlib" }, { "method": "manual", "models": [], "framework": "Lean 4, Mathlib, and repository review" } ], "strongest": "agent", "models": [ "GPT-5.6-sol" ], "spend": "subscription-based", "notes": "The public Challenge selects only the global binary `rank_freeProduct_eq_add` declaration. The AI agent materially assisted with implementation, proof engineering, documentation, and automated checking. The human authors directed the mathematical scope and source attribution, reviewed the code and claims, and are responsible for the final formalization and communication. Lean's kernel and the independent Comparator/NanoDa workflow supplied automated verification." }, "scope": "The principal result is the binary theorem for arbitrary finitely generated groups G and H, stated as Group.rank (G ∗ H) = Group.rank G + Group.rank H. Supporting results include the binary and finite-indexed free-group calculations and the graph-reduction infrastructure that supplies the arbitrary-factor lower bound. The unqualified declaration in `Challenge.lean` is the exact Comparator-selected surface; `Solution.lean` proves the same global declaration by invoking the namespaced implementation theorem. The Challenge has one deliberate proof hole by design; the Solution and implementation l", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "rank_freeProduct_eq_add", "file": "Challenge.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The historical theorem is expressed here using Mathlib's `Group.rank` and the binary free-product type. The proof covers arbitrary finitely generated group factors, while the additional finite-indexed result in the library is stated for free groups. The public Challenge is intentionally smaller than the implementation and selects only the central binary equality; it does not claim the retained Hall or LERF results as new compared statements.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "none" ], "notes": "The selected Challenge/Solution theorem has a small Mathlib-only statement surface. The Solution and implementation library contain no proof holes or nonstandard axioms. The finite labelled-graph reduction always follows the documented sequence of source unfold, subsequent safe fold, and monochromatic-vertex contraction, with connectivity and marking invariants, strong induction, factorwise rank bounds, and the final equality checked by Lean's kernel. The pinned Comparator replay and NanoDa replay are configured to check the exact global selected theorem correspondence independently." }, "canonical": { "repo": "arthur742ramos/grushkoneumanntheorem", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-01-000013", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-01-000013", "commit": "ef1f70db49792bb9bdb0c40eeec48168fca53393", "trust": "high", "theorems": 1, "date": "2026-09-01" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "Arthur742Ramos/hadamard-668-lean/formalization.yaml", "repo": "Arthur742Ramos/hadamard-668-lean", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/Arthur742Ramos/hadamard-668-lean/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "A certified Hadamard matrix of order 668", "description": "An independent Lean 4 and Mathlib formalization of a supplied bordered block-circulant sign matrix of order 668. The main theorem proves that there exists a 668 by 668 integer matrix with entries in {-1, 1}, diagonal row norm 668, and zero inner product between every pair of distinct rows. The formalization is a certificate-specific existence result, not a classification of Hadamard matrices and not a proof of the general Hadamard conjecture.", "authors": [ "Arthur Freitas Ramos", "David Barros Hulak", "Ruy J. G. B. de Queiroz" ], "maintainers": [ "Arthur Freitas Ramos" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Supplied obfuscated certificate payload for a Hadamard matrix of order 668", "id": "https://x.com/__alpoge__/status/2087504785952182273", "authors": [ "Levent Alpöge" ], "type": "web post", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "Hadamard matrices and the Hadamard conjecture", "id": "https://doi.org/10.24033/asens.476", "authors": [ "Jacques Hadamard" ], "type": "journal article", "relationship": "uses-as-background", "endorsement": "not-contacted" } ], "related": [ { "id": "https://github.com/Arthur742Ramos/isabelle-afp-monorepo", "relationship": "independent", "note": "The Isabelle/HOL project `projects/hadamard-matrices-isabelle` contains an earlier kernel-checked formalization of the same supplied certificate. This Lean repository does not import Isabelle code and rechecks the certificate in Lean." } ], "classification": { "arxiv": [ "math.CO", "math.LO" ], "msc2020": [ "05B20", "03B35" ] }, "automation": { "methods": [ { "method": "agent-assisted", "models": [ "GPT-5 Codex" ], "framework": "Lean 4 and Mathlib" } ], "strongest": "other", "models": [ "GPT-5 Codex" ], "spend": "subscription-based", "notes": "The generated data and proof are intended for independent maintainer review." }, "scope": "The compared theorem establishes one concrete real/sign Hadamard matrix of order 668. The proof expands four top rows and 4 times 166 cyclic core rows from four length-166 block families and checks the resulting integer Gram conditions. It does not classify matrices up to equivalence, prove uniqueness, or establish the Hadamard conjecture for all multiples of four.", "sorry_count": 1, "sorry_in_definitions": 0, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The Isabelle development presents the certificate through list-level bordered block-circulant predicates. The Lean version retains that construction but exposes the final result as a Matrix over integers so the Comparator statement is mathematically ordinary. The source payload is supplied rather than discovered here; no classification or unrestricted search claim is made.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "none" ], "notes": "Local Lean builds, exact data-shape checks, and an independent integer checker (`independent_checker.py`) are recorded in the repository. The single sorry is the expected placeholder in the Comparator-facing Challenge.lean statement; the Solution.lean theorem and its proof modules contain no sorry. Palomar mechanical verification, automated review, and registration have not been requested or run; human review beyond this maintainer-prepared snapshot remains pending." }, "canonical": { "repo": "arthur742ramos/hadamard-668-lean", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-29-000009", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-29-000009", "commit": "78fbae8d69b6fdb66705b662469692d60e346ef0", "trust": "high", "theorems": 1, "date": "2026-08-29" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "Arthur742Ramos/HowsonIntersectionTheorem/formalization.yaml", "repo": "Arthur742Ramos/HowsonIntersectionTheorem", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/Arthur742Ramos/HowsonIntersectionTheorem/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Howson's theorem through finite cores and Kurosh decomposition", "description": "Formalizes Howson's theorem for finite-rank free groups: the intersection of two finitely generated subgroups is finitely generated. The proof uses the repository's inclusion-compatible Marshall Hall construction to place one subgroup as a free factor of a finite-index overgroup, applies Schreier's finite-index theorem to the other subgroup, and then uses the actual Kurosh decomposition of the resulting subgroup of a binary free product. The final map from the relevant conjugate-intersection factor is defined from the factor inclusions and proved surjective onto the internal subgroup intersect", "authors": [ "Arthur Freitas Ramos", "David Barros Hulak", "Ruy J. G. B. de Queiroz" ], "maintainers": [ "Arthur Freitas Ramos" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "On the Intersection of Finitely Generated Free Groups", "id": "10.1112/jlms/s1-29.4.428", "authors": [ "A. G. Howson" ], "type": "article", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "Topology of finite graphs", "id": "10.1007/BF02095993", "authors": [ "John R. 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It proves Group.FG (H ⊓ K). The implementation explicitly passes through a finite-index Hall overgroup, the binary free-product bridge, a finitely generated Kurosh factor, and a homomorphism induced by the actual factor inclusion whose image is exactly H ⊓ K. The repository retains the preceding Hall and Kurosh infrastructure as supporting formalization; those earlier named results are not presented as new sources here.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "HowsonChallenge.howson_intersection_fg", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The classical Howson property is often stated for an arbitrary free group. This submission formalizes the finite-rank form, which is the exact scope of the present finite-type development. It proves finite generation of the intersection and does not claim Howson's sharper rank bound. The proof's Kurosh step is not an abstract isomorphism: the selected factor is the actual intersection with a conjugate of the Hall factor, and the final homomorphism is defined from the corresponding subgroup inclusion.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "none" ], "notes": "The Challenge theorem is a small statement surface with one deliberate proof hole excluded from the status counts. Solution.lean and the implementation library contain no proof holes or custom axioms. 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B. de Queiroz" ], "maintainers": [ "Arthur Freitas Ramos" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "An explicit counterexample to the Jacobian conjecture", "id": "https://x.com/__alpoge__/status/2079028340955197566", "authors": [ "Levent Alpöge" ], "type": "web post", "relationship": "formalizes", "endorsement": "" } ], "related": [ { "id": "https://github.com/deancureton/jacobian/tree/0d4a9212d874226ad81ce5a926becddfa94e6a88", "relationship": "independent", "note": "A public Lean formalization of the same map was audited as related work. 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It does not formalize a general theory of polynomial automorphisms, the source's geometric explanation, or the dimension-two Jacobian conjecture.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The source announcement presents the complex three-dimensional map informally. The Lean definitions are polymorphic over fields where possible, while the advertised conjecture-negation theorem specializes to ℂ. The formalization proves the displayed collision rather than a complete description of the fibre and does not formalize later geometric or dynamical claims.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "none" ], "notes": "The map, determinant identity, collision, and final implication were checked in Lean 4 with pinned Mathlib. 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The Schreier proof realizes a subgroup as the vertex group of a finite action-groupoid covering, while the Kurosh proof constructs the Bass--Serre tree, a locally bijective graph-of-groups cover, and its path-lifting normal form. The Kurosh factor-only statement is a free product of the nontrivial quotient vertex stabilizers and the free group of the quotient graph; each vertex stabilizer is identified with an intersection with ", "authors": [ "Arthur Freitas Ramos", "David Barros Hulak", "Ruy J. G. B. de Queiroz" ], "maintainers": [ "Arthur Freitas Ramos" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Die Untergruppen der freien Produkte von beliebigen Gruppen", "id": "EUDML:159700", "authors": [ "A. G. 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It proves an explicit group equivalence with FreeGroup (Fin (1 + H.index * (Fintype.card alpha - 1))) and, internally, a FreeGroupBasis indexed by that type. The Kurosh theorem covers every subgroup H of an arbitrary indexed free product. Its factor-only product is indexed by the nontrivial vertex stabilizers of the quotient Bass--Serre graph together with the quotient graph's free group, and each such stabilizer is proved equal to `H ∩ g G_", "sorry_count": 1, "sorry_in_definitions": 0, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The formalization states group equivalence rather than only equality of abstract ranks, and uses finite free groups indexed by Fin. The positive hypothesis is explicit because natural-number subtraction is truncated; the implementation also proves the equivalent all-n truncation-safe form. The finite covering is the action groupoid on the right coset quotient supplied by Mathlib, with a rose as base. This is the finite Schreier graph construction, not a claim to have formalized the entire topological subgroup-cover correspondence. The Kurosh statement uses the canonical quotient-vertex stabilizer index rather than choosing a separate double-coset representative for every factor; its accompanying factor theorem proves the equivalent intersection description without making those choices part of the public product type.", "alignment": true, "original": true, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "none" ], "notes": "The Challenge theorem is a small, closure-safe statement surface with one deliberate proof hole, while Solution and the implementation library are proof-hole-free. The Schreier covering maps, free-groupoid generators, vertex count, edge count, subgroup identification, and final reindexing equivalence are checked by Lean's kernel. The Kurosh development adds a checked Bass--Serre tree, local cover bijections, path lifting, trivial factor removal, and a kernel proof for the graph-of-groups normal form. 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The development constructs normalized Riemannian volume, solves the mean-zero Poisson problem variationally and upgrades it to a smooth representative, proves the actual integrated Bochner and contracted-Bianchi identities, and closes the sharp algebra.", "authors": [ "Arthur Freitas Ramos", "David Barros Hulak", "Ruy J. G. B. de Queiroz" ], "maintainers": [ "Arthur Freitas Ramos" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "Almost-Schur lemma", "id": "10.1007/s00526-011-0413-z", "authors": [ "Camillo De Lellis", "Peter M. Topping" ], "type": "paper", "relationship": "formalizes", "endorsement": "not-contacted" } ], "related": [ { "id": "https://github.com/Arthur742Ramos/lean-poincare-formalization-plan/tree/12cebb809524d0cd185c6cd7bcb5b73d3562bce1/contracted-bianchi", "relationship": "builds-on", "note": "The eight files in AlmostSchur/CurvatureVendor/ adapt the immutable contracted-bianchi curvature core at this repository commit. Their source identities and adaptations are recorded in AlmostSchur/CurvatureVendor/PROVENANCE.json. The inherited curvature lemmas are supporting infrastructure; the selected result here is the almost-Schur inequality and equality package. GeometryStatements.lean and it" }, { "id": "https://github.com/abenenson/rellich-kondrachov/tree/70f85d4c1bf99c6e7d61e8be4daa6f3664d08d23", "relationship": "adapts", "note": "Thirty-six attributed modules under RellichKondrachov/ are adapted from this immutable Apache-2.0 source snapshot for Lean 4.33.0 compatibility; the exact inventory and notices are checked by scripts/check-vendored.py. The adapted library supplies generic chart Sobolev compactness and measure-transport infrastructure, not the almost-Schur theorem." }, { "id": "https://github.com/leanprover-community/mathlib4", "relationship": "builds-on", "note": "The project uses the pinned Mathlib dependency recorded in lakefile.toml and lake-manifest.json for manifold, measure, analysis and linear-algebra infrastructure." } ], "classification": { "arxiv": [ "math.DG", "math.LO" ], "msc2020": [ "53C21", "53C25", "03B35" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "GPT-5", "GPT-6" ], "framework": "Codex with Lean 4.33.0 and pinned Mathlib" }, { "method": "manual", "models": [], "framework": "Lean, repository and mathematical review" } ], "strongest": "agent", "models": [ "GPT-5", "GPT-6" ], "spend": "", "notes": "The AI-assisted development and mechanical verification are disclosed here without attributing human authorship to the agent. Lean's kernel and the independent Comparator/NanoDa workflow provide mechanical evidence, not expert peer review or source-author endorsement." }, "scope": "Smooth finite-dimensional real Riemannian manifolds with nonempty compact boundaryless topology, Hausdorff and Lindelof hypotheses, a complete model space, and a preconnected manifold. The metric and constructed Levi--Civita connection have the displayed regularity instances. The theorem assumes pointwise nonnegative Ricci curvature and real dimension greater than two. The volume is the normalized smooth metric-density measure constructed in the development; no identification with Hausdorff measure, space-form classification, optimality proof, or Ricci-flow result is claimed.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "AlmostSchurEntry.Geometry.almostSchur", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "AlmostSchur.almostSchur_bound_complete", "file": "AlmostSchur/AlmostSchurFinal.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "AlmostSchur.almostSchur_equality_iff", "file": "AlmostSchur/AlmostSchurEquality.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] } ], "comparator": false, "literature_dependencies": 0, "divergences": "The formalization uses a normalized smooth metric-density measure assembled from coordinate charts and does not assert that it is the dimensional Hausdorff measure of the induced Riemannian distance. The trace-free Ricci norm is the intrinsic squared Hilbert--Schmidt norm of the metric-raised Ricci operator. The proof establishes the contracted-Bianchi and Bochner identities for the constructed Levi--Civita connection instead of taking them as theorem hypotheses. The displayed equality characterization is the Einstein condition in metric-raised trace-free form; no optimality or classification theorem is included. The independent Challenge expresses the result by constructing the connection and volume before stating the conditional inequality and equality theorem. Its curvature is the actual covariant-derivative commutator, Ricci and scalar curvature are explicit orthonormal-frame contractions, and volume is characterized by its chart Gram-determinant density. GeometryComparison.lean proves their agreement with the implementation; no analytic bridge is assumed in the Challenge.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "The selected Challenge theorem contains the complete manifold-level inequality and Einstein equality characterization, together with the construction of its connection and volume. Comparator selects that theorem and all nine geometric definitions. The earlier algebra-only selection was replaced during the semantic audit. The source-based theorem is classical; this project makes no originality, priority, endorsement or independent human-review claim." }, "canonical": { "repo": "arthur742ramos/lean-poincare-formalization-plan", "directory": "almost-schur" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-07-000015", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-07-000015", "commit": "d9f4c4deaea269a1ab1f5f8874697da798cc3db2", "trust": "high", "theorems": 1, "date": "2026-09-07" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "Arthur742Ramos/lean-poincare-formalization-plan/bonnet-myers/formalization.yaml", "repo": "Arthur742Ramos/lean-poincare-formalization-plan", "path": "bonnet-myers/formalization.yaml", "directory": "bonnet-myers", "origins": [ "palomar", "search" ], "url": "https://github.com/Arthur742Ramos/lean-poincare-formalization-plan/blob/HEAD/bonnet-myers/formalization.yaml", "version": "v0.4", "missing": [], "name": "Independent Bonnet--Myers diameter and compactness theorem", "description": "Formalizes the classical Bonnet--Myers theorem for complete connected smooth boundaryless finite-dimensional real Riemannian manifolds. From dimension n >= 2, K > 0, and the pointwise lower bound Ric(a,a) >= (n-1) K g(a,a), the selected theorem constructs the Levi--Civita connection and actual curvature, proves compactness, and gives the sharp induced extended-diameter bound pi/sqrt K. The development proves minimizing-geodesic, parallel-transport, second-variation, index-form, and metric-compactness bridges rather than assuming them.", "authors": [ "Arthur Freitas Ramos", "David Barros Hulak", "Ruy J. G. B. de Queiroz" ], "maintainers": [ "Arthur Freitas Ramos" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Riemannian manifolds with positive mean curvature", "id": "10.1215/S0012-7094-41-00832-3", "authors": [ "Sumner Byron Myers" ], "type": "paper", "relationship": "independently-proves", "endorsement": "not-contacted" } ], "related": [ { "id": "https://github.com/Arthur742Ramos/lean-poincare-formalization-plan/tree/5db025e6d20d8aad714c3d116714545d1823fd8e/curvature", "relationship": "builds-on", "note": "Nineteen unchanged files are vendored from this immutable same-repository snapshot. They provide general covariant-derivative, curvature, bundle-section, and smooth ODE infrastructure, not the Bonnet--Myers minimizing-geodesic, second-variation, comparison, diameter, or compactness proof. Their exact inventory and Git blob identities are checked by scripts/check-vendored.py." }, { "id": "https://github.com/qinz1yang/differential-geometry/tree/1b535dd102b94cc42b107cca27059687888f08b3", "relationship": "independent", "note": "A superseded noncompiled wrapper in this repository formerly depended on this separate Bonnet--Myers implementation. The active proof has no such dependency and does not copy, translate, vendor, or adapt that proof." }, { "id": "https://github.com/leanprover-community/mathlib4/tree/db584cd6d46c92f209a44c0f1c829460d327499d", "relationship": "builds-on", "note": "Pinned Mathlib foundations for manifolds, Riemannian metrics, calculus, ordinary differential equations, integration, linear algebra, topology, and extended metric spaces." } ], "classification": { "arxiv": [ "math.DG", "math.LO" ], "msc2020": [ "53C20", "53C21", "03B35" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "GPT-5" ], "framework": "Codex with Lean 4.33.0 and pinned Mathlib" }, { "method": "manual", "models": [], "framework": "Lean, Git, and mathematical review" } ], "strongest": "agent", "models": [ "GPT-5" ], "spend": "subscription-based", "notes": "The AI-assisted development is disclosed without treating the agent as an author. Lean, Comparator, and NanoDa checks are mechanical evidence and do not constitute independent expert review." }, "scope": "Complete connected smooth boundaryless finite-dimensional real manifolds modeled on a nonzero complete normed vector space, with Hausdorff manifold and tangent bundle, sigma-compact topology, a smooth Riemannian metric, dimension at least two, and positive K. Completeness and diameter use the metric induced by that Riemannian metric. Ricci is the ordinary trace of the actual Levi--Civita curvature endomorphism. The theorem concludes CompactSpace and extended diameter at most ENNReal.ofReal (pi/sqrt K). It does not prove equality rigidity, sphere classification, conjugate-point language, or fin", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "BonnetMyersEntry.bonnet_myers", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "BonnetMyersEntry.completeStatement_proved", "file": "BonnetMyers/Complete.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "BonnetMyersEntry.IntrinsicGeodesic.GlobalGeodesic.GlobalParallelField.globalIndexForm_sine_nonnegative_of_endpoint_minimizing", "file": "BonnetMyers/GlobalIndexNonnegative.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] } ], "comparator": false, "literature_dependencies": 0, "divergences": "The formal statement uses Mathlib's induced extended Riemannian distance and reports the diameter as an ENNReal bound. It assumes an explicitly sigma-compact Hausdorff smooth manifold and Hausdorff tangent bundle to make the underlying manifold and ODE infrastructure available. Ricci is written as the ordinary linear trace of z |-> R(z,a)a, with the curvature sign convention fixed by the displayed covariant-derivative commutator. The result includes compactness and the sharp diameter bound from Myers' theorem but omits fundamental-group finiteness and equality rigidity.", "alignment": true, "original": true, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "none" ], "notes": "The statement, geometric dependency chain, Challenge isolation, exact vendored provenance, proof-hole count, and axiom surface are checked by the repository verification scripts. No independent human mathematical review, Palomar editorial acceptance, intake, or registration is claimed." }, "canonical": { "repo": "arthur742ramos/lean-poincare-formalization-plan", "directory": "bonnet-myers" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-11-000001", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-11-000001", "commit": "0cc7c31bf6e2dac5c0359a432f3d99803f563017", "trust": "high", "theorems": 1, "date": "2026-09-11" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "Arthur742Ramos/lean-poincare-formalization-plan/contracted-bianchi/formalization.yaml", "repo": "Arthur742Ramos/lean-poincare-formalization-plan", "path": "contracted-bianchi/formalization.yaml", "directory": "contracted-bianchi", "origins": [ "palomar", "search" ], "url": "https://github.com/Arthur742Ramos/lean-poincare-formalization-plan/blob/HEAD/contracted-bianchi/formalization.yaml", "version": "v0.4", "missing": [], "name": "Double-contracted second Bianchi identity for metric-compatible torsion-free connections", "description": "Proves the orthonormal double contraction of the actual corrected curvature derivative of a tangent connection on a smooth finite-dimensional real manifold. The Challenge exposes the corrected commutator and the smooth tangent-vector extensions explicitly; the Solution proves the identity for every supplied extension satisfying the stated hypotheses. The derivative symmetries and cyclic Bianchi equations are derived from torsion-freeness and metric compatibility, not assumed as abstract tensor data. No separate Einstein-divergence theorem is selected.", "authors": [ "Arthur Freitas Ramos", "David Barros Hulak", "Ruy J. G. B. de Queiroz" ], "maintainers": [ "Arthur Freitas Ramos" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Lecture Notes on General Relativity", "id": "https://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll3.html", "authors": [ "Sean M. Carroll" ], "type": "lecture notes", "relationship": "adapts", "endorsement": "not-contacted" }, { "title": "Einstein Manifolds", "id": "10.1007/978-3-540-74311-8", "authors": [ "Arthur L. Besse" ], "type": "book", "relationship": "background", "endorsement": "not-contacted" } ], "related": [], "classification": { "arxiv": [ "math.DG", "gr-qc", "math.LO" ], "msc2020": [ "53B20", "53C21", "03B35" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "GPT-5 Codex", "GPT-6 Codex" ], "framework": "Codex with Lean 4.33 and Mathlib" } ], "strongest": "agent", "models": [ "GPT-5 Codex", "GPT-6 Codex" ], "spend": "", "notes": "Fuller contribution and oversight record: https://github.com/Arthur742Ramos/lean-poincare-formalization-plan/blob/12cebb809524d0cd185c6cd7bcb5b73d3562bce1/contracted-bianchi/AGENT-CONTRIBUTION.md is pinned to the immutable commit containing the replacement proof and record." }, "scope": "Smooth Hausdorff real manifolds with a finite-dimensional complete model space; tangent bundle smoothness through order three, connection smoothness through order two, and Riemannian bundle smoothness through order two (including an explicit order-one instance). The selected theorem assumes torsion-freeness and metric compatibility. At the selected point it takes an explicit C^3 extension for each tangent vector, anchored by evaluation at that point, and contracts the resulting corrected curvature derivative in an orthonormal tangent basis. Compactness and sigma-compactness are not required. A", "sorry_count": 1, "sorry_in_definitions": 0, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The selected statement is the explicit double contraction of the corrected curvature derivative, with anchored smooth extensions and an orthonormal basis. The Solution uses the section-level theorem from the local dependency closure for the supplied extensions. It does not assert a separately formalized equality with the differential of scalar curvature or divergence of a Ricci/Einstein field. Source index conventions are adapted to the repository's curvature commutator. The local dependency closure is included in this project; only Mathlib is an external Lean dependency.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "none" ], "notes": "An earlier abstract version of this project was submitted and failed mechanical verification; this stronger geometric replacement is unregistered. Any fresh intake must have a blank existing_id. The mathematical interest is the differential curvature constraint obtained from a torsion-free metric-compatible connection, with an audience in Riemannian geometry and geometric field equations. The proof derives the curvature derivative symmetries, rather than assuming a tensor with those identities. Editorial acceptance is not claimed." }, "canonical": { "repo": "arthur742ramos/lean-poincare-formalization-plan", "directory": "contracted-bianchi" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-06-000004", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-06-000004", "commit": "90d215d81d30a5f67922dce00dfa160b4878e1c5", "trust": "high", "theorems": 1, "date": "2026-09-06" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "Arthur742Ramos/lean-poincare-formalization-plan/curvature/formalization.yaml", "repo": "Arthur742Ramos/lean-poincare-formalization-plan", "path": "curvature/formalization.yaml", "directory": "curvature", "origins": [ "palomar", "search" ], "url": "https://github.com/Arthur742Ramos/lean-poincare-formalization-plan/blob/HEAD/curvature/formalization.yaml", "version": "v0.4", "missing": [], "name": "Diffeomorphism transport of connections and curvature in Lean", "description": "Formalizes the fixed-diffeomorphism transport of a covariant derivative on the tangent bundle of a smooth finite-dimensional manifold. The selected statements expose the tangent pushforward and pullback, define the pulled-back connection and raw curvature commutator, and prove transport of curvature and torsion together with preservation of metric compatibility and the Levi-Civita property. The result is a standard geometric naturality theorem family formalized from the repository's proved source; it is not claimed as an original mathematical discovery or as a proof of Ricci-flow local existen", "authors": [ "Arthur Freitas Ramos", "David Barros Hulak", "Ruy J. G. B. de Queiroz" ], "maintainers": [ "Arthur Freitas Ramos" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Foundations of Differential Geometry, Volume I", "id": "10.1007/978-1-4757-9966-8", "authors": [ "Shoshichi Kobayashi", "Katsumi Nomizu" ], "type": "book", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "Riemannian Manifolds: An Introduction to Curvature", "id": "10.1007/978-3-319-91755-9", "authors": [ "John M. Lee" ], "type": "book", "relationship": "formalizes", "endorsement": "not-contacted" } ], "related": [], "classification": { "arxiv": [ "math.DG", "math.AP", "math.LO" ], "msc2020": [ "53C25", "53E20", "35K15", "03B35" ] }, "automation": { "methods": [ { "method": "agent-assisted", "models": [ "GPT-5 Codex" ], "framework": "Codex with Lean 4.33 and Mathlib" } ], "strongest": "other", "models": [ "GPT-5 Codex" ], "spend": "subscription-based", "notes": "Fuller contribution and oversight record: [AGENT-CONTRIBUTION-05.md](https://github.com/Arthur742Ramos/lean-poincare-formalization-plan/blob/e0f3438b55206b6bf443e07c907d199e3aa3e497/curvature/AGENT-CONTRIBUTION-05.md) is pinned to the immutable preparation commit that contains the record and the synchronized Comparator surface." }, "scope": "The selected Challenge surface is a smooth finite-dimensional Hausdorff manifold with a bundled `C^2` self-diffeomorphism, its invertible tangent map, a covariant derivative on the tangent bundle, and the explicit raw curvature commutator. The selected metric statements quantify over smooth Riemannian metrics and include the pointwise equation identifying the target inner product with the pullback metric. The declarations therefore concern actual manifold connections and Levi-Civita structure, not an unconstrained endomorphism, scalar family, or generic ODE. This package proves geometric trans", "sorry_count": 7, "sorry_in_definitions": 1, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "`Challenge.lean` imports only Mathlib and exposes the tangent-map, covariant-derivative, curvature-commutator, torsion, and metric-compatibility formulas compared by Palomar. `Solution.lean` proves the same declarations by invoking the fully proved manifold transport source in the pinned repository. The source import is explicit in Solution.lean and is not available to Challenge.lean; the mathematical definitions and hypotheses remain visible at the Comparator boundary.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "none" ], "notes": "This revision removes the stale affine, metric-cone, and gauge-reduction surface entirely. It selects the already-proved fixed-diffeomorphism transport of a genuine tangent-bundle connection, raw curvature, torsion, metric compatibility, and Levi-Civita structure. The relationship to the classical sources is stated as standard formalization/adaptation; no originality or Ricci-flow existence claim is made." }, "canonical": { "repo": "arthur742ramos/lean-poincare-formalization-plan", "directory": "curvature" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-02-000007", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-02-000007", "commit": "80b71f3a239ac2b294c91c66d020476f667a1306", "trust": "high", "theorems": 5, "date": "2026-09-03" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "Arthur742Ramos/lean-poincare-formalization-plan/hamilton-ivey-reaction/formalization.yaml", "repo": "Arthur742Ramos/lean-poincare-formalization-plan", "path": "hamilton-ivey-reaction/formalization.yaml", "directory": "hamilton-ivey-reaction", "origins": [ "search" ], "url": "https://github.com/Arthur742Ramos/lean-poincare-formalization-plan/blob/HEAD/hamilton-ivey-reaction/formalization.yaml", "version": "v0.4", "missing": [], "name": "Hamilton-Ivey curvature-ODE pinching", "description": "Proves preservation of the three-dimensional Hamilton-Ivey logarithmic defect along differentiable ordered solutions of the curvature-reaction ODE. From the standard initial least-eigenvalue bound, the proof derives the scalar lower barrier, the exact defect derivative, and strict nonlinear reaction coercivity, then applies a proved invariant-region principle.", "authors": [ "Arthur Freitas Ramos", "David Barros Hulak", "Ruy J. G. B. de Queiroz" ], "maintainers": [ "Arthur Freitas Ramos" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Local pinching estimates in 3-dim Ricci flow", "id": "arXiv:1206.1814", "authors": [ "Bing-Long Chen", "Guoyi Xu", "Zhuhong Zhang" ], "type": "paper", "relationship": "adapts", "endorsement": "not-contacted" } ], "related": [ { "id": "https://github.com/leanprover-community/mathlib4/tree/db584cd6d46c92f209a44c0f1c829460d327499d", "relationship": "builds-on", "note": "Pinned Mathlib real analysis, calculus, topology, ordered algebra, and proof tactics. No external Ricci-flow formalization is reused." } ], "classification": { "arxiv": [ "math.DG", "math.AP", "math.LO" ], "msc2020": [ "53E20", "35K55", "03B35" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "OpenAI GPT-5" ], "framework": "Codex with Lean 4.33.0 and pinned Mathlib" }, { "method": "manual", "models": [], "framework": "Lean, Git, and mathematical review" } ], "strongest": "agent", "models": [ "OpenAI GPT-5" ], "spend": "subscription-based", "notes": "Agent assistance is disclosed without treating the agent as an author." }, "scope": "Differentiable real eigenvalue curves lambda >= mu >= nu on [0,T] solving the diagonal three-dimensional curvature-reaction ODE, with nu < 0, K > 0, and nu(0) >= -K. The proof derives R >= -3K/(1+Kt), and the conclusion preserves the Hamilton-Ivey defect w >= 0. Supporting results prove the expanded ODE derivative identity, scale crossing, the quantitative reaction lower bound (-nu)/9, and strict positivity. 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Independent mathematical review, the required Linux Landrun and hosted renderer replays, Palomar intake, and registration have not occurred." }, "canonical": { "repo": "arthur742ramos/lean-poincare-formalization-plan", "directory": "hamilton-ivey-reaction" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "Arthur742Ramos/lean-poincare-formalization-plan/lichnerowicz-obata/formalization.yaml", "repo": "Arthur742Ramos/lean-poincare-formalization-plan", "path": "lichnerowicz-obata/formalization.yaml", "directory": "lichnerowicz-obata", "origins": [ "palomar", "search" ], "url": "https://github.com/Arthur742Ramos/lean-poincare-formalization-plan/blob/HEAD/lichnerowicz-obata/formalization.yaml", "version": "v0.4", "missing": [], "name": "Lichnerowicz--Obata first eigenvalue and round-sphere rigidity", "description": "Formalizes attainment of the first positive smooth Laplace eigenvalue on closed connected smooth Riemannian manifolds of dimension n >= 2, its sharp lower bound n*K under Ric >= (n-1)*K with K > 0, and equality if and only if the manifold is smoothly Riemannian-isometric to the round sphere of radius 1/sqrt K. The independent statement constructs the metric-compatible torsion-free connection and explicitly defines Ricci curvature, the Riesz gradient, the div-grad Laplacian, eigenvalue minimality, and the radius-scaled induced round metric.", "authors": [ "Arthur Freitas Ramos" ], "maintainers": [ "Arthur Freitas Ramos" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "Géométrie des groupes de transformations", "id": "oclc:1212521", "authors": [ "André Lichnerowicz" ], "type": "book", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "Certain conditions for a Riemannian manifold to be isometric with a sphere", "id": "10.2969/jmsj/01430333", "authors": [ "Morio Obata" ], "type": "paper", "relationship": "formalizes", "endorsement": "not-contacted" } ], "related": [ { "id": "https://github.com/Arthur742Ramos/lean-poincare-formalization-plan/tree/3faf25aefc27842a77c37ca178e8a40a20bb20c7/almost-schur", "relationship": "builds-on", "note": "The almost-schur implementation and its RellichKondrachov sources are vendored byte-for-byte from this immutable snapshot under vendor/almost-schur, with the original license, notices, and provenance inventories preserved. They compile as local libraries into the root .lake/build, avoiding both sibling path dependencies and the registry's Git-subdirectory sandbox bug. It supplies geometric curvatu" }, { "id": "https://github.com/Arthur742Ramos/lean-poincare-formalization-plan/tree/12cebb809524d0cd185c6cd7bcb5b73d3562bce1/contracted-bianchi", "relationship": "builds-on", "note": "The inherited almost-Schur CurvatureVendor adaptation and its provenance inventory credit this immutable curvature core. The independent smooth tangent-extension commutator also descends from this construction. GeometryComparison proofs are implementation identifications, not assumed differential identities or a new selected curvature result." }, { "id": "https://github.com/abenenson/rellich-kondrachov/tree/70f85d4c1bf99c6e7d61e8be4daa6f3664d08d23", "relationship": "builds-on", "note": "The unchanged vendored implementation contains Adam Benenson's attributed 36-module Apache-2.0 adaptation, with its inventory and source notices. It supplies generic Sobolev compactness and measure transport, not spectral attainment, global sphere rigidity, or the selected theorem." }, { "id": "https://github.com/leanprover-community/mathlib4/tree/db584cd6d46c92f209a44c0f1c829460d327499d", "relationship": "builds-on", "note": "Pinned Mathlib infrastructure for manifolds, sphere charts, analysis, measure theory, linear algebra, topology, and ordinary differential equations. The exact revision is recorded in lake-manifest.json." } ], "classification": { "arxiv": [ "math.DG", "math.SP", "math.LO" ], "msc2020": [ "53C21", "58J50", "03B35" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "GPT-5" ], "framework": "Codex with Lean 4.33.0 and pinned Mathlib" }, { "method": "manual", "models": [], "framework": "Maintainer-directed theorem selection and delivery" } ], "strongest": "agent", "models": [ "GPT-5" ], "spend": "", "notes": "AI assistance is disclosed without attributing human authorship to the agent. Kernel and Comparator evidence do not constitute expert review or endorsement by the mathematical source authors." }, "scope": "Nonempty compact preconnected Hausdorff boundaryless smooth real Riemannian manifolds with finite-dimensional complete inner-product model space and the displayed smooth tangent-bundle and metric instances. The statement includes the Lindelof hypothesis, dimension n >= 2, positive K, and the actual pointwise Ricci lower bound. The first positive eigenvalue is characterized by a smooth nonconstant eigenfunction and minimality among all positive smooth eigenvalues. The round metric is explicit through the radius-scaled ambient inclusion of a diffeomorphism to the standard Euclidean unit sphere.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "LichnerowiczObataEntry.Geometry.lichnerowiczObata", "file": "LichnerowiczObataSolution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "LichnerowiczObata.lichnerowicz_obata", "file": "LichnerowiczObata/LichnerowiczObata.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "LichnerowiczObata.obata_round_metric_diffeomorph", "file": "LichnerowiczObata/ObataRoundDiffeomorph.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "LichnerowiczObata.round_metric_diffeomorph_exists_extremal_eigenfunction", "file": "LichnerowiczObata/RoundSphereConverse.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] } ], "comparator": false, "literature_dependencies": 0, "divergences": "The Laplacian uses the div-grad sign convention, so eigenfunctions satisfy laplacian f = -lambda*f. The source's nonnegative Laplacian has the opposite sign. The statement covers closed manifolds, not the full noncompact complete Hessian-rigidity theorem. It asserts attainment and minimality, not the entire spectral decomposition or multiplicities. The sphere's radius is expressed by scaling the standard unit-sphere inclusion; the exact derivative identity specifies its round Riemannian metric and the Diffeomorph type includes a smooth inverse. All nine geometric definitions are inlined inside the independently compared completeStatement body. Its equivalence to the original quantified statement is proved by rfl; closing the parameters avoids the registry renderer's dependent-type proxy bug without changing the mathematics. The one intentional Challenge theorem hole is excluded from the zero-hole implementation/Solution status and is never imported into their proof closure.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "Mechanical verification and source alignment are distinct from independent human review and registry acceptance. This is a new entry candidate, not a version update to any earlier submission. No intake or public registration is authorized by package preparation alone." }, "canonical": { "repo": "arthur742ramos/lean-poincare-formalization-plan", "directory": "lichnerowicz-obata" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-08-000006", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-08-000006", "commit": "5db025e6d20d8aad714c3d116714545d1823fd8e", "trust": "high", "theorems": 1, "date": "2026-09-08" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "Arthur742Ramos/lean-poincare-formalization-plan/ricci-scalar-comparison/formalization.yaml", "repo": "Arthur742Ramos/lean-poincare-formalization-plan", "path": "ricci-scalar-comparison/formalization.yaml", "directory": "ricci-scalar-comparison", "origins": [ "search" ], "url": "https://github.com/Arthur742Ramos/lean-poincare-formalization-plan/blob/HEAD/ricci-scalar-comparison/formalization.yaml", "version": "v0.4", "missing": [], "name": "Sharp Einstein scalar comparison and Ricci-flow lifespan", "description": "Formalizes the exact shrinking Ricci flow generated by positive Einstein initial data on a nonempty compact smooth manifold. The selected theorem proves the homothetic metric formula, the componentwise Ricci-flow equation, equality with the quadratic scalar-curvature comparison profile, the reciprocal Type-I-rate scalar law R(t) = n/(2(T-t)), divergence of scalar curvature to positive infinity as t approaches T from below, the lifetime identity T-t0 = n/(2*R0), the exact positive-definite time interval ending at t0 + 1/(2*lambda), and the obstruction to realizing the collapsed endpoint tensor ", "authors": [ "Arthur Freitas Ramos", "David Barros Hulak", "Ruy J. G. B. de Queiroz" ], "maintainers": [ "Arthur Freitas Ramos" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Lectures on the Ricci Flow", "id": "10.1017/CBO9780511721465", "authors": [ "Peter Topping" ], "type": "book", "relationship": "adapts", "endorsement": "not-contacted" }, { "title": "Three-manifolds with positive Ricci curvature", "id": "10.4310/jdg/1214436922", "authors": [ "Richard S. Hamilton" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" } ], "related": [ { "id": "https://github.com/Arthur742Ramos/lean-poincare-formalization-plan/tree/0cc7c31bf6e2dac5c0359a432f3d99803f563017/curvature", "relationship": "builds-on", "note": "Nineteen unchanged files and two narrowly adapted files are vendored from this immutable same-repository curvature and intrinsic Ricci-flow development. Exact blob identities, the adapted hashes, inventory and licensing are checked by scripts/check-vendored.py. The new scalar comparison, exact time-set and endpoint-obstruction results live in this package." }, { "id": "https://github.com/qinz1yang/differential-geometry/tree/1b535dd102b94cc42b107cca27059687888f08b3", "relationship": "independent", "note": "Independent Lean formalization associated with arXiv:2608.21502. It already includes the general scalar-curvature evolution and maximum principle infrastructure. This package does not import or vendor it and deliberately selects the narrower exact Einstein equality model instead of duplicating its general theorem." }, { "id": "https://github.com/leanprover-community/mathlib4/tree/db584cd6d46c92f209a44c0f1c829460d327499d", "relationship": "builds-on", "note": "Pinned Mathlib foundations for manifolds, tangent bundles, Riemannian metrics, covariant derivatives, finite-dimensional inner-product linear algebra, topology and real calculus." } ], "classification": { "arxiv": [ "math.DG", "math.AP", "math.LO" ], "msc2020": [ "53E20", "53C25", "03B35" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "OpenAI GPT-5" ], "framework": "Codex with Lean 4.33.0 and pinned Mathlib" }, { "method": "manual", "models": [], "framework": "Lean, Git and mathematical review" } ], "strongest": "agent", "models": [ "OpenAI GPT-5" ], "spend": "subscription-based", "notes": "Agent assistance is disclosed without treating the agent as an author. Lean, Comparator and NanoDa provide mechanical evidence rather than expert peer review, source-author endorsement or editorial acceptance." }, "scope": "Nonzero finite-dimensional complete real model spaces and nonempty compact Hausdorff smooth manifolds with C2 tangent bundle and Riemannian metric. A C1 covariant derivative is assumed, together with supplied global C3 tangent-vector extensions anchored at their base points, positive lambda, the torsion-free metric-compatible conditions, and the Einstein identity Ric(g0) = lambda*g0. 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Ricci and scalar curvature are explicit orthonormal contractions of a corrected commutator evaluated on supplied smooth extensions. Compactness is exposed to supply the intrinsic sigma-compact infrastructure, although the Einstein computation is pointwise. The family is a genuine Riemannian metric for every real time by a fallback definition, while all flow and homothetic-equality conclusions are restricted to times before extinction. The reciprocal profile and one-sided divergence describe scalar curvature only for this explicit equality model and do not establish the standard full-curvature-norm Type-I condition. The endpoint theorem rules out equality with the collapsed homothetic tensor; it does not establish uniqueness or exclude every nonhomothetic continuation.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "The statement, Challenge isolation, geometric bridges, overlap boundary, proof-hole policy, axiom surface, source identities and package metadata are mechanically checked. No independent human line-by-line proof review, Palomar editorial acceptance, intake or registration is claimed." }, "canonical": { "repo": "arthur742ramos/lean-poincare-formalization-plan", "directory": "ricci-scalar-comparison" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "Arthur742Ramos/lean-poincare-formalization-plan/schur-rigidity/formalization.yaml", "repo": "Arthur742Ramos/lean-poincare-formalization-plan", "path": "schur-rigidity/formalization.yaml", "directory": "schur-rigidity", "origins": [ "search" ], "url": "https://github.com/Arthur742Ramos/lean-poincare-formalization-plan/blob/HEAD/schur-rigidity/formalization.yaml", "version": "v0.4", "missing": [], "name": "Schur rigidity and geometric contracted Bianchi", "description": "Proves geometric contracted Bianchi for the actual Ricci and scalar fields, the divergence-free Einstein tensor, constancy of a differentiable Einstein factor on connected manifolds of dimension at least three, and the full constant-curvature tensor formula in dimension three. The Mathlib-only Challenge exposes the connection commutator, orthonormal contractions and actual bilinear covariant derivative. Trace differentiation and extension independence are proved rather than assumed.", "authors": [ "Arthur Freitas Ramos", "David Barros Hulak", "Ruy J. G. B. de Queiroz" ], "maintainers": [ "Arthur Freitas Ramos" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Lecture Notes on General Relativity", "id": "https://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll3.html", "authors": [ "Sean M. Carroll" ], "type": "lecture notes", "relationship": "adapts", "endorsement": "not-contacted" }, { "title": "Riemannian Geometry, Spring 2024", "id": "https://www.wim.uni-mannheim.de/media/Lehrstuehle/wim/schmidt/FSS2024/Riemannian_Geometry/Web/RGch5.html", "authors": [ "Ross Ogilvie" ], "type": "lecture notes", "relationship": "adapts", "endorsement": "not-contacted" }, { "title": "Differential geometry lecture notes", "id": "https://roderic.uv.es/bitstreams/99135e48-85cb-43b8-947d-0c0fbb028335/download", "authors": [ "Esther Cabezas-Rivas" ], "type": "lecture notes", "relationship": "adapts", "endorsement": "not-contacted" } ], "related": [ { "id": "https://github.com/Arthur742Ramos/lean-poincare-formalization-plan/tree/90d215d81d30a5f67922dce00dfa160b4878e1c5/contracted-bianchi", "relationship": "builds-on", "note": "Reuses the earlier contracted-bianchi formalization in repository Arthur742Ramos/lean-poincare-formalization-plan at immutable commit 90d215d81d30a5f67922dce00dfa160b4878e1c5, project path contracted-bianchi. Its twelve contracted-bianchi/PoincareCurvature/**/*.lean library files are copied unchanged into schur-rigidity/PoincareCurvature/, retaining their original contributor notices. This inherit" } ], "classification": { "arxiv": [ "math.DG", "math.LO" ], "msc2020": [ "53B20", "53C25", "03B35" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Codex" ], "framework": "Codex with Lean 4.33.0 and pinned Mathlib" } ], "strongest": "agent", "models": [ "Codex" ], "spend": "", "notes": "AI-assisted proofs and integration are disclosed in AGENT-CONTRIBUTION.md. Independent human review by each project author is not recorded." }, "scope": "Smooth Hausdorff real Riemannian manifolds with a finite-dimensional complete model space; tangent-bundle regularity through order three, connection regularity through order two and metric regularity through order two, with explicit lower-order instances. The tangent connection is torsion-free and metric-compatible. Supplied global C3 tangent-vector extensions are anchored at every point; the conclusions hold for every such family. Connectedness and dimension bounds occur only in rigidity theorems. The Einstein factor is manifold-differentiable. No compactness, geodesic completeness, sigma-com", "sorry_count": 4, "sorry_in_definitions": 0, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "Definitions use explicit anchored smooth extensions of tangent vectors and finite orthonormal contractions. Their equivalence to intrinsic connection curvature and covariant differentiation is proved. The Ricci and scalar derivative bridges are not hypotheses. Internal conditional assembly lemmas are discharged in the selected final proofs. The three-dimensional theorem gives one global constant for the full Riemann tensor, but not an isometry classification. Only Mathlib is an external Lean dependency; the earlier curvature closure is copied with provenance.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "none" ], "notes": "Prepared as a new entry under schur-rigidity/comparator.json with blank existing_id. This is a distinct differential-geometric rigidity package, not a re-selection of the earlier double-contracted curvature-derivative theorem. No editorial acceptance, mathematical originality or first-Lean priority claim is made. See VERIFICATION.md for mechanical evidence and SUBMISSION.md for the authorization gate." }, "canonical": { "repo": "arthur742ramos/lean-poincare-formalization-plan", "directory": "schur-rigidity" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "Arthur742Ramos/lean-poincare-formalization-plan/symmetric-tensor-heat/formalization.yaml", "repo": "Arthur742Ramos/lean-poincare-formalization-plan", "path": "symmetric-tensor-heat/formalization.yaml", "directory": "symmetric-tensor-heat", "origins": [ "search" ], "url": "https://github.com/Arthur742Ramos/lean-poincare-formalization-plan/blob/HEAD/symmetric-tensor-heat/formalization.yaml", "version": "v0.4", "missing": [], "name": "Short-time well-posedness for the symmetric tensor heat equation", "description": "Formalizes a finite-atlas Schauder-class existence, uniqueness and a priori estimate theorem for the inhomogeneous heat equation on symmetric covariant two-tensors over a nonempty closed smooth Riemannian manifold. The equation uses the actual Levi-Civita rough Laplacian tr_g(nabla^2 u), expanded in the Mathlib-only Challenge through induced tensor connections and an orthonormal trace. The selected statement exposes its finite nonempty atlas, partition weights and spanning frames, exact local-to-global data, solution and time-derivative reconstruction, faithful nontrivial coefficient represent", "authors": [ "Arthur Freitas Ramos", "David Barros Hulak", "Ruy J. G. B. de Queiroz" ], "maintainers": [ "Arthur Freitas Ramos" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "The Cauchy problem for fully nonlinear parabolic systems on manifolds", "id": "arXiv:1506.05030", "authors": [ "Hong Huang" ], "type": "paper", "relationship": "adapts", "endorsement": "not-contacted" }, { "title": "Three-manifolds with positive Ricci curvature", "id": "10.4310/jdg/1214436922", "authors": [ "Richard S. Hamilton" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Deforming metrics in the direction of their Ricci tensors", "id": "10.4310/jdg/1214509286", "authors": [ "Dennis M. DeTurck" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" } ], "related": [ { "id": "https://github.com/Arthur742Ramos/lean-poincare-formalization-plan/tree/d6ef7f253bb95fa44d1fe61c9b1a52e061ca0951/curvature", "relationship": "builds-on", "note": "Immutable same-repository source subproject reused through an exact mechanically extracted snapshot under vendor/curvature so the focused project is self-contained. The snapshot is compiled as a root library, with no nested Lake path dependency, so every generated file remains beneath the single writable package build root used by Palomar. All vendored implementation sources and contributor notice" }, { "id": "https://github.com/leanprover-community/mathlib4/tree/db584cd6d46c92f209a44c0f1c829460d327499d", "relationship": "builds-on", "note": "Pinned Mathlib foundations for manifolds, Riemannian bundles, covariant derivatives, continuous linear maps, finite-dimensional orthonormal bases, topology and real calculus." } ], "classification": { "arxiv": [ "math.DG", "math.AP", "math.LO" ], "msc2020": [ "53E20", "35K51", "35B45", "03B35" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "OpenAI GPT-5" ], "framework": "Codex with Lean 4.33.0 and pinned Mathlib" }, { "method": "manual", "models": [], "framework": "Lean, Git and mathematical review" } ], "strongest": "agent", "models": [ "OpenAI GPT-5" ], "spend": "subscription-based", "notes": "Agent assistance is disclosed without treating the agent as an author. Lean, Comparator and NanoDa provide mechanical evidence rather than expert peer review, source-author endorsement or editorial acceptance." }, "scope": "Nonzero finite-dimensional complete real model spaces and nonempty compact Hausdorff boundaryless smooth manifolds with a C2 Riemannian tangent bundle, C3 vector-bundle structure and a C1 Levi-Civita connection. Supplied C1/C2 induced connections on covariant two- and three-tensors define the actual orthonormal-trace rough Laplacian. For each 0 < alpha < 1, the conclusion constructs a positive short interval and a finite nonempty atlas. The compared proposition requires nonnegative weights summing to one, spanning local frames, normalized coordinates and time, positive source rescaling, exact ", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "SymmetricTensorHeatEntry.symmetricTensorHeatShortTimeWellPosed", "file": "TensorHeatSolution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] } ], "comparator": false, "literature_dependencies": 0, "divergences": "The classical vector-bundle theorem is represented through explicitly constructed finite-atlas coefficient spaces. Initial data and sources are quantified through those representations, and uniqueness is for the higher-coefficient witness satisfying the constructed classical-solution predicate. The Challenge now exposes exact atlas reconstruction of data, solution and time derivative, active spanning frames, nontrivial injective coefficient readouts, derivative identities, Holder bounds, geometric readouts, symmetry, the initial trace and actual rough-Laplacian PDE. It still does not assert surjectivity of the coefficient encoders onto every intrinsic Holder section. The theorem selects an existential short endpoint rather than Huang's arbitrary fixed finite interval. Merge commit a210bc382e4f1e34f8e7de2384cf3b86cf561938 had a mechanically passing but underconstrained selected statement and was editorially rejected; its evidence is historical and is not used to claim adequacy of this replacement. These boundaries avoid hiding missing representation or continuation theorems in informal wording.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "Mathematical repair is required before intake. The selected readout is symmetrized by construction, and coefficient uniqueness for fixed local traces has not been extended to geometric uniqueness for the ordinary reconstructed field. Data coverage also remains to be established. The historical candidate's editorial rejection is acknowledged rather than treated as fixed by metadata or mechanical checks. No independent human line-by-line proof review, Palomar editorial acceptance, fresh intake or registration is claimed." }, "canonical": { "repo": "arthur742ramos/lean-poincare-formalization-plan", "directory": "symmetric-tensor-heat" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "Arthur742Ramos/MarshallHallTheorem/formalization.yaml", "repo": "Arthur742Ramos/MarshallHallTheorem", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/Arthur742Ramos/MarshallHallTheorem/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Marshall Hall's theorem through finite cores", "description": "Formalizes Marshall Hall's finite-index free-factor theorem for finitely generated subgroups of finite-rank free groups. The proof extracts the finitely many suffix states of a subgroup generating set, completes the resulting partial generator actions to permutations of a finite state set, identifies the connected orbit with a finite core, and extends a geodesic spanning tree to obtain an inclusion-compatible free-product decomposition. It also exposes an explicit finite permutation separator for every finitely generated subgroup and element outside it, yielding the LERF consequence. The repos", "authors": [ "Arthur Freitas Ramos", "David Barros Hulak", "Ruy J. G. B. de Queiroz" ], "maintainers": [ "Arthur Freitas Ramos" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "Subgroups of Finite Index in Free Groups", "id": "10.4153/CJM-1949-017-2", "authors": [ "Marshall Hall, Jr." ], "type": "article", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "Topology of finite graphs", "id": "10.1007/BF02095993", "authors": [ "John R. Stallings" ], "type": "article", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Stallings foldings and subgroups of free groups", "id": "10.1006/jabr.2001.9033", "authors": [ "Ilya Kapovich", "Alexei Myasnikov" ], "type": "article", "relationship": "background", "endorsement": "not-contacted" }, { "title": "On the Number of Generators of a Free Product", "id": "10.1112/jlms/s1-18.1.12", "authors": [ "B. H. Neumann" ], "type": "article", "relationship": "formalizes", "endorsement": "not-contacted" } ], "related": [ { "id": "https://github.com/leanprover-community/mathlib4/blob/v4.32.0/Mathlib/GroupTheory/FreeGroup/NielsenSchreier.lean", "relationship": "builds-on", "note": "Uses Mathlib's free-group representation and Nielsen--Schreier infrastructure; the finite-core separation construction is developed in this repository." }, { "id": "https://github.com/leanprover-community/mathlib4/blob/v4.32.0/Mathlib/GroupTheory/FreeGroup/Reduce.lean", "relationship": "builds-on", "note": "Uses the canonical reduced-word encoding and normalization lemmas to enumerate suffix states of finite words." }, { "id": "https://github.com/leanprover-community/mathlib4/blob/v4.32.0/Mathlib/GroupTheory/Coset/Basic.lean", "relationship": "builds-on", "note": "Uses the left-coset quotient and its exact relation to subgroup membership." }, { "id": "https://github.com/leanprover-community/mathlib4/blob/v4.32.0/Mathlib/Logic/Equiv/Fintype.lean", "relationship": "builds-on", "note": "Uses `Equiv.extendSubtype` to complete each finite partial generator action to a permutation." }, { "id": "https://github.com/Arthur742Ramos/GraphCoveringTheory", "relationship": "other", "note": "The preceding covering-graph and Schreier development motivated the finite-state covering formulation, but is not imported by this repository." }, { "id": "https://github.com/Arthur742Ramos/HowsonIntersectionTheorem", "relationship": "other", "note": "A downstream Howson intersection development using the Hall and Kurosh layers; it is a separate formalization and is not the selected finite-separator result here." } ], "classification": { "arxiv": [ "math.GR" ], "msc2020": [ "20E07", "20F65", "05C25" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "GPT-5 Codex" ], "framework": "Lean 4 and Mathlib" } ], "strongest": "agent", "models": [ "GPT-5 Codex" ], "spend": "subscription-based", "notes": "The public statements are isolated from the finite-core implementation modules. The `lerf-comparator.json` surface selects the explicit finite permutation action, while `comparator.json` preserves the earlier Hall/Grushko surface." }, "scope": "The main Hall theorem covers every finite generating type alpha and every finitely generated subgroup H of FreeGroup alpha. It produces a subgroup K containing H with nonzero natural-number index, a complement, and a multiplicative equivalence H * complement ≃* K whose left injection is the actual inclusion of H. The same finite-state engine yields an explicit finite permutation separator for any element outside H, and its basepoint stabilizer yields the LERF witness. 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It proves existence and uniqueness of the Nash product maximizer, shows the maximizer satisfies the four axioms, and proves Nash's characterization: any bargaining solution satisfying the four axioms selects the Nash product maximizer. This is a reusable formalization artifact, not a claim of new mathematics.", "authors": [ "Arthur Freitas Ramos" ], "maintainers": [ "Arthur Freitas Ramos" ], "license": "BSD-3-Clause", "role": "", "substantive": "", "sources": [ { "title": "The Bargaining Problem", "id": "https://doi.org/10.2307/1907266", "authors": [ "John F. 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Challenge.lean repeats the compared statements with four deliberate proof holes; those statement holes are excluded from the implementation sorry_count.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "Two-player bargaining only; feasible utility sets are compact and convex subsets of the plane; the Nash solution maximizes the product of utility gains over individually rational outcomes.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "none" ], "notes": "The implementation library and Solution are proof-hole-free under the pinned local build. Challenge contains four deliberate Palomar statement holes, excluded from the implementation sorry_count. 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Pure Nash equilibrium is defined over utility preorders by the absence of a strict improvement; weak-inequality characterizations are restricted to linear orders. The library names the one-way improvement condition a generalized ordinal potential and reserves ordinal potential for the standard bidirectional strict-sign condition. It proves a global potential-maximizer certificate for pure Nash equilibrium, t", "authors": [ "Arthur Freitas Ramos" ], "maintainers": [ "Arthur Freitas Ramos" ], "license": "BSD-3-Clause", "role": "", "substantive": "", "sources": [ { "title": "Equilibrium points in n-person games", "id": "https://doi.org/10.1073/pnas.36.1.48", "authors": [ "John F. Nash, Jr." ], "type": "article", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "Non-cooperative Games", "id": "https://doi.org/10.2307/1969529", "authors": [ "John F. Nash" ], "type": "article", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Potential Games", "id": "https://doi.org/10.1006/game.1996.0044", "authors": [ "Dov Monderer", "Lloyd S. Shapley" ], "type": "article", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "A class of games possessing pure-strategy Nash equilibria", "id": "https://doi.org/10.1007/BF01737559", "authors": [ "Robert W. Rosenthal" ], "type": "article", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Flows and Decompositions of Games: Harmonic and Potential Games", "id": "https://doi.org/10.1287/moor.1110.0500", "authors": [ "Ozan Candogan", "Ishai Menache", "Asuman Ozdaglar", "Pablo A. 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The Lean development changes the definitions and proof architecture, separates generalized from standard ordinal potentials, and adds the explicit improvement-path interface. 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The upstream source is retained with an exact commit and license notice; this project contributes the MixedGame/MixedProfile bridge." } ], "classification": { "arxiv": [ "math.LO", "cs.LO" ], "msc2020": [ "91A10" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "GPT-5 Codex" ], "framework": "Lean 4 and Mathlib" } ], "strongest": "agent", "models": [ "GPT-5 Codex" ], "spend": "subscription-based", "notes": "AI assistance was used for proof engineering. The final definitions, statements, and proofs are checked by Lean's kernel." }, "scope": "The selected surface contains six theorems. First, every finite game with a standard ordinal potential has a pure Nash profile that globally maximizes the potential. Second, under the same standard bidirectional condition, pure Nash profiles are exactly potential local maxima. Third, the weaker generalized one-way condition still makes every nonempty better-response path strictly ascend and therefore rules out directed better-response cycles; the same finite argument gives weak acyclicity, so every legal starting profile reaches a pure Nash profile by some finite better-response path. A checke", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The Lean development uses a native Game structure, finite move sets, function-update deviations, a finite maximum over legal profiles, and explicit finite sums for the mixed layer. Its potential layer deliberately distinguishes the generalized one-way condition from the standard bidirectional condition and exposes finite better-response paths. The mixed layer formalizes native finite expected payoffs, support and Dirac lemmas, the normalized excess-map reduction, concrete matching pennies and rock-paper-scissors equilibria, and the closed/compact/convex continuous mixed-profile setup. The final existence theorem uses the included, attributed Scarf/Brouwer fixed-point source and contributes a native arbitrary-finite-type bridge; it is not a file-level translation of the Isabelle entry.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "none" ], "notes": "The implementation library and Solution are proof-hole-free under the pinned local build. Challenge contains two deliberate Palomar statement holes, excluded from the implementation sorry_count. The fixed-point core is included under the MIT notice in THIRD_PARTY_NOTICES.md and builds without proof placeholders. Hosted Comparator verification, Palomar editorial review, and registration are separate states and are not inferred from local compilation. This is a reusable formalization artifact, not a claim of a new economic theorem or first Lean proof of Nash existence. 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It proves that the Shapley value satisfies all four and proves uniqueness via unanimity games and Möbius inversion over the Boolean lattice. This is a reusable formalization artifact, not a claim of new mathematics.", "authors": [ "Arthur Freitas Ramos" ], "maintainers": [ "Arthur Freitas Ramos" ], "license": "BSD-3-Clause", "role": "", "substantive": "", "sources": [ { "title": "A Value for n-Person Games", "id": "https://doi.org/10.1515/9781400881970-018", "authors": [ "Lloyd S. Shapley" ], "type": "incollection", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "The Shapley Value: Essays in Honor of Lloyd S. Shapley", "id": "https://doi.org/10.1017/CBO9780511528446", "authors": [ "Alvin E. Roth" ], "type": "book", "relationship": "background", "endorsement": "not-contacted" } ], "related": [], "classification": { "arxiv": [ "math.LO", "cs.GT" ], "msc2020": [ "91A12" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "gpt-6-luna (Codex CLI)" ], "framework": "Lean 4 and Mathlib" } ], "strongest": "agent", "models": [ "gpt-6-luna (Codex CLI)" ], "spend": "subscription-based", "notes": "AI assistance was used for proof engineering. The final definitions, statements, and proofs are checked by Lean's kernel." }, "scope": "The library formalizes finite player types, normalized real-valued characteristic functions, the factorial Shapley formula, its four axioms, and the uniqueness characterization using unanimity games and Möbius inversion over the Boolean lattice. Challenge.lean repeats the compared statements with six deliberate proof holes; those statement holes are excluded from the implementation sorry_count.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "Finite player types; real-valued characteristic functions with v ∅ = 0 bundled in the Game structure; explicit finite sums.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "none" ], "notes": "The implementation library and Solution are proof-hole-free under the pinned local build. Challenge contains six deliberate Palomar statement holes, excluded from the implementation sorry_count. 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It gives an executable finite folding construction for every finite list of words in the rank-two free group, with exact subgroup-membership correctness. The least deterministic fold congruence is computed by finite exhaustive search; there is no complexity claim or implementation of core trimming, subgroup bases, intersections, or index algorithms.", "authors": [ "Arthur Freitas Ramos" ], "maintainers": [ "Arthur Freitas Ramos" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Topology of Finite Graphs", "id": "https://doi.org/10.1007/BF02095993", "authors": [ "John R. Stallings" ], "type": "paper", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "Stallings foldings and the subgroup structure of free groups", "id": "https://doi.org/10.1006/jabr.2001.9033", "authors": [ "Ilya Kapovich", "Alexei Myasnikov" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" } ], "related": [ { "id": "https://github.com/leanprover-community/mathlib4/tree/db584cd6d46c92f209a44c0f1c829460d327499d", "relationship": "builds-on", "note": "Immutable source project mathlib4 at commit db584cd6d46c92f209a44c0f1c829460d327499d; reuses Mathlib/GroupTheory/FreeGroup/Reduce.lean and its imported free-group definitions for FreeGroup (Fin 2), word evaluation, canonical reduced representatives, and the quotient soundness results. No earlier formalization project or same-repository subproject is vendored. The new selected result is Stallings.f" } ], "classification": { "arxiv": [ "math.GR", "math.LO" ], "msc2020": [ "20E05", "68Q45", "03B35" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "GPT-6 Codex" ], "framework": "Codex with Lean 4.33.0 and pinned Mathlib" } ], "strongest": "agent", "models": [ "GPT-6 Codex" ], "spend": "subscription-based", "notes": "AI-assisted Lean development with human-directed scope. The selected theorem is proved in Solution.lean; Challenge.lean contains the standalone Mathlib-only statement and its deliberate Comparator placeholder." }, "scope": "For a finite list S of signed words over two generators, constructs the finite flower graph, computes the intersection of all Boolean fold congruences, and returns a partial deterministic inverse automaton. Proves that its based-loop subgroup is exactly the subgroup generated by S and that its executable traversal of FreeGroup.toWord decides membership. The reference implementation is exhaustive rather than optimized. It retains auxiliary isolated states; core trimming, subgroup bases and rank, finite-index and index computation, subgroup intersections, and Hall-type finite-cover extensions ar", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The mathematical output is a finite inverse automaton with the same subgroup membership semantics as Stallings folding. For a compact reference implementation, folding is computed as the least deterministic equivalence relation over a finite state set instead of a sequence of local union-find folds. The construction does not trim the core and does not formalize the paper's basis, intersection, finite-index, or covering extensions.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "none" ], "notes": "The Lean development and compiled Challenge statement surface were self-audited. The repository contains a reproducible pinned Palomar mechanical preflight and hosted renderer replay workflow; its run report is the authority for the checked commit. No independent mathematical review or Palomar editorial review is claimed. Preparing the preflight does not initiate intake or registration." }, "canonical": { "repo": "arthur742ramos/stallings-folding", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-23-000004", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-23-000004", "commit": "117dde0a8415d3da1787e15ad22a6223a744432d", "trust": "high", "theorems": 1, "date": "2026-09-23" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "Arthur742Ramos/thomistic-causal-arguments-lean/formalization.yaml", "repo": "Arthur742Ramos/thomistic-causal-arguments-lean", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/Arthur742Ramos/thomistic-causal-arguments-lean/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Thomistic causal arguments in Lean", "description": "A standalone Lean 4 and Mathlib development of carrier-restricted causal and grounding regress arguments associated with the Second Way. It separates acyclicity, well-foundedness, chain termination, local and global foundation, and plural full grounding; proves the exact implication atlas; supplies finite checker and cycle-analyzer interfaces; and formalizes explicit Cohoe-style and Romero--Pérez-style reconstruction packages. The selected theorem proves that, under explicit essential-ordering, grounding, and foundation bridges, every caused target has a first cause that is not derivative in t", "authors": [ "Arthur Freitas Ramos", "David Barros Hulak", "Ruy J. G. B. de Queiroz" ], "maintainers": [ "Arthur Freitas Ramos" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Summa Theologiae, Prima Pars, question 2, article 3", "id": "https://www.newadvent.org/summa/100203.htm", "authors": [ "Thomas Aquinas" ], "type": "book", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "There Must Be a First: Why Thomas Aquinas Rejects Infinite, Essentially Ordered, Causal Series", "id": "https://doi.org/10.1080/09608788.2013.816934", "authors": [ "Caleb Cohoe" ], "type": "journal article", "relationship": "uses-as-background", "endorsement": "not-contacted" }, { "title": "New Remarks on the Cosmological Argument", "id": "https://doi.org/10.1007/s11153-012-9337-6", "authors": [ "Gustavo E. Romero", "Daniela Pérez" ], "type": "journal article", "relationship": "uses-as-background", "endorsement": "not-contacted" }, { "title": "Viciousness and Circles of Ground", "id": "https://doi.org/10.1111/meta.12072", "authors": [ "Ricki Bliss" ], "type": "journal article", "relationship": "uses-as-background", "endorsement": "not-contacted" }, { "title": "Grounding, Well-Foundedness, and Terminating Chains", "id": "https://doi.org/10.1007/s11406-022-00593-x", "authors": [ "Olley Pearson" ], "type": "journal article", "relationship": "uses-as-background", "endorsement": "not-contacted" } ], "related": [], "classification": { "arxiv": [ "math.LO", "cs.LO" ], "msc2020": [ "03B35" ] }, "automation": { "methods": [ { "method": "agent-assisted", "models": [ "GPT-5 Codex" ], "framework": "Lean 4 and Mathlib" }, { "method": "manual", "models": [], "framework": "Lean 4 and Mathlib" } ], "strongest": "manual", "models": [ "GPT-5 Codex" ], "spend": "subscription-based", "notes": "Generated compiler outputs are excluded from the repository. The Challenge placeholder is isolated from the proof-bearing Solution and production library." }, "scope": "The selected theorem is Palomar.cohoe_nonDerivative_firstCauseOf. It proves a non-derivative first-cause result for every caused target under an essentially ordered series, full foundation, endpoint, nonemptiness, and ancestry bridge package. The production library additionally proves the dependency/foundation interface, unrestricted and finite implication lattices, explicit separation models, executable finite checker soundness, cycle-candidate soundness, two reconstruction packages, and representative premise-deletion certificates. The development does not formalize a full theology, a unique", "sorry_count": 1, "sorry_in_definitions": 0, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The Lean package uses cohesive modules and proposition-level relations, with a separate Bool executable layer for finite input. The selected theorem preserves the explicit carrier, ancestry, essential-ordering, full-grounding, endpoint, nonemptiness, and ancestry-coverage premises; implementation names are idiomatic Lean names and the public registry surface is intentionally smaller than the complete research library.", "alignment": true, "original": false, "review": { "status": "self-assessed with clean-build and local comparator-shape checks", "bucket": "self-assessed", "reviewers": [ "Arthur Freitas Ramos" ], "notes": "The repository has been checked with pinned Lean/Mathlib builds, a self-contained Challenge import boundary, an identical Solution statement, explicit public-text hygiene checks, and deterministic finite regressions. Independent registry verification and editorial review remain pending." }, "canonical": { "repo": "arthur742ramos/thomistic-causal-arguments-lean", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "Arthur742Ramos/TopologicalComputationalPaths/formalization.yaml", "repo": "Arthur742Ramos/TopologicalComputationalPaths", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/Arthur742Ramos/TopologicalComputationalPaths/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Quotient Fundamental Groups, Winding, and Smith Exactness", "description": "Follow-up formalization of a concrete winding-word/Smith certificate for quotient fundamental groups of finite tori. The selected declaration is a kernel-checked topology-to-arithmetic synthesis: covering-map images, the sharp product-quotient boundary, rectangular cokernel exactness, and induced finite-torus matrix maps are combined with a fixed list syntax whose realization is concatenation of standard topological loops. Trace length is List.length, and the zero/one normal-form laws are proved from that syntax. Classical ingredients and priority are not claimed as new.", "authors": [ "Arthur Freitas Ramos", "Ruy J. G. B. de Queiroz", "Anjolina Grisi de Oliveira", "Tiago M. L. de Veras", "David Barros Hulak" ], "maintainers": [ "Arthur Freitas Ramos" ], "license": "MIT", "role": "substantive-development", "substantive": "", "sources": [ { "title": "Discreteness and Homogeneity of the Topological Fundamental Group", "id": "https://arxiv.org/abs/0904.4739", "authors": [ "Jack S. Calcut", "John D. McCarthy" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "On Fundamental Groups with the Quotient Topology", "id": "https://arxiv.org/abs/1304.6453", "authors": [ "Jeremy Brazas", "Paul Fabel" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Algebraic Topology", "id": "https://pi.math.cornell.edu/~hatcher/AT/AT.pdf", "authors": [ "Allen Hatcher" ], "type": "book", "relationship": "background", "endorsement": "n/a" }, { "title": "Finitely Generated Abelian Groups and Similarity of Matrices over a Field", "id": "https://doi.org/10.1007/978-1-4471-2730-7", "authors": [ "Christopher Norman" ], "type": "book", "relationship": "background", "endorsement": "n/a" }, { "title": "Topological Semantics for Scoped Computational Paths", "id": "https://github.com/Arthur742Ramos/ComputationalPathsLean/blob/2a2baa1f31c68f0e696021db91f8381dd2854652/paper/topological/main.tex", "authors": [ "Arthur Freitas Ramos", "Ruy J. G. B. de Queiroz", "Anjolina Grisi de Oliveira", "Tiago M. L. de Veras" ], "type": "paper", "relationship": "background", "endorsement": "participated" }, { "title": "Topological Smith exactness with a concrete winding-word bridge", "id": "", "authors": [ "Arthur Freitas Ramos", "Ruy J. G. B. de Queiroz", "Anjolina Grisi de Oliveira", "Tiago M. L. de Veras", "David Barros Hulak" ], "type": "original-proof", "relationship": "other", "endorsement": "participated" } ], "related": [ { "id": "https://github.com/Arthur742Ramos/TopologicalComputationalPaths/commit/e5d8495df101bab49bdd21b236a37040fced8097", "relationship": "builds-on", "note": "Accepted baseline package; not selected by this follow-up." }, { "id": "https://github.com/Arthur742Ramos/ComputationalPathsLean/commit/2a2baa1f31c68f0e696021db91f8381dd2854652", "relationship": "background", "note": "Pinned parent development and manuscript." }, { "id": "https://github.com/leanprover-community/mathlib4/commit/81a5d257c8e410db227a6665ed08f64fea08e997", "relationship": "builds-on", "note": "Pinned Mathlib APIs used by the challenge and replay." } ], "classification": { "arxiv": [ "math.AT", "math.GN" ], "msc2020": [ "55Q05", "54B15", "15A03", "20K01" ] }, "automation": { "methods": [ { "method": "manual", "models": [], "framework": "" }, { "method": "agent", "models": [ "Codex" ], "framework": "Lean 4.32.0 and Lake" } ], "strongest": "agent", "models": [ "Codex" ], "spend": "not tracked", "notes": "Lake, repository gates, and Comparator replay are recorded." }, "scope": "The selected declaration is TopologicalComputationalPathsFollowup.topological_smith_exactness in FollowupSolution.lean. It is a Nonempty TopologicalSmithExactnessCertificate with nine top-level fields: monodromy-stabilizer image equality for covering maps; the sharp product-quotient obstruction; rectangular winding-lattice cokernel short exactness; a complete finite-torus winding classifier with actual induced matrix-map image, injectivity, and surjectivity criteria; matrix composition; rectangular finite-cokernel profiles; arbitrary-rank Smith decomposition with explicit ZMod 0 free direction", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "TopologicalComputationalPathsFollowup.topological_smith_exactness", "file": "FollowupSolution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [ { "statement": "Circle and finite-product winding classification", "source": "Algebraic Topology" }, { "statement": "Smith invariant factors and determinant index", "source": "Finitely Generated Abelian Groups and Similarity of Matrices over a Field" }, { "statement": "Quotient-topological fundamental-group context", "source": "On Fundamental Groups with the Quotient Topology" } ] } ], "comparator": true, "literature_dependencies": 3, "divergences": "The selected theorem is a bundled certificate, not a claim that winding, fundamental groups, or Smith normal form were newly discovered. The concrete_winding_word_bridge is a fixed list language: trace length is List.length, realization is concatenation of actual standard torus loops, representation is proved against the winding standard representative, and matrix naturality is stated for the induced topological map. The shortest-trace conclusion is therefore derived for this syntax: zero winding has a zero-length word and nonzero winding has a one-letter word, with the lower bound proved from the homotopy/winding theorem. Smith image tests are stated for actual quotient maps and for geometric word representatives. Finite cardinality and prime-support statements are gated by finite-cokernel hypotheses; ZMod 0 records free factors. Classical ingredient results and repository-only APIs outside this declaration are not attributed to the selected theorem.", "alignment": true, "original": true, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "Local Lean and Comparator checks establish mechanical reproducibility, not independent peer review, novelty, or registration." }, "canonical": { "repo": "arthur742ramos/topologicalcomputationalpaths", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-29-000005", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-29-000005", "commit": "8254c40d0de03ff469c7c9c05087b8bc154e9a87", "trust": "high", "theorems": 1, "date": "2026-08-29" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "Arthur742Ramos/two-dimensional-lambda-calculus-lean/formalization.yaml", "repo": "Arthur742Ramos/two-dimensional-lambda-calculus-lean", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/Arthur742Ramos/two-dimensional-lambda-calculus-lean/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Certified naturality and the obstruction to unrestricted naturality in a typed two-dimensional calculus", "description": "We formalize the boundary of naturality in the labelled conversion calculus of A Theory of a Two-Dimensional Typed Lambda Calculus. Canonical evaluation of inductively presented homotopies constructs naturality squares, and transport along finite presentations preserves two-paths and recovers the naturality boundary. In contrast, an explicit pointwise family on Fin 3 admits no naturality two-cell or finite pasting: a six-element dihedral-group invariant separates its two boundary paths while respecting every declared two-cell generator. This independently reconstructed refutation is classical,", "authors": [ "Arthur F. Ramos" ], "maintainers": [ "Arthur F. Ramos" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "A Theory of a Two-Dimensional Typed Lambda Calculus", "id": "", "authors": [ "Daniel O. Martínez-Rivillas", "Arthur F. Ramos", "Ruy J. G. B. de Queiroz" ], "type": "paper", "relationship": "formalizes", "endorsement": "participated" } ], "related": [], "classification": { "arxiv": [ "math.LO" ], "msc2020": [ "03B40", "03B38" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "OpenAI GPT-6", "OpenAI GPT-5" ], "framework": "OpenAI Codex" } ], "strongest": "agent", "models": [ "OpenAI GPT-5", "OpenAI GPT-6" ], "spend": "subscription-based", "notes": "The mathematical source and Palomar scope were selected with the human; Codex produced and checked the Lean implementation and repository package." }, "scope": "Formalizes Definitions 2.1, 2.2, 3.1, 5.1, 5.2, 5.5, and 6.3; Theorems 5.3 and 6.2; finite-presentation transport results corresponding to Proposition 6.4 and Theorems 6.6 through 6.8; Theorem 7.3; Corollary 7.5; and its nontrivial-loop consequence. Also proves unrestricted-naturality nonexistence through a new classical invariant, rather than porting the claimed constructive Idris proof. Adds standalone typed syntax, representation adequacy, substitution soundness, and a parity-preserving conversion interpretation. Does not claim normalization or denotational completeness; a semantic higher-l", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The Lean encoding keeps all object types in one universe level at a time and places inspectable evidence one universe higher. As in the manuscript's Idris host encoding, beta and eta make endpoint expressions definitionally equal in the host theory while preserving their constructor tags as data. The new unrestricted-naturality refutation is a classical Lean proof via a noncommutative invariant; it is not a port or verification of the claimed constructive Idris proof, whose sources were not provided. Syntax.lean is an additional standalone encoding. Its adequacy theorem concerns raw-tree representation, not full abstraction or denotational completeness. The interpretation preserves conversion parity and the displayed separation. The core proper-subclass obstruction remains valid. Transport and direct naturality cover arbitrary finite global presentations. The transport comparison uses step naturality and is not an independent proof of it. No claim outside the stated scope is represented as a Lean theorem.", "alignment": true, "original": false, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [ "OpenAI Codex" ], "notes": "The agent audited every Step2 constructor against Definition 3.1, checked every evaluator branch and transport boundary, checked parity preservation case by case, kept the Palomar Challenge independent of candidate-local imports, and ran the repository verification scripts. Lean's #print axioms reports no axioms for the 35 audited core declarations; the core Comparator export configuration permits propext because the exported Lean environment declares it and NanoDa requires that declaration to be allowlisted. The 17 extension declarations are audited separately against exactly the standard propext, Quot.sound, and Classical.choice allowlist. Independent Comparator/NanoDa configurations preserve this distinction. submission-comparator.json selects the union of the core and extension declara" }, "canonical": { "repo": "arthur742ramos/two-dimensional-lambda-calculus-lean", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-14-000010", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-14-000010", "commit": "7aa1f961347b19deab3718521627b98f14357bc7", "trust": "high", "theorems": 33, "date": "2026-09-14" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "Arthur742Ramos/verified-regret-matching-lean/formalization.yaml", "repo": "Arthur742Ramos/verified-regret-matching-lean", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/Arthur742Ramos/verified-regret-matching-lean/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Verified regret matching and finite-time equilibrium computation", "description": "For finite nonempty action sets and unit-interval full-information payoffs, regret matching and regret matching+ have cumulative external regret at most the square root of the product of the number of actions and the horizon, against every fixed pure or mixed comparator. In finite two-player zero-sum matrix games, simultaneous self-play gives an explicit finite-time approximate Nash guarantee for uniformly averaged strategies. A finite-vector implementation is proved to reproduce the abstract averages. Square-root-free accuracy budgets and finite rational payoff inequalities certify approximat", "authors": [ "Arthur Freitas Ramos" ], "maintainers": [ "Arthur Freitas Ramos" ], "license": "BSD-3-Clause", "role": "", "substantive": "", "sources": [ { "title": "MIT 6.S890, Lecture 5: Learning in Games: Algorithms (Part I)", "id": "https://www.mit.edu/~gfarina/2024/6S890f24_L05_algo1/L05.pdf", "authors": [ "Gabriele Farina" ], "type": "other", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "MIT 6.S890, Lecture 4: Learning in Games: Foundations", "id": "https://www.mit.edu/~gfarina/2024/6S890f24_L04_regret/L04.pdf", "authors": [ "Gabriele Farina" ], "type": "other", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "Solving Large Imperfect Information Games Using CFR+", "id": "https://arxiv.org/abs/1407.5042", "authors": [ "Oskari Tammelin" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" } ], "related": [ { "id": "https://doi.org/10.1007/978-3-319-89884-1_20", "relationship": "independent", "note": "Verified Learning Without Regret is prior Coq work on multiplicative weights. No code is imported or copied. This package is not claimed to be the first verified no-regret algorithm." }, { "id": "https://github.com/gametheoryinlean/EconCSLib/tree/cef01c709a7d238f076b45818f2ff2629518efe3", "relationship": "other", "note": "Related minimax and learning work; inspected regret/approachability hits were staged knowledge documents, not a matching Lean RM implementation. The search is scoped and does not establish global priority." }, { "id": "https://github.com/danlyng/Econlib/tree/003655ccf010cdf44c4f67d6675167b54ce0e9df", "relationship": "other", "note": "Inspected Blackwell theory is contraction/Bellman, not approachability. No code is imported or copied." }, { "id": "https://github.com/Arthur742Ramos/correlated-equilibrium-lean", "relationship": "other", "note": "Earlier companion proves conditional external/swap-regret transport to approximate equilibria. The present standalone package proves a concrete RM/RM+ algorithm's quantitative bound; it does not depend on that project." } ], "classification": { "arxiv": [ "cs.GT" ], "msc2020": [ "91A10", "91A26" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "GPT-5 Codex" ], "framework": "Lean 4 and Mathlib" } ], "strongest": "agent", "models": [ "GPT-5 Codex" ], "spend": "", "notes": "AI assistance was used for proof engineering. The final definitions, statements, and proofs are checked by Lean." }, "scope": "Nine fixed-definition statements cover probability validity, both potential steps, pure and mixed external regret, actual finite-time self-play, executable refinement, generic squared accuracy budgets, rational-to-real certificate soundness, and an evaluated rational example.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "Full-information mixed expected rewards are maximized. The matrix entries lie in [0,1]; the column learner receives one minus its matrix cost. Updates are simultaneous, with zero initial scores and uniform zero-positive-mass fallback. The guarantee uses unweighted averages of the first T pre-update strategies, with T positive for all average guarantees. No sampled-action, internal/swap-regret, general-sum Nash, last-iterate, monotone-error, floating-point, or extensive-form CFR guarantee is supplied. Exact rational denominators may grow rapidly; the CLI restricts execution to T at most 8. The 800-round example is an a priori symbolic theorem, not an execution benchmark.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "Agent-assisted self-review only; no independent human or Palomar editorial approval is claimed. Nine deliberate theorem holes occur only in the isolated challenge and are excluded from the implementation counts. All algorithm definitions are fixed and there are no definition holes. See READINESS_REVIEW.md and VALIDATION.md for evidence and outstanding gates." }, "canonical": { "repo": "arthur742ramos/verified-regret-matching-lean", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-08-000002", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-08-000002", "commit": "52e2c3dbbd8308b11271daf2e4c5a80aebae1df7", "trust": "high", "theorems": 9, "date": "2026-09-08" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "ashebson/kinin-lean/formalization.yaml", "repo": "ashebson/kinin-lean", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/ashebson/kinin-lean/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Kinnim in Lean", "description": "A Lean 4 formalization of all sixteen units of Mishnah Kinnim, with a machine-checked physical optimization theorem for the Koppel--Reiss reading of Kinnim 3:2. For any list of pair counts grouped by owner, the priest has already acted on exactly half of the mixed birds above and half below. A physical world records one action level for every individual bird of every owner. Lean proves that every such world is compatible, by permutation, with the fixed half-split action list. Counting valid birds directly from each owner's closed-pair quota, the theorem proves that the exact minimum over all p", "authors": [ "Aryeh Shebson" ], "maintainers": [ "Aryeh Shebson" ], "license": "MIT", "role": "", "substantive": "", "sources": [ { "title": "Seder Kinnim: A Mathematical Commentary on Tractate Kinnim", "id": "", "authors": [ "Moshe Koppel" ], "type": "book", "relationship": "formalizes", "endorsement": "no-response" }, { "title": "A Mathematical Proof of Kinnim 3:2", "id": "doi:10.2307/40914641", "authors": [ "Philip Reiss" ], "type": "paper", "relationship": "formalizes", "endorsement": "not-contacted" } ], "related": [], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05A17", "90C27" ] }, "automation": { "methods": [ { "method": "agent", "models": [], "framework": "OpenAI Codex and ChatGPT" } ], "strongest": "agent", "models": [], "spend": "subscription-based; not tracked", "notes": "Human-directed, AI-assisted formalization. Coherent milestones were built, reviewed for sorry/custom axioms, and merged through GitHub pull requests." }, "scope": "The repository represents every unit of Mishnah Kinnim and proves the Kinnim 3:2 optimization result at substantially greater depth than the other units. The Comparator challenge records the direct physical theorem: compatibility of complete per-owner individual-bird assignments with the fixed action list, a universal smallest-majority lower bound, and a concrete assignment attaining it. Several other units begin from documented interpretive rules encoded as definitions; Lean proves their consequences but does not establish that those rules are the uniquely correct readings of the Hebrew sourc", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The compared theorem formalizes Reiss's optimization domain directly: complete owner-by-owner assignments of individual birds to the two action levels, with exactly half above globally. It names levels and closed-pair quotas rather than the full ritual vocabulary. Global temporal interleaving is quotiented by permutation; Lean proves compatibility with the concrete fixed action list and proves that owner-level reordering cannot affect the valid-bird count. Other known textual-to-model boundaries are enumerated in SOURCES.md.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "The author and AI agents checked statement/source alignment, compiled the full project repeatedly, searched for sorry/admit/custom axioms, and strengthened the central proof from aggregate arithmetic to physical ownership worlds. Moshe Koppel was sent a source archive for interpretive review; no response had been incorporated at submission time." }, "canonical": { "repo": "ashebson/kinin-lean", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-22-000007", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-22-000007", "commit": "d8ca202e6ccd655bd9c49dad5b9ee724cd3c2f54", "trust": "high", "theorems": 1, "date": "2026-08-22" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "AviKndr/universal-words/lean/formalization.yaml", "repo": "AviKndr/universal-words", "path": "lean/formalization.yaml", "directory": "lean", "origins": [ "palomar" ], "url": "https://github.com/AviKndr/universal-words/blob/HEAD/lean/formalization.yaml", "version": "v0.4", "missing": [], "name": "Universality of the words x^r y^s in symmetric groups (Kourovka Notebook Problem 10.32)", "description": "Answer to Problem 10.32 of the Kourovka Notebook, posed by Yu. I. Merzlyakov in 1986 and still open in the Notebook's 21st edition (2026), in both directions. Positive direction: there are constants C and N0 such that for all nonzero integers r, s with r or s odd, the word x^r y^s is universal on the symmetric group S_n (every permutation is a value) whenever n >= N0 and n >= C * log m(r,s), where m(r,s) is the product of the distinct primes dividing rs. This is the S_n analogue of the logarithmic Brenner-Evans-Silberger bound for A_n; the best previous S_n bound (Droste, 1986) was linear in m", "authors": [ "Avi Kondareddy" ], "maintainers": [ "Avi Kondareddy" ], "license": "MIT", "role": "", "substantive": "", "sources": [ { "title": "The universality of the words x^r y^s in symmetric groups: a logarithmic bound answering Kourovka Notebook Problem 10.32", "id": "", "authors": [ "Avi Kondareddy" ], "type": "original-proof", "relationship": "other", "endorsement": "" }, { "title": "Unsolved problems in group theory. The Kourovka Notebook, Problem 10.32", "id": "arXiv:1401.0300", "authors": [], "type": "other", "relationship": "background", "endorsement": "" }, { "title": "The universality of words x^r y^s in alternating groups", "id": "Proc. Amer. Math. Soc. 96 (1986), no. 1, 23-28", "authors": [ "J. L. Brenner", "R. J. Evans", "D. M. Silberger" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "On the universality of words for the alternating groups", "id": "Proc. Amer. Math. Soc. 96 (1986), no. 1, 18-22", "authors": [ "M. Droste" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Representation of permutations as products of two cycles", "id": "Discrete Math. 285 (2004), 323-327", "authors": [ "M. Herzog", "G. Kaplan", "A. Lev" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Cycles comme produit de deux permutations de classes données", "id": "Discrete Math. 38 (1982), 129-142", "authors": [ "G. Boccara" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Approximate formulas for some functions of prime numbers", "id": "Illinois J. Math. 6 (1962), 64-94", "authors": [ "J. B. Rosser", "L. Schoenfeld" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Sharper bounds for the Chebyshev functions theta(x) and psi(x)", "id": "Math. Comp. 29 (1975), 243-269", "authors": [ "J. B. Rosser", "L. Schoenfeld" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "The large sieve", "id": "Mathematika 20 (1973), 119-134", "authors": [ "H. L. Montgomery", "R. C. Vaughan" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Representation of an odd number as a sum of three primes", "id": "C. R. (Dokl.) Acad. Sci. URSS 15 (1937), 169-172", "authors": [ "I. M. Vinogradov" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Sieve Methods", "id": "Academic Press, 1974", "authors": [ "H. Halberstam", "H.-E. Richert" ], "type": "book", "relationship": "background", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.GR", "math.NT" ], "msc2020": [ "20B30", "20D60", "11P32", "05A05" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "claude-fable-5", "claude-opus-5" ], "framework": "Claude Code" } ], "strongest": "agent", "models": [ "claude-fable-5", "claude-opus-5" ], "spend": "", "notes": "Both the mathematics (the proof presented in the accompanying paper) and the Lean formalization were produced by the agent under the author's direction. No claim in this file has been verified by an independent human mathematician." }, "scope": "Whole-paper formalization: every proposition, lemma and theorem of the paper's proof (sections 2-8) is formalized and proved, with the eight classical inputs stated as named propositions and taken as hypotheses. The only deliberately unformalized step is Corollary 8.2's classical fact that theta(n) < n for infinitely many n, so the optimality *interpretation* (constant 1 unattainable) rests on that unformalized classical fact, while the construction itself (not_universal, L_eq_theta) is fully machine-checked.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "UniversalWords.boccara_three", "UniversalWords.boccara_three (Unconditional.lean only)", "UniversalWords.brun_titchmarsh", "UniversalWords.brun_titchmarsh (Unconditional.lean only)", "UniversalWords.hkl", "UniversalWords.hkl (Unconditional.lean only)", "UniversalWords.pi_lower", "UniversalWords.pi_lower (Unconditional.lean only)", "UniversalWords.pi_upper", "UniversalWords.pi_upper (Unconditional.lean only)", "UniversalWords.rs_window_mass", "UniversalWords.rs_window_mass (Unconditional.lean only)", "UniversalWords.sieve_r2", "UniversalWords.sieve_r2 (Unconditional.lean only)", "UniversalWords.vinogradov", "UniversalWords.vinogradov (Unconditional.lean only)", "propext" ], "nonstandard_axioms": [ "UniversalWords.boccara_three", "UniversalWords.boccara_three (Unconditional.lean only)", "UniversalWords.brun_titchmarsh", "UniversalWords.brun_titchmarsh (Unconditional.lean only)", "UniversalWords.hkl", "UniversalWords.hkl (Unconditional.lean only)", "UniversalWords.pi_lower", "UniversalWords.pi_lower (Unconditional.lean only)", "UniversalWords.pi_upper", "UniversalWords.pi_upper (Unconditional.lean only)", "UniversalWords.rs_window_mass", "UniversalWords.rs_window_mass (Unconditional.lean only)", "UniversalWords.sieve_r2", "UniversalWords.sieve_r2 (Unconditional.lean only)", "UniversalWords.vinogradov", "UniversalWords.vinogradov (Unconditional.lean only)" ], "results": [ { "declaration": "UniversalWords.kourovka_10_32_conditional", "file": "UniversalWords/Main.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [ { "statement": "RSWindowMassStatement (hypothesis of the implication)", "source": "Rosser-Schoenfeld 1962, Theorem 9; Rosser-Schoenfeld 1975, Corollary to Theorem 6" }, { "statement": "HKLStatement (hypothesis of the implication)", "source": "Herzog-Kaplan-Lev 2004, Theorem 7" }, { "statement": "BoccaraStatement (hypothesis of the implication)", "source": "Boccara 1982" }, { "statement": "PiUpperStatement, PiLowerStatement (hypotheses of the implication)", "source": "Rosser-Schoenfeld 1962, eqs. (3.6), (3.5)" }, { "statement": "BrunTitchmarshStatement (hypothesis of the implication)", "source": "Montgomery-Vaughan 1973, Theorem 2" }, { "statement": "VinogradovStatement (hypothesis of the implication)", "source": "Vinogradov 1937; Vaughan, The Hardy-Littlewood Method, 2nd ed., Theorem 3.4" }, { "statement": "SieveR2Statement (hypothesis of the implication)", "source": "Halberstam-Richert 1974, Theorem 3.11" } ] }, { "declaration": "UniversalWords.not_universal", "file": "UniversalWords/LowerBound.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "UniversalWords.kourovka_10_32", "file": "UniversalWords/Unconditional.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound", "UniversalWords.rs_window_mass", "UniversalWords.hkl", "UniversalWords.boccara_three", "UniversalWords.pi_upper", "UniversalWords.pi_lower", "UniversalWords.brun_titchmarsh", "UniversalWords.vinogradov", "UniversalWords.sieve_r2" ], "sorry_count": 0, "comparator": false, "literature": [ { "statement": "The eight classical statements above, assumed as axioms", "source": "See the source entries for each; documented on each axiom in UniversalWords/Unconditional.lean" } ] } ], "comparator": true, "literature_dependencies": 8, "divergences": "The eight classical inputs are stated in the specialised forms the proof uses rather than full generality: Rosser-Schoenfeld as a single window- mass bound and two pi(x) bounds; Boccara specialised to cycle type (q1,q2,q3,1) against a full cycle; Vinogradov in count form with an ineffective threshold; Halberstam-Richert with the constant absorbing the twin-prime constant. The formal constants (C, N0) are constructed from the hypotheses' existential constants rather than being the paper's numerically optimised C = 4.65, so the formal theorem asserts existence of constants rather than that particular value, and the threshold N0 inherits the ineffectivity of the Vinogradov hypothesis's threshold. Corollary 8.2's appeal to theta(n) < n infinitely often is not formalized.", "alignment": true, "original": true, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [ "Claude (adversarial referee rounds by the same agent family that produced the work)" ], "notes": "The paper underwent six adversarial referee rounds plus a communication audit, all performed by the producing agent (not independent). The Lean development is checked by `lake build` at the pinned toolchain and Mathlib revision; the axiom audit in UniversalWords.lean certifies the dependency sets stated above. No independent human peer review has taken place." }, "canonical": { "repo": "avikndr/universal-words", "directory": "lean" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-27-000006", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-27-000006", "commit": "90e3cbf743db257ecc13639f024c52fb197d6a48", "trust": "high", "theorems": 2, "date": "2026-08-27" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "AxiomMath/BerkovichUncu/formalization.yaml", "repo": "AxiomMath/BerkovichUncu", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/AxiomMath/BerkovichUncu/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "BerkovichUncu", "description": "`BerkovichUncu` is a formal proof of the statistic-transfer lemmas behind the bijection of *A bijective proof of a partition theorem of Berkovich and Uncu* (arXiv:2608.05142). It proves Lemma 2.5, Lemma 2.10, Lemma 2.14 and Lemma 2.17, which track the residue and multiplicity statistics through the four maps $\\varphi$, $\\Phi$, $\\psi^{-1}$ and Glaisher's $G$. The Lean files were generated by AxiomProver, Axiom Math's in-house theorem proving system.", "authors": [ "Kenny Lau" ], "maintainers": [ "Kenny Lau" ], "license": "MIT", "role": "substantive-development", "substantive": "", "sources": [ { "title": "A bijective proof of a partition theorem of Berkovich and Uncu", "id": "https://arxiv.org/abs/2608.05142", "authors": [ "Michal Mogielnicki", "Ken Ono", "Niels Voss", "Jujian Zhang" ], "type": "article", "relationship": "formalizes", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05A17", "05A19", "68V15" ] }, "automation": { "methods": [ { "method": "autonomous", "models": [ "AxiomProver" ], "framework": "" } ], "strongest": "autonomous", "models": [ "AxiomProver" ], "spend": "", "notes": "" }, "scope": "Lemma 2.5, Lemma 2.10, Lemma 2.14 and Lemma 2.17, for every $n$. The bijections themselves are supplied to `problem.lean` as data, and the assembly of the four lemmas into the bijection $F$ and the resulting cardinality statement is carried out in `problem.lean`.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "BerkovichUncu.phi_stat", "file": "BerkovichUncu/solution.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "BerkovichUncu.Phi_stat", "file": "BerkovichUncu/solution.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "BerkovichUncu.psiInv_oddMultiplicity", "file": "BerkovichUncu/solution.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "BerkovichUncu.G_oddPart", "file": "BerkovichUncu/solution.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "", "alignment": true, "original": false, "review": { "status": "author-verified", "bucket": "author-verified", "reviewers": [ "Kenny Lau" ], "notes": "The formal challenge in `BerkovichUncu/problem.lean` was checked against the source paper by an author of that paper." }, "canonical": { "repo": "axiommath/berkovichuncu", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "AxiomMath/Biswal/formalization.yaml", "repo": "AxiomMath/Biswal", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/AxiomMath/Biswal/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Biswal", "description": "`Biswal` is a formal proof of the coefficient results for Chebyshev quotients in *Chebyshev quotients, Demazure multiplicities, and Dyck-path models* (arXiv:2604.25246). For the generating function $F_{\\lambda,m,\\mu}(x) = \\sum_{r \\ge 0} a_r x^r$ it proves Theorem 1.1, which splits the quotient into a polynomial case and an eventually positive case; Theorem 1.2, the signed summation formula for $a_r$; Theorem 1.3, the factorization into Dyck-path counts $D_m(a,b;u)$ and the resulting nonnegativity; and Lemma 3.3, the coefficient formula for $p_n$. 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Theorem 1.1 (2)(b) is stated over $\\mathbb{Q}$, where the positivity of $a_r$ can be expressed.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "genFun_eq_one_of_m_eq_one", "file": "Biswal/theorem1/solution.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "genFun_is_polynomial", "file": "Biswal/theorem1/solution.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "genFun_coeff_eventually_pos", "file": "Biswal/theorem1/solution.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "polyP_coeff", "file": "Biswal/theorem2and3/solution.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "thm_comb_cauchy", "file": "Biswal/theorem2and3/solution.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "thm_comb", "file": "Biswal/theorem2and3/solution.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "thm_manifest_factorization", "file": "Biswal/theorem2and3/solution.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "thm_manifest_coeff_formula", "file": "Biswal/theorem2and3/solution.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "thm_manifest_D_nonneg", "file": "Biswal/theorem2and3/solution.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "thm_manifest_coeff_nonneg", "file": "Biswal/theorem2and3/solution.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "", "alignment": true, "original": false, "review": { "status": "author-verified", "bucket": "author-verified", "reviewers": [ "Kenny Lau" ], "notes": "The formal challenge in `Biswal/theorem1/problem.lean` and `Biswal/theorem2and3/problem.lean` was checked against the source paper by an author of that paper." }, "canonical": { "repo": "axiommath/biswal", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "AxiomMath/challenge_3/formalization.yaml", "repo": "AxiomMath/challenge_3", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/AxiomMath/challenge_3/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Challenge_3", "description": "`Challenge_3` is a formal proof of the appendix results of *Dominant Zeros of Nekrasov-Okounkov Polynomials* (arXiv:2606.15394). Writing $q_n$ for the coefficients of $Q(t) = 1 - P(t)S(t)$ and $X^{\\pm}_N$ for the even- and odd-area marked rectangles, it proves the signed count $q_n = |X^+_N| - |X^-_N|$ of Lemma B.2, the injection $X^-_N \\hookrightarrow X^+_N$ of Lemma B.3, the nonnegativity $q_n \\ge 0$ of Theorem B.1, and the strict positivity of the coefficients of $1/S(t)$ of Corollary B.4. The Lean files were generated by AxiomProver, Axiom Math's in-house theorem proving system.", "authors": [ "Kenny Lau" ], "maintainers": [ "Kenny Lau" ], "license": "MIT", "role": "substantive-development", "substantive": "", "sources": [ { "title": "Dominant Zeros of Nekrasov--Okounkov Polynomials", "id": "https://arxiv.org/abs/2606.15394", "authors": [ "Bernhard Heim", "Markus Neuhauser", "with an appendix by Ken Ono" ], "type": "article", "relationship": "formalizes", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.CO", "math.NT" ], "msc2020": [] }, "automation": { "methods": [ { "method": "autonomous", "models": [ "AxiomProver" ], "framework": "" } ], "strongest": "autonomous", "models": [ "AxiomProver" ], "spend": "", "notes": "" }, "scope": "Lemma B.2, Lemma B.3 and Theorem B.1 for every $n \\ge 1$, in the `lemma_b2` run, and Corollary B.4 for every $m \\ge 0$, in the `lemma_b3` run. The `lemma_b3` run restates and reproves Lemma B.2, Lemma B.3 and Theorem B.1 in its own module, since it was given them as input.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "ColoredRectangles.q_signed_count", "file": "Challenge_3/lemma_b2/solution.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "ColoredRectangles.injection", "file": "Challenge_3/lemma_b2/solution.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "ColoredRectangles.qn_nonneg", "file": "Challenge_3/lemma_b2/solution.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "ColoredRectangles.cQ_pos", "file": "Challenge_3/lemma_b3/solution.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "", "alignment": true, "original": false, "review": { "status": "author-verified", "bucket": "author-verified", "reviewers": [ "Kenny Lau" ], "notes": "The formal challenge in `Challenge_3/lemma_b2/problem.lean` and `Challenge_3/lemma_b3/problem.lean` was checked against the source paper by an author of that paper." }, "canonical": { "repo": "axiommath/challenge_3", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "AxiomMath/fel-polynomial/formalization.yaml", "repo": "AxiomMath/fel-polynomial", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/AxiomMath/fel-polynomial/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "FelConjecture", "description": "`FelConjecture` is a formal proof of Fel's conjecture on syzygies of numerical semigroups. For a numerical semigroup $S = \\langle d_1, \\dots, d_m \\rangle \\subseteq \\mathbb{N}$ with $\\gcd(d_1, \\dots, d_m) = 1$ and gap set $\\Delta = \\mathbb{N} \\setminus S$, it proves that for every $p \\ge 0$, $K_p(S) = \\sum_{r=0}^{p} \\binom{p}{r} T_{p-r}(\\sigma) G_r(S) + \\frac{2^{p+1}}{p+1} T_{p+1}(\\delta)$, where $G_r(S) = \\sum_{g \\in \\Delta} g^r$ are the gap power sums, $K_p(S)$ is the normalized alternating power sum of the syzygy degrees appearing in the Hilbert numerator $Q_S$, and $T_n$ are Fel's universal", "authors": [ "Kenny Lau" ], "maintainers": [ "Kenny Lau" ], "license": "MIT", "role": "substantive-development", "substantive": "", "sources": [ { "title": "Fel's Conjecture on Syzygies of Numerical Semigroups", "id": "https://arxiv.org/abs/2602.03716", "authors": [ "Evan Chen", "Chris Cummins", "GSM", "Dejan Grubisic", "Leopold Haller", "Letong Hong", "Andranik Kurghinyan", "Kenny Lau", "Hugh Leather", "Seewoo Lee", "Aram Markosyan", "Ken Ono", "Manooshree Patel", "Gaurang Pendharkar", "Vedant Rathi", "Alex Schneidman", "Volker Seeker", "Shubho Sengupta", "Ishan Sinha", "Jimmy Xin", "Jujian Zhang" ], "type": "article", "relationship": "formalizes", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.CO", "math.AC", "math.NT" ], "msc2020": [ "20M14", "11P81" ] }, "automation": { "methods": [ { "method": "autonomous", "models": [ "AxiomProver" ], "framework": "" } ], "strongest": "autonomous", "models": [ "AxiomProver" ], "spend": "", "notes": "" }, "scope": "Fel's conjecture for every numerical semigroup $S$, every generating set of $S$, and every $p \\ge 0$. The Hilbert numerator $Q_S$ and the gap polynomial $\\Phi_S$ are constructed explicitly in `problem.lean`.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "NumericalSemigroup.semigroupGapDecomposition", "file": "FelConjecture/solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "NumericalSemigroupGenerators.numeratorIdentity", "file": "FelConjecture/solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "fels_conjecture", "file": "FelConjecture/solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "", "alignment": true, "original": false, "review": { "status": "author-verified", "bucket": "author-verified", "reviewers": [ "Kenny Lau" ], "notes": "The formal problem statement in `FelConjecture/problem.lean` was checked against the source paper by an author of that paper." }, "canonical": { "repo": "axiommath/fel-polynomial", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "AxiomMath/HJOa3Small/formalization.yaml", "repo": "AxiomMath/HJOa3Small", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/AxiomMath/HJOa3Small/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "hjoa3small", "description": "`hjoa3small` is a formal proof of the small cases at $a = 3$ of the Huang-Jiang-Oblomkov conjecture treated in *Algebraic geometric framework of Rogers-Ramanujan identities* (arXiv:2608.15219). It proves Theorem 6, the sum-to-sum conjecture at $b = 4, 5, 7, 8$; Theorem 7, the sum-to-sum conjecture at $q = 1$; and Theorem 3, the HJO conjecture for $a = 3$ and $b = 4, 5, 7, 8$, assuming Warnaar's Theorem 1.1. The Lean files were generated by AxiomProver, Axiom Math's in-house theorem proving system.", "authors": [ "Kenny Lau" ], "maintainers": [ "Kenny Lau" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "Algebraic geometric framework of Rogers--Ramanujan identities", "id": "https://arxiv.org/abs/2608.15219", "authors": [ "Yifeng Huang", "Kenny Lau", "Ken Ono", "Peter Paule" ], "type": "article", "relationship": "formalizes", "endorsement": "" }, { "title": "The $A_2$ Andrews-Gordon identities and cylindric partitions", "id": "https://arxiv.org/abs/2111.07550", "authors": [ "S. O. Warnaar" ], "type": "article", "relationship": "background", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.NT", "math.CO", "math.RT" ], "msc2020": [] }, "automation": { "methods": [ { "method": "autonomous", "models": [ "AxiomProver" ], "framework": "" } ], "strongest": "autonomous", "models": [ "AxiomProver" ], "spend": "", "notes": "" }, "scope": "Theorem 6 for $b = 4, 5, 7, 8$, Theorem 7 for every level, and Theorem 3 conditional on Warnaar's Theorem 1.1, which `Challenge/Basic.lean` assumes whole as `AndrewsGordon.Theorem11` rather than in the four instances the proof uses.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "HJOA3.Challenge.thm_sum", "file": "Solution/Basic.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "HJOA3.Challenge.thm_q1", "file": "Solution/Basic.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "HJOA3.Challenge.thm_main", "file": "Solution/Basic.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [ { "statement": "`AndrewsGordon.Theorem11`: Warnaar's Theorem 1.1, both families at every level - the modulus-$(3\\mathbf{k}+2)$ identity (1.10) and the modulus-$(3\\mathbf{k}+1)$ identity (1.11).", "source": "The $A_2$ Andrews-Gordon identities and cylindric partitions, Theorem 1.1, (1.10)-(1.11)" } ] } ], "comparator": true, "literature_dependencies": 1, "divergences": "", "alignment": true, "original": false, "review": { "status": "author-verified", "bucket": "author-verified", "reviewers": [ "Kenny Lau" ], "notes": "The formal challenge in `Challenge/Basic.lean` was checked against the source paper by an author of that paper." }, "canonical": { "repo": "axiommath/hjoa3small", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "AxiomMath/kaprekar4/formalization.yaml", "repo": "AxiomMath/kaprekar4", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/AxiomMath/kaprekar4/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Kaprekar4", "description": "`Kaprekar4` is a formal proof of the structural classification of four-digit Kaprekar dynamics in odd bases, and of the cycle counts it yields, from *Four-digit Kaprekar dynamics in odd bases* (arXiv:2606.20439). It proves Theorem 1.1, Corollary 1.2 and Corollary 1.3. The Lean files were generated by AxiomProver, Axiom Math's in-house theorem proving system.", "authors": [ "Kenny Lau" ], "maintainers": [ "Kenny Lau" ], "license": "MIT", "role": "substantive-development", "substantive": "", "sources": [ { "title": "Four-digit Kaprekar dynamics in odd bases", "id": "https://arxiv.org/abs/2606.20439", "authors": [ "Evan Chen", "Ken Ono", "Richard E. Schwartz", "Dinesh S. Thakur" ], "type": "article", "relationship": "formalizes", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.NT", "math.CO" ], "msc2020": [ "11A63", "05A15", "37P99" ] }, "automation": { "methods": [ { "method": "autonomous", "models": [ "AxiomProver" ], "framework": "" } ], "strongest": "autonomous", "models": [ "AxiomProver" ], "spend": "", "notes": "" }, "scope": "Theorem 1.1, Corollary 1.2 and Corollary 1.3, for every odd base $B > 3$. The dynamics are formalized in the difference coordinates $(d_1, d_2)$, which `problem.lean` justifies from the four-digit description.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "thm_structural", "file": "Kaprekar4/solution.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "cor_length", "file": "Kaprekar4/solution.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "cor_prime", "file": "Kaprekar4/solution.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "", "alignment": true, "original": false, "review": { "status": "author-verified", "bucket": "author-verified", "reviewers": [ "Kenny Lau" ], "notes": "The formal challenge in `Kaprekar4/problem.lean` was checked against the source paper by an author of that paper." }, "canonical": { "repo": "axiommath/kaprekar4", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "AxiomMath/lattice-triangle/formalization.yaml", "repo": "AxiomMath/lattice-triangle", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/AxiomMath/lattice-triangle/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "LatticeTriangle", "description": "`LatticeTriangle` is a formal proof of the analytic engine behind the paucity of lattice triangles. For $\\eta \\in (0, 1/6)$ and $\\theta \\in (0, 1)$, it proves that the proportion of pairs $(p, q)$ in the truncated obtuse region $H_n(\\eta)$ satisfying $\\gcd(q, P^+(n)) = 1$ and $S(p, q) < 5$ tends to $0$ as $n \\to \\infty$ along those $n$ with $P^+(n) \\ge n^\\theta$, where $P^+(n)$ is the largest prime factor of $n$ and $S(p, q) = \\#\\{a \\in (\\mathbb{Z}/n)^\\times : ap \\in I_{2p-1},\\ aq \\in I_{2q-1}\\}$. The Lean files were generated by AxiomProver, Axiom Math's in-house theorem proving system.", "authors": [ "Kenny Lau" ], "maintainers": [ "Kenny Lau" ], "license": "MIT", "role": "substantive-development", "substantive": "", "sources": [ { "title": "On the paucity of lattice triangles", "id": "https://arxiv.org/abs/2603.23928", "authors": [ "David Kurniadi Angdinata", "Evan Chen", "Ken Ono", "Jiaxin Zhang", "Jujian Zhang" ], "type": "article", "relationship": "formalizes", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.DS", "math.CO", "math.NT" ], "msc2020": [ "37E35", "30F60" ] }, "automation": { "methods": [ { "method": "autonomous", "models": [ "AxiomProver" ], "framework": "" } ], "strongest": "autonomous", "models": [ "AxiomProver" ], "spend": "", "notes": "" }, "scope": "Theorem 6.1, which the paper calls the main engine, for all $\\eta \\in (0, 1/6)$ and $\\theta \\in (0, 1)$. `problem.lean` states it as a self-contained analytic claim.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "analyticEngine_lower_bound", "file": "LatticeTriangle/solution.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "", "alignment": true, "original": false, "review": { "status": "author-verified", "bucket": "author-verified", "reviewers": [ "Kenny Lau" ], "notes": "The formal problem statement in `LatticeTriangle/problem.lean` was checked against the source paper by an author of that paper." }, "canonical": { "repo": "axiommath/lattice-triangle", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "AxiomMath/parity-differential/formalization.yaml", "repo": "AxiomMath/parity-differential", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/AxiomMath/parity-differential/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "ParityDifferential", "description": "`ParityDifferential` is a formal proof of the combinatorial identity behind the parity of $k$-differentials in genus zero and one. For odd $k \\ge 3$ and $m = (k-1)/2$, let $N_k(n)$ count the pairs $(b_1, b_2)$ with $1 \\le b_1, b_2 \\le m$, $b_1 + b_2 \\ge m + 1$ and $b_2 \\equiv n b_1 \\pmod k$, and let $F_k(a) = \\sum_{i=1}^{m} \\lfloor (ai + m)/k \\rfloor$. It proves $N_k(n) = F_k(n+1) - F_k(n)$. The Lean files were generated by AxiomProver, Axiom Math's in-house theorem proving system.", "authors": [ "Kenny Lau" ], "maintainers": [ "Kenny Lau" ], "license": "MIT", "role": "substantive-development", "substantive": "", "sources": [ { "title": "Parity of $k$-differentials in genus zero and one", "id": "https://arxiv.org/abs/2602.03722", "authors": [ "Dawei Chen", "Evan Chen", "Kenny Lau", "Ken Ono", "Jujian Zhang" ], "type": "article", "relationship": "formalizes", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.NT", "math.AG", "math.GT" ], "msc2020": [ "11A07", "14H10", "32G15" ] }, "automation": { "methods": [ { "method": "autonomous", "models": [ "AxiomProver" ], "framework": "" } ], "strongest": "autonomous", "models": [ "AxiomProver" ], "spend": "", "notes": "" }, "scope": "Lemma 2.5, the combinatorial core of the proof of Theorem 1.2, for every odd $k \\ge 3$ and every $n$. `problem.lean` defines $N_k$ and $F_k$ outright and states the identity between them.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "main_theorem", "file": "ParityDifferential/solution.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "", "alignment": true, "original": false, "review": { "status": "author-verified", "bucket": "author-verified", "reviewers": [ "Kenny Lau" ], "notes": "The formal problem statement in `ParityDifferential/problem.lean` was checked against the source paper by an author of that paper." }, "canonical": { "repo": "axiommath/parity-differential", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "AxiomMath/partial-regularity/formalization.yaml", "repo": "AxiomMath/partial-regularity", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/AxiomMath/partial-regularity/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "PartialRegularity", "description": "`PartialRegularity` is a formal proof of the main theorem of *Almost all primes are partially regular* (arXiv:2602.05090): for fixed $\\alpha > 1/2$, the number of primes $p \\le X$ that are not $M_\\alpha(p)$-regular is $O(X/(\\log X)^{2\\alpha})$ as $X \\to +\\infty$. The Lean files were generated by AxiomProver, Axiom Math's in-house theorem proving system.", "authors": [ "Kenny Lau" ], "maintainers": [ "Kenny Lau" ], "license": "MIT", "role": "substantive-development", "substantive": "", "sources": [ { "title": "Almost all primes are partially regular", "id": "https://arxiv.org/abs/2602.05090", "authors": [ "Evan Chen", "Chris Cummins", "Ben Eltschig", "Dejan Grubisic", "Leopold Haller", "Letong Hong", "Andranik Kurghinyan", "Kenny Lau", "Hugh Leather", "Seewoo Lee", "Aram Markosyan", "Ken Ono", "Manooshree Patel", "Gaurang Pendharkar", "Vedant Rathi", "Alex Schneidman", "Volker Seeker", "Shubho Sengupta", "Ishan Sinha", "Jimmy Xin", "Jujian Zhang" ], "type": "article", "relationship": "formalizes", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11Y40", "11R18" ] }, "automation": { "methods": [ { "method": "autonomous", "models": [ "AxiomProver" ], "framework": "" } ], "strongest": "autonomous", "models": [ "AxiomProver" ], "spend": "", "notes": "" }, "scope": "Theorem 1.1 for every $\\alpha > 1/2$. The bound is stated as an `IsBigO` relation along `atTop` rather than with the explicit constant $C_\\alpha$ of the paper. The main run counts primes $p \\le X$ over real $X$ with a `Finset` filter; the `extension` run counts odd primes over natural $X$ with a `Set`.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "irregular_primes_asymptotic", "file": "PartialRegularity/solution.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "irregularPrimes_isBigO", "file": "extension/ext_solution.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "", "alignment": true, "original": false, "review": { "status": "author-verified", "bucket": "author-verified", "reviewers": [ "Kenny Lau" ], "notes": "The formal challenge in `PartialRegularity/problem.lean` and `extension/ext_problem.lean` was checked against the source paper by an author of that paper." }, "canonical": { "repo": "axiommath/partial-regularity", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "AxiomMath/PartitionElliptic/formalization.yaml", "repo": "AxiomMath/PartitionElliptic", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/AxiomMath/PartitionElliptic/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "PartitionElliptic", "description": "`PartitionElliptic` is a formal proof of the two identities of *Two Key Identities for Formal Verification*, the spec accompanying Ken Ono's *The partition function and elliptic curves*. It proves Key Formula 1 and Key Formula 2. The Lean files were generated by AxiomProver, Axiom Math's in-house theorem proving system.", "authors": [ "Kenny Lau" ], "maintainers": [ "Kenny Lau" ], "license": "MIT", "role": "substantive-development", "substantive": "", "sources": [ { "title": "Two Key Identities for Formal Verification", "id": "", "authors": [], "type": "article", "relationship": "formalizes", "endorsement": "" } ], "related": [], "classification": { "arxiv": [], "msc2020": [] }, "automation": { "methods": [ { "method": "autonomous", "models": [ "AxiomProver" ], "framework": "" } ], "strongest": "autonomous", "models": [ "AxiomProver" ], "spend": "", "notes": "" }, "scope": "Key Formula 1 and Key Formula 2, as identities between complex numbers. `problem.lean` takes the values of the modular objects at a fixed $\\tau$ as given data subject to $E_4 \\ne 0$, $J \\ne 1728$ and $\\Phi_Y \\ne 0$, exactly as the spec's ground data does.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "key_formula_one", "file": "output/solution.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "key_formula_two", "file": "output/solution.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "", "alignment": true, "original": false, "review": { "status": "author-verified", "bucket": "author-verified", "reviewers": [ "Kenny Lau" ], "notes": "The formal challenge in `output/problem.lean` was checked against the source paper by an author of that paper." }, "canonical": { "repo": "axiommath/partitionelliptic", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "AxiomMath/PrimeGapsLib/formalization.yaml", "repo": "AxiomMath/PrimeGapsLib", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/AxiomMath/PrimeGapsLib/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "PrimeGapsLib", "description": "`PrimeGapsLib` is an evolving codebase aimed at housing results about gaps in primes. The flagship result thus far is that pairs of primes are separated by at most 246 infinitely often, conditional on the Bombieri-Vinogradov theorem. The 246 proof was generated collaboratively between human formalisers at Axiom Math and AxiomProver, their in-house theorem proving system. Community feedback and contributions are welcome!", "authors": [ "Evan Chen", "Sidharth Hariharan", "Kenny Lau", "Bhavik Mehta", "Ken Ono", "Ashvin Swaminathan", "Jesse Thorner", "Yunzhou Xie" ], "maintainers": [ "Kenny Lau", "Ashvin Swaminathan" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Variants of the Selberg sieve", "id": "https://doi.org/10.1186/s40687-014-0012-7", "authors": [ "D. H. J. Polymath" ], "type": "article", "relationship": "formalizes", "endorsement": "" }, { "title": "Small gaps between primes", "id": "https://doi.org/10.4007/annals.2015.181.1.7", "authors": [ "James Maynard" ], "type": "article", "relationship": "background", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11N05", "11N35", "11N36", "11P32" ] }, "automation": { "methods": [ { "method": "autonomous", "models": [ "AxiomProver" ], "framework": "" }, { "method": "manual", "models": [], "framework": "" } ], "strongest": "autonomous", "models": [ "AxiomProver" ], "spend": "", "notes": "Foundational definitions formalised manually, some definitions formalised autonomously, most theorems formalised autonomously" }, "scope": "Developing a consolidated Lean library for results on gaps between primes, the flagship result being that the Bombieri-Vinogradov theorem implies that that there are infinitely many pairs of primes separated by no more than 246.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "bombieriVinogradov_implies_prime_gap_le_246", "file": "PrimeGaps/Bounded246.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [ { "statement": "The Bombieri-Vinogradov Theorem", "source": "Small gaps between primes (p.384)" } ] }, { "declaration": "bombieriVinogradov_implies_nth_prime_gap_le_246", "file": "PrimeGaps/Bounded246.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [ { "statement": "The Bombieri-Vinogradov Theorem", "source": "Small gaps between primes (p.384)" } ] } ], "comparator": true, "literature_dependencies": 2, "divergences": "", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Sidharth Hariharan", "Kenny Lau", "Bhavik Mehta", "Ashvin Swaminathan", "Yunzhou Xie" ], "notes": "Foundational definitions underwent multiple rounds of review and modification to ensure reusable and maintainable design. The proof strategy outlined in the blueprint underwent multiple revisions to stay as adjacent as possible to the project dependencies (Mathlib and PNT+). The development of this formalisation took place in a repository internal to Axiom Math. Pull requests to that repository were not typically merged without thorough vetting from the reviewers. Before the contents of the internal repository were moved to this public repository, there were more rounds of code review for both quality and fidelity." }, "canonical": { "repo": "axiommath/primegapslib", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-18-000002", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-18-000002", "commit": "1faa7b14e82ddebc2772dfb9153922f01b106477", "trust": "high", "theorems": 2, "date": "2026-08-18" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "AxiomMath/QBinomialTrace/formalization.yaml", "repo": "AxiomMath/QBinomialTrace", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/AxiomMath/QBinomialTrace/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "QBinomialTrace", "description": "`QBinomialTrace` is a formal proof of the dominance results that reduce the Han-Xiong conjecture on integer traces of fractional Gaussian binomial coefficients to a family of unit fractions, from *On a conjecture of Han and Xiong for fractional Gaussian binomial coefficients* (arXiv:2608.00323). It proves Theorem 1.3, Corollary 1.4 and Theorem 1.5. The Lean files were generated by AxiomProver, Axiom Math's in-house theorem proving system.", "authors": [ "Kenny Lau" ], "maintainers": [ "Kenny Lau" ], "license": "MIT", "role": "substantive-development", "substantive": "", "sources": [ { "title": "On a conjecture of Han and Xiong for fractional Gaussian binomial coefficients", "id": "https://arxiv.org/abs/2608.00323", "authors": [ "Ken Ono" ], "type": "article", "relationship": "formalizes", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05A20", "11P84", "68V15", "68V20" ] }, "automation": { "methods": [ { "method": "autonomous", "models": [ "AxiomProver" ], "framework": "" } ], "strongest": "autonomous", "models": [ "AxiomProver" ], "spend": "", "notes": "" }, "scope": "Theorem 1.3 for every pair of positive rationals $u \\le r$ in lowest terms whose denominators satisfy the hypotheses (6), Corollary 1.4 for every rational $r \\ge 1/2$, and Theorem 1.5 in all three of its parts - the core dominance (8), the equivalence with the unit family (9), and the finite range (10) for fixed $k$ - for every positive rational $r$ and every $k \\ge 1$.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "support_dominance", "file": "QBinomialTrace/solution.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "half_line_dominance", "file": "QBinomialTrace/solution.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "core_dominance", "file": "QBinomialTrace/solution.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "HX_iff_unit_family", "file": "QBinomialTrace/solution.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "HXk_iff_UFk", "file": "QBinomialTrace/solution.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "finite_unit_family_check", "file": "QBinomialTrace/solution.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "", "alignment": true, "original": false, "review": { "status": "author-verified", "bucket": "author-verified", "reviewers": [ "Kenny Lau" ], "notes": "The formal challenge in `QBinomialTrace/problem.lean` was checked against the source paper by an author of that paper." }, "canonical": { "repo": "axiommath/qbinomialtrace", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "AxiomMath/quadratic-dinv/formalization.yaml", "repo": "AxiomMath/quadratic-dinv", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/AxiomMath/quadratic-dinv/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "QuadraticDinv", "description": "`QuadraticDinv` is a formal proof that the symmetric bilinear form of a quadratic form generalizing rational `dinv` computes the cross-`dinv` statistic. For coprime $0 < a < b$, let $G$ be the gap set of the numerical semigroup $\\langle a, b \\rangle$, realized as a Young diagram by $g(x,y) = ab - ax - by$, let $Q$ be the quadratic form on $\\mathbb{R}^G$ with kernel $K(d) = 1_{d \\ge 0} - 1_{d \\ge a} - 1_{d \\ge b} + 1_{d \\ge a+b}$, and let $B$ be its associated symmetric bilinear form. It proves that $B(\\mathbf{1}_D, \\mathbf{1}_E) = \\mathbf{dinv}(D, E)$ for all subdiagrams $D, E \\subseteq G$. Th", "authors": [ "Kenny Lau" ], "maintainers": [ "Kenny Lau" ], "license": "MIT", "role": "substantive-development", "substantive": "", "sources": [ { "title": "A quadratic form generalization of rational dinv", "id": "https://arxiv.org/abs/2604.13238", "authors": [ "Yifeng Huang" ], "type": "article", "relationship": "formalizes", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.CO" ], "msc2020": [] }, "automation": { "methods": [ { "method": "autonomous", "models": [ "AxiomProver" ], "framework": "" } ], "strongest": "autonomous", "models": [ "AxiomProver" ], "spend": "", "notes": "" }, "scope": "Theorem 1.2, which the paper calls its main engine, for all coprime $0 < a < b$ and all subdiagrams $D, E$ of the gap set.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "RationalDinv.bilinForm_eq_crossDinv", "file": "QuadraticDinv/solution.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "", "alignment": true, "original": false, "review": { "status": "author-verified", "bucket": "author-verified", "reviewers": [ "Kenny Lau" ], "notes": "The formal problem statement in `QuadraticDinv/problem.lean` was checked against the source paper by an author of that paper." }, "canonical": { "repo": "axiommath/quadratic-dinv", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "AxiomMath/ramanujan-tau-misses-primes/formalization.yaml", "repo": "AxiomMath/ramanujan-tau-misses-primes", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/AxiomMath/ramanujan-tau-misses-primes/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "RamanujanTauMissesPrimes", "description": "`RamanujanTauMissesPrimes` is a formal proof that the $abc$ Conjecture implies Ramanujan's tau function misses almost all primes. Writing $S(X) = \\#\\{\\ell \\le X : \\ell \\text{ prime},\\ |\\tau(n)| = \\ell \\text{ for some } n \\ge 1\\}$, it proves $S(X) = O(X^{13/22})$, together with the reduction of $S(X)$ to the counts $E_2(X)$, $E_4(X)$ of near-solutions to $x^{11} - y^2$ and $5x^{22} - u^2$, and the $abc$ bounds $E_2(X) \\ll_\\eta X^{4/9+\\eta}$ and $E_4(X) \\ll_\\eta X^{1/5+\\eta}$. The Lean files were generated by AxiomProver, Axiom Math's in-house theorem proving system.", "authors": [ "Kenny Lau" ], "maintainers": [ "Kenny Lau" ], "license": "MIT", "role": "substantive-development", "substantive": "", "sources": [ { "title": "ABC implies that Ramanujan's tau function misses almost all primes", "id": "https://arxiv.org/abs/2603.29970", "authors": [ "David Kurniadi Angdinata", "Evan Chen", "Chris Cummins", "Ben Eltschig", "Dejan Grubisic", "Leopold Haller", "Letong Hong", "Andranik Kurghinyan", "Kenny Lau", "Hugh Leather", "Seewoo Lee", "Simon Mahns", "Aram H. Markosyan", "Rithikesh Muddana", "Ken Ono", "Manooshree Patel", "Gaurang Pendharkar", "Vedant Rathi", "Alex Schneidman", "Volker Seeker", "Shubho Sengupta", "Ishan Sinha", "Jimmy Xin", "Jujian Zhang" ], "type": "article", "relationship": "formalizes", "endorsement": "" }, { "title": "Prime values of Ramanujan's tau function", "id": "https://arxiv.org/abs/2311.12073", "authors": [ "B. Xiong" ], "type": "article", "relationship": "background", "endorsement": "" }, { "title": "La conjecture de Weil. I", "id": "https://doi.org/10.1007/BF02684373", "authors": [ "P. Deligne" ], "type": "article", "relationship": "background", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11F11", "11F30" ] }, "automation": { "methods": [ { "method": "autonomous", "models": [ "AxiomProver" ], "framework": "" } ], "strongest": "autonomous", "models": [ "AxiomProver" ], "spend": "", "notes": "" }, "scope": "Theorem 1, Lemma 4 and Lemma 5, conditional on the $abc$ Conjecture and on Xiong's Proposition 5.4, both of which are hypotheses of the Lean statements. `problem.lean` axiomatizes $\\tau$ as a structure `RamanujanTau` carrying Hecke multiplicativity, the Hecke recurrence, parity, Deligne's bound and the non-unit property, and every result is stated for an arbitrary such structure.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "reduction_lemma", "file": "RamanujanTauMissesPrimes/solution.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [ { "statement": "`ABC`: for every $\\varepsilon > 0$ there is $K_\\varepsilon > 0$ such that $c \\le K_\\varepsilon \\operatorname{rad}(abc)^{1+\\varepsilon}$ for all coprime positive $a + b = c$.", "source": "abc Conjecture (unnumbered), as stated in the paper" }, { "statement": "`Proposition5_4`: for large $N$ and $3 \\le k < \\log N / (2\\log 2)$, $\\#(\\mathcal{P} \\cap X_{2k} \\cap [-N, N]) \\ll N^{1/2}$; and $X_{2k} \\cap [-N,N] = \\varnothing$ for $k \\ge \\log N / (2 \\log 2)$.", "source": "Proposition 9 (Xiong); Prime values of Ramanujan's tau function, Prop. 5.4" }, { "statement": "`RamanujanTau`: $\\tau$ is taken as any function satisfying Hecke multiplicativity and the Hecke recurrence, parity, Deligne's bound, and $\\tau(n) \\ne \\pm 1$ for $n \\ge 2$.", "source": "Equations (2.1), Proposition 6, and Deligne's bound as recorded by Xiong" } ] }, { "declaration": "abc_bound_E2", "file": "RamanujanTauMissesPrimes/solution.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [ { "statement": "`ABC`, as above. This result does not involve $\\tau$.", "source": "abc Conjecture (unnumbered), as stated in the paper" } ] }, { "declaration": "abc_bound_E4", "file": "RamanujanTauMissesPrimes/solution.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [ { "statement": "`ABC`, as above. This result does not involve $\\tau$.", "source": "abc Conjecture (unnumbered), as stated in the paper" } ] }, { "declaration": "main_theorem", "file": "RamanujanTauMissesPrimes/solution.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [ { "statement": "`ABC`, `Proposition5_4` and `RamanujanTau`, as for `reduction_lemma`.", "source": "abc Conjecture; Proposition 9 (Xiong); Equations (2.1), Proposition 6, Deligne" } ] } ], "comparator": true, "literature_dependencies": 6, "divergences": "", "alignment": true, "original": false, "review": { "status": "author-verified", "bucket": "author-verified", "reviewers": [ "Kenny Lau" ], "notes": "The formal problem statement in `RamanujanTauMissesPrimes/problem.lean` was checked against the source paper by an author of that paper." }, "canonical": { "repo": "axiommath/ramanujan-tau-misses-primes", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "AxiomMath/record-compositions/formalization.yaml", "repo": "AxiomMath/record-compositions", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/AxiomMath/record-compositions/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "RecordCompositions", "description": "`RecordCompositions` is a formal proof of the closed formulas for the record composition count $N(\\alpha)$ of down-up alternating permutations, and of their counterpart in the algebra of noncommutative symmetric functions, from *Record compositions of alternating permutations and noncommutative symmetric functions* (arXiv:2607.12873). It proves Theorem 1.1, Theorem 1.2 and Theorem 1.3. The Lean files were generated by AxiomProver, Axiom Math's in-house theorem proving system.", "authors": [ "Kenny Lau" ], "maintainers": [ "Kenny Lau" ], "license": "MIT", "role": "substantive-development", "substantive": "", "sources": [ { "title": "Record compositions of alternating permutations and noncommutative symmetric functions", "id": "https://arxiv.org/abs/2607.12873", "authors": [ "Evan Chen", "Ken Ono", "Michal Mogielnicki" ], "type": "article", "relationship": "formalizes", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.CO", "math.RT" ], "msc2020": [ "05A05", "05A15", "05A19" ] }, "automation": { "methods": [ { "method": "autonomous", "models": [ "AxiomProver" ], "framework": "" } ], "strongest": "autonomous", "models": [ "AxiomProver" ], "spend": "", "notes": "" }, "scope": "Theorem 1.1, Theorem 1.2 and Theorem 1.3, for every $n \\ge 1$ and every composition $\\alpha \\vDash n$, with Theorem 1.3 over any field of characteristic zero. `problem.lean` models `NSym` as the free algebra on generators representing the noncommutative power sums $\\Psi_k$, and `Sym` as a polynomial ring on generators representing the power sums $p_k$, with $\\chi$ the map between them.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "N_eq_rhsBinom", "file": "RecordCompositions/solution.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "N_eq_rhsRat", "file": "RecordCompositions/solution.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "factorial_smul_Anc_eq", "file": "RecordCompositions/solution.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "chi_Anc_eq", "file": "RecordCompositions/solution.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "", "alignment": true, "original": false, "review": { "status": "author-verified", "bucket": "author-verified", "reviewers": [ "Kenny Lau" ], "notes": "The formal challenge in `RecordCompositions/problem.lean` was checked against the source paper by an author of that paper." }, "canonical": { "repo": "axiommath/record-compositions", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "AxiomMath/RR_a3/formalization.yaml", "repo": "AxiomMath/RR_a3", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/AxiomMath/RR_a3/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "RRA3", "description": "`RRA3` is a formal certificate for *Modularity of Point Counts for the Curves $X^a = Y^b$: New Rogers-Ramanujan Identities* (arXiv:2608.05480). It proves the implication \"Facts 1-8 => Theorem 1.1 (1)-(2)\". The Lean files were generated by AxiomProver, Axiom Math's in-house theorem proving system.", "authors": [ "Kenny Lau" ], "maintainers": [ "Kenny Lau" ], "license": "MIT", "role": "substantive-development", "substantive": "", "sources": [ { "title": "Modularity of Point Counts for the Curves $X^a = Y^b$: New Rogers-Ramanujan Identities", "id": "https://arxiv.org/abs/2608.05480", "authors": [ "Kenny Lau", "Ken Ono" ], "type": "article", "relationship": "formalizes", "endorsement": "" }, { "title": "q-supernomial coefficients: from riggings to ribbons", "id": "https://doi.org/10.1007/978-1-4612-0087-1_16", "authors": [ "A. Schilling" ], "type": "article", "relationship": "background", "endorsement": "" }, { "title": "Supernomial coefficients, polynomial identities and q-series", "id": "https://doi.org/10.1023/A:1009780810189", "authors": [ "A. Schilling", "S. O. Warnaar" ], "type": "article", "relationship": "background", "endorsement": "" }, { "title": "Hall-Littlewood functions and the $A_2$ Rogers-Ramanujan identities", "id": "https://doi.org/10.1016/j.aim.2004.12.001", "authors": [ "S. O. Warnaar" ], "type": "article", "relationship": "background", "endorsement": "" }, { "title": "The $A_2$ Andrews-Gordon identities and cylindric partitions", "id": "https://doi.org/10.1090/btran/147", "authors": [ "S. O. Warnaar" ], "type": "article", "relationship": "background", "endorsement": "" }, { "title": "A quadratic form generalization of rational dinv", "id": "https://doi.org/10.1007/s40687-026-00631-0", "authors": [ "Y. Huang" ], "type": "article", "relationship": "background", "endorsement": "" }, { "title": "The Theory of Partitions", "id": "https://doi.org/10.1017/CBO9780511608650", "authors": [ "G. E. Andrews" ], "type": "book", "relationship": "background", "endorsement": "" }, { "title": "The Diophantine Frobenius Problem", "id": "https://doi.org/10.1093/acprof:oso/9780198568209.001.0001", "authors": [ "J. L. Ramirez Alfonsin" ], "type": "book", "relationship": "background", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.NT", "math.CO", "math.RT" ], "msc2020": [ "11P84", "05A17", "05A30", "05E05", "17B67", "20M14" ] }, "automation": { "methods": [ { "method": "autonomous", "models": [ "AxiomProver" ], "framework": "" } ], "strongest": "autonomous", "models": [ "AxiomProver" ], "spend": "", "notes": "" }, "scope": "A formal certificate that Facts 1-8 imply Theorem 1.1 (1)-(2). The $q$-supernomial is opaque, constrained only by Facts 1-3.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "ProblemHJOa3.thm_main", "file": "RRA3/solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [ { "statement": "`Fact1` (rigged-configuration expansion): Schilling's fermionic formula for the $A_1$ completely symmetric $q$-supernomial with two-part content.", "source": "q-supernomial coefficients: from riggings to ribbons, (2.1)-(2.3); Proposition 2.3 of the paper" }, { "statement": "`Fact2Bundle`, Fact 2 specialized to the two component orders used in the proof (multitableau expansion and order-independence): $\\widetilde{S}_{\\lambda\\mu}(q) = \\sum_T q^{\\mathrm{inv}(T)}$ over tuples of one-row semistandard $\\{1,2\\}$-tableaux with row lengths the parts of $\\mu$ and content $\\lambd", "source": "q-supernomial coefficients: from riggings to ribbons, Section 3.2 and Theorem 4.1; Proposition 2.1 of the paper" }, { "statement": "`Fact3` (content symmetry): $\\widetilde{S}_{\\lambda\\mu}(q)$ is invariant under permuting the parts of the content $\\lambda$; in particular $\\widetilde{S}_{(A,B),\\mu} = \\widetilde{S}_{(B,A),\\mu}$.", "source": "Supernomial coefficients, polynomial identities and q-series, Section 2; equation (11) of the paper" }, { "statement": "`Fact4` (Warnaar's $A_2$ Andrews-Gordon identities): the two fermionic-sum $=$ theta-product evaluations, the modulus-$(3k+2)$ identity for $k \\ge 2$ and the modulus-$(3k+4)$ identity for $k \\ge 1$.", "source": "The $A_2$ Andrews-Gordon identities and cylindric partitions, (1.10)-(1.11), building on Hall-Littlewood functions and the $A_2$ Rogers-Ramanujan identities; Proposition 2.4 of the paper" }, { "statement": "`Fact5` ($q$-binomial reciprocity): for $0 \\le j \\le N$, $\\binom{N}{j}_{q^{-1}} = q^{-j(N-j)} \\binom{N}{j}_q$.", "source": "Classical" }, { "statement": "`Fact6` (elementary Gaussian and Pochhammer identities): the two $q$-Pascal recurrences, the factorization $(q)_j (q)_{N-j} \\binom{N}{j}_q = (q)_N$, vanishing of $\\binom{N}{j}_q$ for $j > N$, and the telescoping of a chain of Gaussian polynomials against $1/(q)_{r_1}$ into a product of Pochhammer re", "source": "Classical; The Theory of Partitions, Ch. 3" }, { "statement": "`Fact7` (Sylvester's gap facts): for coprime $a, b > 1$, the gap set of the numerical semigroup $\\langle a, b \\rangle$ has cardinality $(a-1)(b-1)/2$, has maximum (the Frobenius number) $ab - a - b$, and satisfies the symmetry that for $0 \\le s \\le ab-a-b$, $s$ is a gap if and only if $ab - a - b - ", "source": "Classical; The Diophantine Frobenius Problem, Ch. 2" }, { "statement": "`Fact8` (positivity of the HJO quadratic form): on the cone $\\mathcal{C}_{a,b}$ the quadratic form $Q_{a,b}$ is positive definite and satisfies $\\|n\\|_\\infty^2 \\le |G| \\, Q_{a,b}(n)$, so each coefficient of $Z_{a,b}(q)$ receives finitely many contributions and $Z_{a,b}(q)$ is a well-defined formal p", "source": "A quadratic form generalization of rational dinv (Theorem 1.3)" } ] } ], "comparator": true, "literature_dependencies": 8, "divergences": "", "alignment": true, "original": false, "review": { "status": "author-verified", "bucket": "author-verified", "reviewers": [ "Kenny Lau" ], "notes": "The formal challenge in `RRA3/problem.lean` was checked against the source paper by an author of that paper." }, "canonical": { "repo": "axiommath/rr_a3", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "AxiomMath/zeta-h123/formalization.yaml", "repo": "AxiomMath/zeta-h123", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/AxiomMath/zeta-h123/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "ZetaH123", "description": "`ZetaH123` is a formal proof of Thakur's hypotheses H1, H2 and H3 on power sums over $\\mathbb{F}_q[t]$, from *Thakur's hypotheses on power sums of $\\mathbb{F}_q[t]$* (arXiv:2606.16239). It proves Theorem 1.1 (H1 for prime fields), Theorem 1.3 (H2 for prime fields), Theorem 1.8 (H3 for any prime power), and Lemma 4.1, the monomial expansion of $S_d(k)$ they rest on. The Lean files were generated by AxiomProver, Axiom Math's in-house theorem proving system.", "authors": [ "Kenny Lau" ], "maintainers": [ "Kenny Lau" ], "license": "MIT", "role": "substantive-development", "substantive": "", "sources": [ { "title": "Thakur's hypotheses on power sums of $\\mathbb{F}_q[t]$", "id": "https://arxiv.org/abs/2606.16239", "authors": [ "Evan Chen", "Ken Ono" ], "type": "article", "relationship": "formalizes", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.NT", "math.CO" ], "msc2020": [ "11M38", "11T55", "11G09" ] }, "automation": { "methods": [ { "method": "autonomous", "models": [ "AxiomProver" ], "framework": "" } ], "strongest": "autonomous", "models": [ "AxiomProver" ], "spend": "", "notes": "" }, "scope": "Theorem 1.1 for every prime $q = p$ and all $d, k \\ge 0$; Theorem 1.3 for every prime $q = p$, $d > 1$ and $k > 0$; Theorem 1.8 for every prime power $q = p^f$, $d \\ge 1$ and $k > 0$; and Lemma 4.1 for every prime power. 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Missing historical costs, timings, prompts, and hardware are not inferred." }, "scope": "For 0 < theta < 1, the project proves the +1 and -i Fourier eigenmeasure lines with sharp inclusive thresholds 1/8 and 3/8, vanishing of the -1 and +i sectors, explicit generalized eta-product coefficients, and the 2/1/0 dimension law for the full admissible space. It connects coefficient data to locally finite tempered atomic Radon measures. 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The Palomar boundary represents a measure by positive and negative polynomial-growth coefficient sequences and states the Fourier relation on every Schwartz test function. The implementation proves equivalence with the manuscript's locally finite tempered atomic-Radon formulation for theta > 0. Support means containment, not equality. The endpoint theta = 0, distinct support shifts, arbitrary locally finite carriers, and higher-level analogues are outside scope. The optional valence appendix is formalized but is not used by the compared principal proof.", "alignment": true, "original": false, "review": { "status": "author-verified", "bucket": "author-verified", "reviewers": [ "Li Shen" ], "notes": "Li Shen reviewed, refined, and verified the mathematical result and the initial verification software. This is author verification, not independent external peer review. 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No metered API usage.", "notes": "Every proof is checked by the Lean kernel and Mathlib; the production library contains no sorry and no custom axiom. The Mathlib-only comparator challenge files (Audit/*/Challenge.lean) each contain exactly one intentional statement-level sorry -- the body of the theorem being challenged -- which the corresponding Audit/*/Solution.lean fills and leanprover/comparator confirms (\"Your solution is okay!\"). All seven passed on 2026-08-13 from commit b0943d9. DISCLOSURE, tooling modification: the comparator is invoked through docker/landrun-argv-shim.sh rather than calling landrun directly. landrun" }, "scope": "A Lean 4 / Mathlib formalization of absorption cutoff and stationary singularities for Gaussian random dynamical systems with coordinatewise tanh nonlinearity and nearest-grid rounding: the fixed-width vanishing-mesh cutoff profile, with the full cutoff predicate checking center divergence, eventual window positivity, and the little-o window condition; the fixed-precision dimension cutoff; both clauses of qualitative metastability (including fixed-dimensional absorption from every point of the full finite rounded state space); the supercritical dimension cutoff in scalar and reconstructed vect", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "AbsorptionCutoff.MainTheorems.rounded_gaussian_nearest_cutoff", "file": "AbsorptionCutoff/MainTheorems.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "AbsorptionCutoff.MainTheorems.subcritical_dimension_cutoff", "file": "AbsorptionCutoff/MainTheorems.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "AbsorptionCutoff.MainTheorems.rounded_qualitative_metastability", "file": "AbsorptionCutoff/MainTheorems.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "AbsorptionCutoff.MainTheorems.rounded_fixed_dimension_absorption", "file": "AbsorptionCutoff/MainTheorems.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "AbsorptionCutoff.MainTheorems.gaussian_process_cutoff", "file": "AbsorptionCutoff/MainTheorems.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "AbsorptionCutoff.MainTheorems.gaussian_vector_cutoff", "file": "AbsorptionCutoff/MainTheorems.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "AbsorptionCutoff.MainTheorems.nd_power_singularity", "file": "AbsorptionCutoff/MainTheorems.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "Recorded item by item in CORRESPONDENCE.md, and all of them are packaging, proof route, or cases where the Lean result is stronger. After unfolding the standing notation and equivalent regime formulations, the manuscript's hypotheses imply those of the corresponding Lean declaration, and its conclusion implies the manuscript's conclusion. Summary: the fixed-width theorem is stated along positive mesh sequences rather than as a continuous limit; the renewal terminal term and the signed kernel are handled by different but equivalent routes; several manuscript intermediates (the transfer operator, the linear Markov-additive equation, the operator packaging of the forcing) are not mirrored as standalone declarations because the theorem route consumes their content directly; and forcing continuity is proved for every bounded measurable angular test rather than only continuous ones.", "alignment": true, "original": false, "review": { "status": "author-verified", "bucket": "author-verified", "reviewers": [ "Benny Avelin" ], "notes": "The Lean kernel and Mathlib check every production-library proof term (no sorry, no custom axiom). The seven audited declarations were additionally checked by leanprover/comparator against Mathlib-only restatements. This checks the chosen formal statements, not their semantic correspondence with the manuscript; that correspondence was reviewed by the author and recorded in CORRESPONDENCE.md. This is not external peer review." }, "canonical": { "repo": "bennyavelin/absorptioncutoff", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "bjbraams/lean-CA/formalization.yaml", "repo": "bjbraams/lean-CA", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/bjbraams/lean-CA/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "lean-CA: foundations of complex analysis of one variable in Lean 4", "description": "A Lean 4 formalization, built on Mathlib, of classical complex analysis of one variable. The 45 compared results cover global primitives and logarithms, Cauchy and residue theory for cycles, Laurent expansions, the argument principle and Rouché, Hurwitz and local mapping, Montel and Vitali, Runge approximation and Mittag–Leffler, Cauchy–Pompeiu, Weierstrass and Hadamard factorization, Blaschke and Riesz factorization, normalized Riemann mapping, Schwarz–Pick and Koebe estimates, harmonic and subharmonic functions, the Dirichlet problem and Green functions under an exterior-disc condition, refl", "authors": [ "Bastiaan J. Braams" ], "maintainers": [ "Bastiaan J. Braams" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "Functions of One Complex Variable I (second edition)", "id": "", "authors": [ "John B. Conway" ], "type": "book", "relationship": "formalizes", "endorsement": "" }, { "title": "Functions of One Complex Variable II", "id": "", "authors": [ "John B. Conway" ], "type": "book", "relationship": "formalizes", "endorsement": "" }, { "title": "Complex Analysis (fourth edition)", "id": "", "authors": [ "Serge Lang" ], "type": "book", "relationship": "formalizes", "endorsement": "" }, { "title": "Complex Analysis", "id": "", "authors": [ "Theodore W. Gamelin" ], "type": "book", "relationship": "formalizes", "endorsement": "" }, { "title": "Complex Analysis", "id": "", "authors": [ "Elias M. Stein", "Rami Shakarchi" ], "type": "book", "relationship": "formalizes", "endorsement": "" }, { "title": "Basic Complex Analysis: A Comprehensive Course in Analysis, Part 2A", "id": "", "authors": [ "Barry Simon" ], "type": "book", "relationship": "formalizes", "endorsement": "" }, { "title": "Classical Analysis in the Complex Plane", "id": "", "authors": [ "Robert B. Burckel" ], "type": "book", "relationship": "formalizes", "endorsement": "" }, { "title": "An Introduction to Complex Analysis in Several Variables (second edition)", "id": "", "authors": [ "Lars Hörmander" ], "type": "book", "relationship": "formalizes", "endorsement": "" }, { "title": "An Introduction to Complex Analysis", "id": "", "authors": [ "Ravi P. Agarwal", "Kanishka Perera", "Sandra Pinelas" ], "type": "book", "relationship": "background", "endorsement": "" }, { "title": "Complex Analysis (third edition)", "id": "", "authors": [ "Joseph Bak", "Donald J. Newman" ], "type": "book", "relationship": "background", "endorsement": "" }, { "title": "Complex Function Theory", "id": "", "authors": [ "Maurice Heins" ], "type": "book", "relationship": "background", "endorsement": "" }, { "title": "Theory of Complex Functions", "id": "", "authors": [ "Reinhold Remmert" ], "type": "book", "relationship": "background", "endorsement": "" } ], "related": [ { "id": "https://github.com/leanprover-community/mathlib4", "relationship": "builds-on", "note": "The pinned Mathlib release supplies the underlying topology, integration, power series, local holomorphic theory, Schwarz lemma, meromorphic orders and divisors, harmonic theory, Jensen formula and functional analysis. This repository extends those foundations." }, { "id": "https://github.com/vbeffara/RMT4/tree/69a9efe77e912647d651aa7368856955b24dca2f", "relationship": "other", "note": "Vincent Beffara: Montel, both Hurwitz theorems and Riemann mapping, closely overlapping the compared results. See CREDITS.md for differences in domain hypotheses and normalization." }, { "id": "https://github.com/leanprover-community/mathlib4/pull/33505", "relationship": "other", "note": "Yury Kudryashov: Riemann mapping, Hurwitz and a holomorphic argument principle. The Hurwitz declaration names coincide; the reviewed PR assumes a countably generated filter, whereas this project only requires a nontrivial filter. PR status and inspected revision are recorded in CREDITS.md." }, { "id": "https://github.com/leanprover-community/mathlib4/pull/43100", "relationship": "other", "note": "JJYYY-JJY, with credited coauthor ajirving: higher Cauchy formulas at an interior point of a disc, including Banach-valued versions. Closely related to the compared cauchy_derivatives statement." }, { "id": "https://github.com/leanprover-community/mathlib4/pull/29588", "relationship": "other", "note": "Roman Kvasnytskyi: scalar residues defined using circle integrals, independence of radius and simple-pole formulas; related to this project’s Banach-valued residue API." }, { "id": "https://github.com/leanprover-community/mathlib4/pull/39232", "relationship": "other", "note": "Submitted by jerwaynejones; the source credits Jeremy Tan. A rectangular contour residue formula for specified simple poles and a holomorphic remainder, overlapping a special case of cycle residue theory." }, { "id": "https://github.com/leibniz-rs/PrimeNumberTheoremAnd/tree/8dc50485d7166be58b05ee0d54216c06a4b3aef9", "relationship": "other", "note": "Matteo Cipollina: Weierstrass elementary factors, the same small-disc factor estimate, finite-order zero counting and Hadamard factorization. The Hadamard theorem is a close mathematical match with different growth and genus conventions. The fork also contains the PNT+ contributors’ rectangular simple-pole residue theorem. Inspected source was not rebuilt in the attribution review." }, { "id": "https://github.com/will1491/RiemannDynamics/tree/b3fa37cc0f18a23ea66b654ea3f73eb472129010", "relationship": "other", "note": "Will (Ziang) Li: compact-support Cauchy–Pompeiu, differentiating the Cauchy transform and solving the antiholomorphic derivative equation. These overlap the scalar case of the project’s Banach-valued, parameter-dependent theory. Its Montel theorem imports Beffara’s result." }, { "id": "https://github.com/leanprover-community/mathlib4/pull/42630", "relationship": "other", "note": "Will (Ziang) Li: Wirtinger derivatives and the Cauchy–Riemann characterization, related to dbarAlong." }, { "id": "https://github.com/kebekus/ProjectVD/tree/8dd096581f1578e2d0b75b099c8314a198902f1b", "relationship": "other", "note": "Stefan Kebekus: Wirtinger operators, and harmonic, meromorphic, divisor and Jensen foundations contributed to Mathlib. Related ongoing Poisson–Jensen, growth and reciprocal-disc-factor work is distinguished from identical theorem matches in CREDITS.md." }, { "id": "https://github.com/phasetr/ising-model/tree/bb1a2dd19c8b70f33201daf130729ddb4790b66f", "relationship": "other", "note": "The phasetr/ising-model contributors: scalar Montel extraction and Vitali–Porter with an interior accumulation point, close matches to the selected finite-dimensional-valued statements." }, { "id": "https://github.com/girving/ray/tree/753f7131cf96f4651294de4398368abf136c34de", "relationship": "other", "note": "Geoffrey Irving: normalized holomorphic logarithms on discs, holomorphic parameter integrals and locally uniform analytic limits. The selected logarithm theorem treats simply connected open domains." }, { "id": "https://github.com/vbeffara/Curvint/tree/00b0bf1cfa3106fa92329f5f52e449186db0c9d3", "relationship": "other", "note": "Vincent Beffara: covering spaces of local primitives, contour integration by lifting and primitives on star-convex domains. Related infrastructure, not a claim that the selected simply connected primitive theorem is present there." }, { "id": "https://github.com/leanprover-community/mathlib4/pull/26950", "relationship": "other", "note": "Junyan Xu: étalé-space and monodromy infrastructure toward primitives. The reviewed primitive file contains no declarations; this is credit for a related approach, not a completed theorem match." }, { "id": "https://github.com/leanprover-community/mathlib4/pull/33368", "relationship": "other", "note": "Yury Kudryashov: unit-disc shifts, agreeing with discMobius after changing the parameter’s sign. PR #33381 contains related Schwarz estimates." }, { "id": "https://github.com/leanprover-community/mathlib4/pull/26479", "relationship": "other", "note": "thefundamentaltheor3m: Cauchy–Goursat for unbounded rectangles, related to the broader library’s improper contour theory, outside the selected challenge." } ], "classification": { "arxiv": [ "math.CV" ], "msc2020": [ "30C35", "30C55", "30D15", "30D45", "30E10", "30E20", "31A05" ] }, "automation": { "methods": [ { "method": "manual", "models": [], "framework": "" }, { "method": "agent", "models": [ "gpt-6-astra" ], "framework": "Codex" }, { "method": "agent", "models": [ "claude-fable-5-1", "claude-opus-5-5" ], "framework": "Claude Code" } ], "strongest": "agent", "models": [ "claude-fable-5-1", "claude-opus-5-5", "gpt-6-astra" ], "spend": "", "notes": "Human-directed development using GPT and Claude through terminal coding agents. The listed identifiers record the latest models used; historical model versions and costs have not been reconstructed. Coding agents assisted mathematical development, Lean code design, debugging, documentation and review, using terminal builds under AGENTS.md. Lean proofs were produced by AI/LLM exclusively. Challenge.lean gives independent statements over Mathlib and explicit supporting definitions; Solution.lean proves them using the substantive library in this repository." }, "scope": "The submission selects 45 principal results from ComplexAnalysis; supporting Analysis and Topology modules are used by the proofs but contribute no separately selected theorem. The library contains further contour, approximation, continuation and growth theory described in SYNOPSIS.md. This is not a complete formalization of the cited books. Full Picard theory, general boundary extension of conformal maps and a full Hardy-space inner–outer theory are outside the submission. The counts below concern the substantive library and Solution.lean; Challenge.lean intentionally has 45 theorem-proof hol", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The comparison preserves the library hypotheses. Runge disjointness is written as A ∩ K = ∅. Cauchy and residue cycles are finite families of closed C¹ paths with vanishing index outside the domain; the cycle residue theorem allows isolated singularities of any type, with finitely many specified exceptional points. The Laurent theorem uses a closed annulus on which the function is holomorphic on a neighborhood. Montel and Vitali allow finite-dimensional complex Banach targets; the iterated-derivative convergence statement is pointwise. Mittag–Leffler takes principal parts holomorphic on the whole punctured plane at each specified singularity. The sequential Weierstrass factorization uses nonzero zeros tending to infinity, with multiplicities; Hadamard uses an intrinsic countable index type, including finite and empty cases. HasOrderLE is the explicit bound A exp(B |z|^ρ); the genus k satisfies ρ < k + 1 and the polynomial degree is at most k, rather than defining the order by a limsup. The Blaschke condition counts multiplicities at nonzero zeros. Riesz factorization gives a zero-free bounded factor and is not a complete Hardy-space inner–outer factorization. Koebe growth is the up", "alignment": true, "original": false, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [], "notes": "Validated on 24 September 2026 with Lean and Mathlib v4.35.0-rc2. The library passes lake build; the submission passes lake build Challenge Solution and lake env lean on each new Lean file. The only proof holes are the 45 deliberate Challenge theorem bodies; the library and Solution contain none. The metadata passes the v0.4 schema. All 45 solution theorems use only propext, Classical.choice and Quot.sound. Comparator accepts all 45 comparisons and the supporting definitions, and Lean’s kernel, NanoDa and con-ron accept the exported solution. The ordinary sandbox build could not mount this checkout’s intentional .lake symlink, so that source-to-export sandbox stage remains unvalidated locally. Instead, the bundled leanexport exported the successfully built modules using Comparator’s theore" }, "canonical": { "repo": "bjbraams/lean-ca", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "bjbraams/lean-codes/formalization.yaml", "repo": "bjbraams/lean-codes", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/bjbraams/lean-codes/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "B. C. Carlson's approach to special functions of applied mathematics", "description": "Lean formalization of selected results in B. C. Carlson's theory of special functions as complex Dirichlet averages. The registry selection establishes joint entire-parameter continuation of holomorphic averages on convex open scalar domains; joint holomorphy and native-integral agreement for the Gamma-regularized R and L functions on principal slit-node domains; Euler inversion and Euler-Poisson equations; and Gamma-regularized adaptations of both quadratic R transformations for all complex exponent and Dirichlet parameters with positive-real-part unsquared variables. The L function is relate", "authors": [ "Bastiaan J Braams" ], "maintainers": [ "Bastiaan J Braams" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Special Functions of Applied Mathematics", "id": "Academic Press, 1977", "authors": [ "B. C. Carlson" ], "type": "book", "relationship": "formalizes", "endorsement": "" }, { "title": "Special Functions of Applied Mathematics", "id": "Academic Press, 1977", "authors": [ "B. C. Carlson" ], "type": "book", "relationship": "adapts", "endorsement": "" }, { "title": "Dirichlet averages of x^t log x", "id": "SIAM Journal on Mathematical Analysis 18 (1987), 550–565", "authors": [ "B. C. Carlson" ], "type": "paper", "relationship": "formalizes", "endorsement": "" }, { "title": "Lecture notes on several complex variables", "id": "Lecture notes, 2013", "authors": [ "Harold P. Boas" ], "type": "other", "relationship": "adapts", "endorsement": "" }, { "title": "Continuous Multivariate Distributions, Volume 1: Models and Applications", "id": "Wiley, 2000; DOI 10.1002/0471722065", "authors": [ "Samuel Kotz", "N. Balakrishnan", "Norman L. Johnson" ], "type": "book", "relationship": "formalizes", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.CA", "math.CV" ], "msc2020": [ "33C65", "32A05" ] }, "automation": { "methods": [ { "method": "agent", "models": [], "framework": "Codex" } ], "strongest": "agent", "models": [], "spend": "", "notes": "Human-directed, agent-assisted mathematical development, proof writing, debugging, and review using Codex and Lean validation. Historical model versions and per-session costs are not reconstructed here. The metadata does not claim an independent human peer review." }, "scope": "All five mathematical libraries are included in the proof-side imports. Comparator selects 10 Carlson/Dirichlet-average continuation theorems and one R-function construction. Statement.lean and Solution.lean retain their 21 theorem declarations unchanged; the other 11 are supporting material, not separately submitted claims. Their presence does not assert that they are already available in Mathlib. comparator.json specifies the selection; the README explains its mathematical scope and research interest.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The ambient measure is coordinate Lebesgue measure on the entire sum-one hyperplane, restricted only when integrating over the simplex. R and L are Gamma-regularized and use principal branches. Native agreement on slit nodes requires the whole node convex hull to avoid the cut. Quadratic formulas retain Carlson's positive-real-part unsquared variables, allowing slit-plane squared and transformed nodes as in his stated formulas. They are adaptations to Gamma regularization with no exclusions on t or beta, not claims that ordinary R is finite at all parameter poles. Probability normalization and natural mixed moments exclude the empty index type. General nonconvex simply connected continuation, multiply connected and Riemann-surface extensions, and contour formula 6.8-7 remain open.", "alignment": false, "original": false, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [], "notes": "Lean builds and declaration axiom audits have been used during development. The maintainer reports that Palomar mechanical verification succeeded for the preceding submission with the same Lean declarations. Its automated editorial review declined registration because the separate foundational selections lacked a research-interest account and the quadratic adaptations needed clearer provenance. This revision narrows the Comparator selection and revises the documentation without changing any Lean file. External verification and editorial review of this revised snapshot remain pending; no independent human peer review or registration is claimed." }, "canonical": { "repo": "bjbraams/lean-codes", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-15-000004", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-15-000004", "commit": "b7ed9a789ef8f1dea50cf82e29b7aded2b07b097", "trust": "high", "theorems": 10, "date": "2026-09-15" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "bjbraams/lean-LCS/formalization.yaml", "repo": "bjbraams/lean-LCS", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/bjbraams/lean-LCS/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "lean-LCS: locally convex spaces, duality, and closed graph theorems in Lean 4", "description": "A Lean 4 formalization, built on Mathlib, of the basic theory of locally convex topological vector spaces over the real or complex numbers. It covers barrelled, quasi-barrelled, bornological and ultrabornological spaces; the closed graph and open mapping theorems for barrelled spaces, and Pták's closed graph and open mapping theorems; duality theory with the bipolar, Alaoglu–Bourbaki, Mackey, Mackey–Arens, Banach–Dieudonné and Krein–Šmulian theorems and Grothendieck's completeness theorem; semi-reflexive, reflexive and Montel spaces with their permanence properties; completeness of strong dual", "authors": [ "Bastiaan J. Braams" ], "maintainers": [ "Bastiaan J. Braams" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Topological Vector Spaces (second edition)", "id": "doi:10.1007/978-1-4612-1468-7", "authors": [ "Helmut H. Schaefer", "Manfred P. Wolff" ], "type": "book", "relationship": "formalizes", "endorsement": "" }, { "title": "Topological Vector Spaces I", "id": "doi:10.1007/978-3-642-64988-2", "authors": [ "Gottfried Köthe" ], "type": "book", "relationship": "formalizes", "endorsement": "" }, { "title": "Topological Vector Spaces II", "id": "doi:10.1007/978-1-4684-9409-9", "authors": [ "Gottfried Köthe" ], "type": "book", "relationship": "formalizes", "endorsement": "" }, { "title": "Topological Vector Spaces, Chapters 1–5", "id": "doi:10.1007/978-3-642-61715-7", "authors": [ "N. Bourbaki" ], "type": "book", "relationship": "formalizes", "endorsement": "" }, { "title": "Topological Vector Spaces (second edition)", "id": "isbn:9781584888666", "authors": [ "Lawrence Narici", "Edward Beckenstein" ], "type": "book", "relationship": "background", "endorsement": "" }, { "title": "Closed Graph Theorems and Webbed Spaces", "id": "", "authors": [ "Marc De Wilde" ], "type": "book", "relationship": "background", "endorsement": "" }, { "title": "Completeness and the open mapping theorem", "id": "", "authors": [ "Vlastimil Pták" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "A Course in Functional Analysis (second edition)", "id": "", "authors": [ "John B. Conway" ], "type": "book", "relationship": "background", "endorsement": "" }, { "title": "Introduction to Topological Vector Spaces (lecture notes)", "id": "", "authors": [ "Bill Casselman" ], "type": "other", "relationship": "background", "endorsement": "" } ], "related": [ { "id": "https://github.com/leanprover-community/mathlib4", "relationship": "builds-on", "note": "Extends Mathlib's library of locally convex spaces (seminorms, von Neumann bounded sets, barrelled spaces and Banach–Steinhaus, polars, weak and strong duals, Montel spaces, Hahn–Banach separation, Krein–Milman, test functions, Schwartz space). The new space classes are defined through seminorms in the style of Mathlib's BarrelledSpace." }, { "id": "https://github.com/leanprover-community/mathlib4/pull/26345", "relationship": "adapts", "note": "Open Mathlib PR (C. Hoskin) proving the bipolar theorem for a pairing. The project's bipolar theorem was written after reading it; the statement is reformulated and the proof is organized differently. See the notes in LocallyConvexSpaces/Bipolar.lean." }, { "id": "https://github.com/leanprover-community/mathlib4/pull/41166", "relationship": "adapts", "note": "Open Mathlib PRs #40983 and #41166 (K. H. Wilson) generalize the open mapping theorem to F-spaces. The project contains adapted duplicates of support lemmas of #40983, under the same names, and its successive-approximation argument follows the organization of the proof in #41166; see the notes in TopologicalVectorSpaces/Basic.lean and TopologicalGroups/NearlyOpen.lean." }, { "id": "https://github.com/leanprover-community/mathlib4/pull/26339", "relationship": "independent", "note": "Draft Mathlib PR (C. Hoskin) towards the Banach–Dieudonné lemma for normed spaces; the project proves it independently for first-countable topological vector spaces." }, { "id": "https://github.com/leanprover-community/mathlib4/pull/43747", "relationship": "independent", "note": "Open Mathlib PR (yuanyi-350) characterizing dense real submodules by annihilators; the project's Transpose.lean proves the pointwise statement over ℝ or ℂ independently." }, { "id": "https://github.com/leanprover-community/mathlib4/pull/34106", "relationship": "independent", "note": "Draft Mathlib PR (A. Dedecker) on quotient group seminorms; the project's QuotientSeminorm.lean is the independently obtained analogue for seminorms over a field." }, { "id": "https://github.com/mrdouglasny/gaussian-field", "relationship": "independent", "note": "Independent Lean development (M. R. Douglas, Y. Tanimoto) of nuclear spaces and Gaussian measures on duals of nuclear Fréchet spaces; nuclear spaces are not treated in lean-LCS." } ], "classification": { "arxiv": [ "math.FA" ], "msc2020": [ "46A03", "46A08", "46A30", "46A20", "46A25", "46A13", "46A04" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "gpt-6-astra" ], "framework": "Codex" }, { "method": "agent", "models": [ "claude-fable-5-1", "claude-opus-5-5" ], "framework": "Claude Code" } ], "strongest": "agent", "models": [ "claude-fable-5-1", "claude-opus-5-5", "gpt-6-astra" ], "spend": "subscription-based", "notes": "Human-directed, agent-assisted mathematical development, proof writing, debugging and review, validated throughout by Lean builds. The author chose the material, the conventions and the sources and directed the work; coding agents wrote most of the Lean text. Historical model versions and per-session costs are not reconstructed here. Challenge.lean copies the needed library definitions verbatim and states the compared theorems with sorry; Solution.lean restates them and proves each by the corresponding library theorem." }, "scope": "Formalized: the results listed in the description, together with supporting theory of gauges, seminorm families, completions, quotients, final locally convex topologies, transposes, local Banach spaces and projective limits, Banach disks and fast convergence, and permanence of webbedness. Scalars are ℝ or ℂ except where a statement holds over a more general normed field. Not formalized: (DF)-spaces, nuclear and Schwartz-type spaces in general, tensor products, and non-archimedean spaces. Specializations to Banach spaces are left to Mathlib.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "Bornological, quasi-barrelled and ultrabornological spaces are defined through seminorms, in the style of Mathlib's BarrelledSpace; theorems prove the equivalence with the classical definitions by absorbing sets for real and complex spaces. The test spaces in the definition of ultrabornological spaces are complete seminormed spaces in the universe of the space. Local convexity is expressed as LocallyConvexSpace ℝ E. A Fréchet space is a Hausdorff, complete, first-countable locally convex space. Results without Hausdorffness are stated for complete, first-countable locally convex spaces. A reflexive space is a semi-reflexive space whose canonical map into the strong bidual is inducing. Strict inductive limits are countable and carry the final locally convex topology. De Wilde's theorems are stated for sequentially closed graphs, with webs as in Köthe II §35. Milman's theorem is stated for real spaces. The Alaoglu–Bourbaki theorem is stated over a proper nontrivially normed field.", "alignment": true, "original": false, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [], "notes": "The library builds with Lean and Mathlib v4.35.0-rc2 without errors, warnings or sorry, with warnings treated as errors in continuous integration, and passes Mathlib's environment linters. MainResults.lean checks the axioms of the headline theorems. The compared declarations use only propext, Quot.sound and Classical.choice. Before the official run, the definitions and statements of Challenge.lean were checked to elaborate to the same terms as the library's by comparing fully elaborated expressions. Comparator (lake comparator of Lean v4.35.0-rc2, with the con-ron, NanoDa and Lean kernels) accepts Solution.lean for all 52 compared theorems. The code was reviewed by coding agents for generality, Mathlib naming, docstrings and provenance. Palomar's editorial review is pending. No independent" }, "canonical": { "repo": "bjbraams/lean-lcs", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-25-000007", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-25-000007", "commit": "d2a9c3112132820d7d1bf2fbc340a9db3cd6213a", "trust": "high", "theorems": 52, "date": "2026-09-25" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "bjbraams/lean-SCV/formalization.yaml", "repo": "bjbraams/lean-SCV", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/bjbraams/lean-SCV/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Several complex variables: local theory, Hartogs phenomena, Weierstrass theory, domains of holomorphy and pseudoconvexity", "description": "A Lean 4 formalization, on top of Mathlib, of the classical function theory of several complex variables on open subsets of finite-dimensional complex normed spaces, with values in complex Banach spaces wherever the statements allow. The compared statements cover: Cauchy's integral formula on polydiscs, Osgood's lemma, the equivalence of holomorphy and analyticity, the Cauchy-Riemann equations, Cauchy estimates, the identity theorem, the maximum modulus principle, holomorphic dependence of integrals and the Cauchy-Pompeiu identity; the theorems of Weierstrass, Montel and Vitali and an estimate", "authors": [ "Bastiaan J Braams" ], "maintainers": [ "Bastiaan J Braams" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Holomorphic Functions and Integral Representations in Several Complex Variables", "id": "Graduate Texts in Mathematics 108, Springer, 1986", "authors": [ "R. Michael Range" ], "type": "book", "relationship": "adapts", "endorsement": "" }, { "title": "An Introduction to Complex Analysis in Several Variables", "id": "2nd edition, North-Holland Mathematical Library 7, North-Holland, 1973", "authors": [ "Lars Hörmander" ], "type": "book", "relationship": "adapts", "endorsement": "" }, { "title": "From Holomorphic Functions to Complex Manifolds", "id": "Graduate Texts in Mathematics 213, Springer, 2002", "authors": [ "Klaus Fritzsche", "Hans Grauert" ], "type": "book", "relationship": "adapts", "endorsement": "" }, { "title": "Lectures on Holomorphic Functions of Several Complex Variables", "id": "Jagiellonian University manuscript, 1997-2021", "authors": [ "Piotr Jakóbczak", "Marek Jarnicki" ], "type": "other", "relationship": "adapts", "endorsement": "" }, { "title": "Introduction to Complex Analysis in Several Variables", "id": "Birkhäuser, 2005", "authors": [ "Volker Scheidemann" ], "type": "book", "relationship": "adapts", "endorsement": "" }, { "title": "Several Complex Variables", "id": "University of Amsterdam lecture notes, version of 23 August 2017", "authors": [ "Jaap Korevaar", "Jan Wiegerinck" ], "type": "other", "relationship": "adapts", "endorsement": "" }, { "title": "Lecture Notes on Several Complex Variables", "id": "Draft of 3 December 2013", "authors": [ "Harold P. Boas" ], "type": "other", "relationship": "adapts", "endorsement": "" }, { "title": "Complex Analytic Geometry: From the Localization Viewpoint", "id": "World Scientific, 2024", "authors": [ "Tatsuo Suwa" ], "type": "book", "relationship": "adapts", "endorsement": "" }, { "title": "Introduction to Complex Analysis, Part II: Functions of Several Variables", "id": "Translations of Mathematical Monographs 110, American Mathematical Society, 1991", "authors": [ "Boris V. Shabat" ], "type": "book", "relationship": "background", "endorsement": "" }, { "title": "Tasty Bits of Several Complex Variables: A Whirlwind Tour of the Subject", "id": "Version 4.4, 31 May 2026", "authors": [ "Jiří Lebl" ], "type": "other", "relationship": "background", "endorsement": "" }, { "title": "Potential Theory in the Complex Plane", "id": "London Mathematical Society Student Texts 28, Cambridge University Press, 1995", "authors": [ "Thomas Ransford" ], "type": "book", "relationship": "background", "endorsement": "" } ], "related": [ { "id": "https://github.com/leanprover-community/mathlib4", "relationship": "builds-on", "note": "The development uses Mathlib throughout, in particular its one-variable Cauchy theory, torus integrals, analytic functions, the analytic identity principle, the local maximum modulus principle, circle averages and the Laplacian. Mathlib's formal Weierstrass preparation theorem for power series over complete local rings is not used; the analytic division and preparation theorems here are proved for" }, { "id": "https://palomar-registry.org/entry?id=PALOMAR-2026-08-29-000010&version=1", "relationship": "independent", "note": "Bochao Kong, classical-complex-wpt (repository BochaoKong/classical-complex-wpt, commit b4a7273fe5c9752753c52e10494097569089642d, registered 29 August 2026): the complex-analytic Weierstrass preparation theorem at the origin of C^n, with uniqueness of the factors as germs. It overlaps with item 41 here (SCV.weierstrass_preparation). The present development was carried out independently of it and d" }, { "id": "https://palomar-registry.org/entry?id=PALOMAR-2026-08-30-000001&version=1", "relationship": "independent", "note": "Bochao Kong, nullstellensatz (repository BochaoKong/nullstellensatz, commit 029697242294aea7989bf5d391199696a43490df, registered 30 August 2026), built on the entry above: Rückert's basis theorem for the ring of analytic germs, finite projection for analytic hypersurface germs, the holomorphic constant-rank normal form, and Rückert's local analytic Nullstellensatz. Its basis theorem overlaps with " } ], "classification": { "arxiv": [ "math.CV" ], "msc2020": [ "32A10", "32A05", "32D05", "32D15", "32D20", "32B05", "32U05", "68V20" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "gpt-6-astra" ], "framework": "Codex" }, { "method": "agent", "models": [ "claude-fable-5-1" ], "framework": "Claude Code" } ], "strongest": "agent", "models": [ "claude-fable-5-1", "gpt-6-astra" ], "spend": "", "notes": "Human-directed, agent-assisted mathematical development, proof writing, debugging and review, validated throughout by Lean builds. The author chose the material, the conventions and the sources and directed the work; coding agents wrote most of the Lean text. Historical model versions and per-session costs are not reconstructed here. Challenge.lean is generated from Solution.lean by the script make_challenge.py, which removes the import of the library and the bridging lemmas and replaces every proof by sorry, so that the statements and definitions of the two files are textually identical." }, "scope": "The compared statements are the 67 theorems of Challenge.lean. They follow the project's catalogue of main theorems (SCVMainTheorems.md, 65 items in sections A-K); the number in each docstring of Challenge.lean is the catalogue item. Every item is represented except items 42 and 43 (factors of distinguished polynomials; comparison of polynomials over the germ ring with germs), which are algebraic steps towards items 44 and 45 and would require restating a larger block of definitions. An item with several assertions is represented by its principal assertion. Not covered by the project: the Levi", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "SCV.cauchy_formula_polydisc", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.osgood", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.differentiableOn_iff_analyticOnNhd", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.analyticOnNhd_iff_cauchyRiemann", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.identity_theorem", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.maximum_modulus", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.cauchy_pompeiu", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.cauchy_estimates", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.analyticOnNhd_integral", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.weierstrass_convergence", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.isClosed_holomorphicMaps", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.montel", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.vitali", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.holomorphic_Lp_bound", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.inverse_mapping", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.implicit_mapping", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.IsCompleteReinhardt.isReinhardt_and_isPathConnected", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.isLogarithmicallyConvex_iff_geometric", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.convergenceDomain_properties", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.exists_powerSeries_eqOn_of_isCompleteReinhardt", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.exists_convergenceDomain_eq", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.existsUnique_laurent_expansion", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.exists_extension_completeReinhardtHull", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.exists_extension_balancedHull", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.hartogs_taylor_expansion", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.hartogs_cylinder_extension", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.hartogs_separate_analyticity", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.exists_extension_diff_singleton", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.frequently_zero_punctured", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.riemann_extension_first", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.hartogs_compact_hole", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.IsAnalyticSet.interior_eq_empty_and_isConnected_sdiff", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.isRegularAnalyticSetAt_iff_exists_equations", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.exists_extension_across_coordinatePlane", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.riemann_extension_second", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.analyticGermRing_isDomain_isLocalRing", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.exists_regular_coordinate_change", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.taylorSeries_mul_and_eq_zero_iff", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.coordinatePower_division", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.weierstrass_division", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.weierstrass_preparation", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.analyticGermRing_isNoetherianRing_ufd", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.isOpen_isRelPrime_locus", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.exists_regularPoint_zeroSet", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.injOn_biholomorphic", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.cartan_uniqueness", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.biholomorphic_circular_linear", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.exists_ball_automorphism", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.not_biholomorphic_polydisc_ball", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.common_extension", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.holomorphicHull_idem_and_isBounded", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.isHolomorphicallyConvex_of_completeReinhardt", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.isDomainOfHolomorphy_of_convex_and_pi", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.isHolomorphicallyConvex_iff_escaping", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.isDomainOfHolomorphy_iff_hull_radius", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.cartan_thullen", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.bochner_tube", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.SubharmonicOn.eqOn_const_of_isMaxOn", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.subharmonicOn_iff_laplacian_nonneg", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.plurisubharmonicOn_iff_leviForm_nonneg", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.IsDomainOfHolomorphy.isPseudoconvex", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.IsPseudoconvex.continuity_principle", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.IsDomainOfHolomorphy.isLeviPseudoconvexAt", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.leviForm_comp_analytic", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.isLeviPseudoconvexAt_iff_of_defining", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.exists_local_peak_function", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SCV.runge_domains", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "Statements are phrased with Mathlib's predicates rather than those of any one source. (1) Holomorphic is DifferentiableOn over the complex numbers and analytic is AnalyticOnNhd; they agree on open subsets of finite-dimensional spaces (item 3), and hypotheses use whichever the library proves directly. (2) Coordinate spaces carry the supremum norm, so metric balls are polydiscs; the hull-radius form of Thullen's lemma (item 55) is for this norm. (3) Domains are not assumed connected or nonempty unless stated; dimension zero and empty index types are included. (4) Holomorphic and polynomial hulls are defined through all real upper bounds instead of suprema. (5) IsDomainOfHolomorphy is a generalized continuation-obstruction predicate defined for arbitrary sets; openness, connectedness and nonemptiness are separate hypotheses. It tests common continuation through a fixed nonempty open overlap and is vacuously true when the set has empty interior. In item 53, the product assertion is proved for arbitrary plane factors with this meaning. Its specialization to nonempty connected open plane factors is the classical product theorem for domains of holomorphy. (6) Subharmonic and plurisubharmo", "alignment": false, "original": false, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [], "notes": "The library, Challenge.lean and Solution.lean build with Lean 4.34.0 and the pinned Mathlib without errors, warnings or sorry in the library. The compared declarations use only propext, Quot.sound and Classical.choice. The declarations of Challenge.lean, including all auxiliary definitions, were checked to have the same types and values in the Solution environment by comparing expression hashes, which emulates Comparator's declaration check. An official Comparator and NanoDa run and Palomar's editorial review are pending. No independent human peer review is claimed." }, "canonical": { "repo": "bjbraams/lean-scv", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-18-000008", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-18-000008", "commit": "336759b3964011aa0d6e1111e32276b1476fb7f0", "trust": "high", "theorems": 67, "date": "2026-09-18" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "bjrtx/ordinal-step-up/formalization.yaml", "repo": "bjrtx/ordinal-step-up", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/bjrtx/ordinal-step-up/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Erdős–Rado negative stepping-up for ordinal order types", "description": "A Lean 4 formalisation of the Erdős–Rado negative stepping-up lemma in its order-type form, for arities three and above and for at least two colours. Fix an ordinal kappa; a colour count mu ≥ 2; and a target order type theta = omega^alpha with alpha > 0, so that theta is additively indecomposable. The theorem proved is, for every natural number r: if some colouring of the (r+3)-subsets of kappa by ordinals below mu admits no monochromatic subset of kappa of order type theta, then some colouring of the (r+4)-subsets of 2^kappa admits none either, using the same number of colours. The constructi", "authors": [ "Bruno Jartoux" ], "maintainers": [ "Bruno Jartoux" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "\"A partition calculus in set theory\" by Erdős and Rado for readers from the twenty-first century", "id": "https://eprints.illc.uva.nl/id/eprint/1797", "authors": [ "David Joël de Graaf" ], "type": "thesis", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "Combinatorial Set Theory: Partition Relations for Cardinals", "id": "ISBN:978-0-444-86157-2", "authors": [ "Paul Erdős", "András Hajnal", "Attila Máté", "Richard Rado" ], "type": "book", "relationship": "background", "endorsement": "" }, { "title": "A partition calculus in set theory", "id": "doi:10.1090/S0002-9904-1956-10036-0", "authors": [ "Paul Erdős", "Richard Rado" ], "type": "article", "relationship": "background", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.LO", "math.CO" ], "msc2020": [ "03E02", "05D10" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "claude-opus-5" ], "framework": "Claude Code" } ], "strongest": "agent", "models": [ "claude-opus-5" ], "spend": "subscription-based usage", "notes": "iterative development via agent-human 'conversation'" }, "scope": "Formalization of Theorem 4.18 in (de Graaf 2021), with no sorry and no axioms beyond propext, Classical.choice and Quot.sound.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "OrdinalStepUp.ErdosRado.noHomog_stepUp", "file": "OrdinalStepUp/Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "Colourings are total functions on all of Finset Ordinal, rather than partial functions defined on the r-subsets of the ground set; their values elsewhere are immaterial and unconstrained. Since the colourings are total, kappa is not an ambient type but a bound on the sets quantified over: subsets of Set.Iio kappa downstairs, and of its powerset upstairs. Order types upstairs are taken with respect to an arbitrary well-order on the powerset, obtained from well-ordering; the proof never uses which. The source's theta^- correction, needed there because a successor-length homogeneous set has a top point contributing no discrepancy, is unnecessary here: indecomposability forces theta to be a limit, so every point of such a set has a successor in it.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "" }, "canonical": { "repo": "bjrtx/ordinal-step-up", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-08-000003", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-08-000003", "commit": "e46994fca709e3983061ef8d52a824b29a613ee4", "trust": "high", "theorems": 1, "date": "2026-09-08" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "blawton/fried-conj-lean/formalization.yaml", "repo": "blawton/fried-conj-lean", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/blawton/fried-conj-lean/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Resonance-cluster analysis for a proposed counterexample to Fried's conjecture", "description": "A Lean 4 formalization of the theorem FriedCrossing.fried_counterexample_of_resolvent_inputs, the machine-checked reduction behind a proposed codimension-one failure of Fried's identity ζ(0) = τ_R for acyclic characters on a closed hyperbolic 3-manifold with b₁ = 1. For a simple illustration of the method with figures, see docs/counterexample_intuitive_view.pdf. The Mechanism: Near (g_hyp, ρ_triv), along the conformal family g_τ = e^{−2τb}g_hyp with characters χ_θ, the Ruelle resonances near 0 form two small clusters: a rank-2 cluster in degree 1, encoded by a 2×2 matrix M(θ,τ), and a rank-4 c", "authors": [ "Benjamin Lawton" ], "maintainers": [ "Benjamin Lawton" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "A codimension-one failure of Fried's identity for acyclic characters on nearly hyperbolic 3-manifolds", "id": "", "authors": [ "Benjamin Lawton" ], "type": "original-proof", "relationship": "other", "endorsement": "participated" }, { "title": "The Ruelle zeta function at zero for nearly hyperbolic 3-manifolds", "id": "", "authors": [ "Mihajlo Cekić", "Benjamin Delarue", "Semyon Dyatlov", "Gabriel P. Paternain" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "The Fried conjecture in small dimensions", "id": "", "authors": [ "Nguyen Viet Dang", "Colin Guillarmou", "Gabriel Rivière", "Shu Shen" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Dynamical torsion for contact Anosov flows", "id": "", "authors": [ "Yann Chaubet", "Nguyen Viet Dang" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Topology of Pollicott–Ruelle resonant states", "id": "", "authors": [ "Nguyen Viet Dang", "Gabriel Rivière" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Selberg Zeta and Theta Functions", "id": "", "authors": [ "Ulrich Bunke", "Martin Olbrich" ], "type": "book", "relationship": "background", "endorsement": "n/a" }, { "title": "Power spectrum of the geodesic flow on hyperbolic manifolds", "id": "", "authors": [ "Semyon Dyatlov", "Frédéric Faure", "Colin Guillarmou" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Analytic torsion and closed geodesics on hyperbolic manifolds", "id": "", "authors": [ "David Fried" ], "type": "paper", "relationship": "background", "endorsement": "n/a" } ], "related": [], "classification": { "arxiv": [ "math.DS", "math.SP" ], "msc2020": [ "37C30", "37D20", "58J52", "57Q10" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "claude-fable-5-1" ], "framework": "Claude Code" }, { "method": "manual", "models": [], "framework": "" } ], "strongest": "agent", "models": [ "claude-fable-5-1" ], "spend": "subscription-based Claude Code usage; per-project spend not recorded separately", "notes": "Mechanical checks run locally before submission: lake build of the whole suite; #print axioms on the compared declaration ([propext, Classical.choice, Quot.sound]); leanprover/comparator (tag v4.33.0) on comparator.json with Challenge.lean generated from B1s/fried_statement_defs_9_14.lean. The statement's definitions are kept in one module because comparator compares definitions by value and Lean's per-module cache of lifted auxiliary proofs (numerals ⇒ Nat.AtLeastTwo instances) otherwise gives the same definition different auxiliary constants in the two environments. scripts/ledger_check.py c" }, "scope": "One compared declaration, FriedCrossing.fried_counterexample_of_resolvent_inputs, proved in B1s/fried_counterexample_main.lean from the leg files fried_crossing_9_03, fried_crossing_rate_9_07, fried_crossing_firstvariation_9_08, fried_crossing_purezeros_9_08, fried_cluster_matrices_9_10, resolvent_scale_9_11, cluster_from_resolvent_9_11, with every definition in its statement collected in B1s/fried_statement_defs_9_14.lean, which Challenge.lean reproduces verbatim. The theorem is conditional: its sixteen binders are the cited facts listed in the description and in the docstrings of Challenge.l", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "FriedCrossing.fried_counterexample_of_resolvent_inputs", "file": "B1s/fried_counterexample_main.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The formal theorem is universally quantified over abstract data satisfying the sixteen hypotheses; the informal claim is existential (a manifold, a character family and a metric at which ζ(0; g_σ) ≠ τ_R). The gap between them is exactly the geometric identification, which is not formalized: nothing in Lean asserts that data satisfying all sixteen binders exist, and the binder docstrings are where each fact is attributed to its source. Specific points. (i) M and A are defined as REAL PARTS of the complex frame matrices A(p)⁻¹B(p); that the imaginary parts vanish (conjugation symmetry of the cluster) is interpretive and not consumed by the proof. (ii) The frames are assumed nondegenerate (hdet₁, hdet₂: det A(p) ≠ 0 for all p); informally this is the choice of a dual frame at (0,0) with the parameters retracted into the region where it stays invertible. (iii) The resolvent families are stated globally in the parameter (θ,τ,λ) ∈ (ℝ×ℝ)×ℂ; informally they are the pull-backs by smooth retractions of (θ,τ) into a neighbourhood of (0,0) and of λ into an annulus around the fixed contour where the resolvent exists (CDDP §4.2). (iv) hstates (two linearly independent states at s*(θ)) is derived", "alignment": true, "original": true, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [], "notes": "No independent human mathematician has reviewed the development or the informal account. The author has read the cited papers against each ledger row (the [V]/[D]/[def] status column of the README) but has not independently verified informal-to-formal fidelity beyond that. Model-side checks: the construction was attacked along five lines by a second model instance and re-derived from the cited texts by a third with no access to the earlier derivations, and the three agree (informal account, Disclosure). What is established mechanically: the compared declaration is proved from the three standard axioms, and comparator matches it constant by constant against Challenge.lean." }, "canonical": { "repo": "blawton/fried-conj-lean", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-17-000006", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-17-000006", "commit": "610d659ea15af869553f6e3e79c432880d9b0ab0", "trust": "high", "theorems": 1, "date": "2026-09-17" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "BochaoKong/classical-complex-wpt/formalization.yaml", "repo": "BochaoKong/classical-complex-wpt", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/BochaoKong/classical-complex-wpt/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Classical Complex-Analytic Weierstrass Preparation", "description": "A Mathlib-only Lean 4 formalization of the classical complex-analytic Weierstrass preparation theorem at the origin. For every number of base variables and every preparation degree, including the zero-dimensional and degree-zero cases, it constructs the distinguished monic polynomial and nonvanishing analytic unit and proves uniqueness of all factors as germs.", "authors": [ "BochaoKong" ], "maintainers": [ "BochaoKong" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Tasty Bits of Several Complex Variables", "id": "https://www.jirka.org/scv/scv.pdf", "authors": [ "Jiří Lebl" ], "type": "textbook", "relationship": "independently-proves", "endorsement": "not-contacted" } ], "related": [ { "id": "https://github.com/leanprover-community/mathlib4/blob/81a5d257c8e410db227a6665ed08f64fea08e997/Mathlib/RingTheory/PowerSeries/WeierstrassPreparation.lean", "relationship": "independent", "note": "Mathlib's existing theorem is algebraic preparation for formal power series over complete local rings. It neither implies nor is used as a substitute for the analytic convergence theorem proved here." } ], "classification": { "arxiv": [ "math.CV" ], "msc2020": [ "32A05", "32A10", "32B05" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "OpenAI GPT-5-family Codex (exact backend identifier not exposed)" ], "framework": "Codex multi-agent v2" } ], "strongest": "agent", "models": [ "OpenAI GPT-5-family Codex (exact backend identifier not exposed)" ], "spend": "subscription-based; not tracked", "notes": "Codex performed the implementation and automated audits under user direction. No human mathematical or code review is claimed by this metadata." }, "scope": "Proves at the origin the classical complex-analytic Weierstrass preparation theorem for f : C^n x C -> C from an exact distinguished-variable order expressed by iterated derivatives. It constructs analytic lower coefficients vanishing at zero, an analytic unit nonzero at zero, an eventual-neighborhood factorization, and full coefficient and unit germ uniqueness, uniformly for all natural n and d. An explicit open-neighborhood corollary is included. An arbitrary-base-point translation and a public general division theorem are not included.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The compared theorem is the origin-based germ formulation and therefore does not retain the source theorem's chosen polydisc or its clause locating every polynomial zero inside the distinguished disc. It includes the natural d=0 unit case and n=0 one-variable case. Analyticity and equality of germs are encoded by AnalyticAt C and EventuallyEq at the neighborhood filter. The public unit condition is nonvanishing at the origin; continuity yields, and a proved corollary exposes, nonvanishing on a smaller open neighborhood. The proof architecture is independent of the cited contour proof.", "alignment": true, "original": true, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [ "OpenAI Codex agents (automated; no external human reviewer)" ], "notes": "The frozen statement received an independent semantic and adversarial audit. Separate Codex agents audited the weighted convergence bridge, the normalization/division construction, and full germ uniqueness. No external human peer review or Palomar editorial review has occurred." }, "canonical": { "repo": "bochaokong/classical-complex-wpt", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-29-000010", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-29-000010", "commit": "b4a7273fe5c9752753c52e10494097569089642d", "trust": "high", "theorems": 1, "date": "2026-08-29" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "BochaoKong/nullstellensatz/formalization.yaml", "repo": "BochaoKong/nullstellensatz", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/BochaoKong/nullstellensatz/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Foundational Theorems in Local Complex-Analytic Geometry", "description": "A Lean 4 formalization of Rückert's Noetherianity theorem for local holomorphic-germ rings, algebraic and genuinely geometric finite projection for analytic hypersurface germs, the holomorphic constant-rank normal form, and Rückert's local analytic Nullstellensatz in finite-family and arbitrary-ideal radical forms.", "authors": [ "BochaoKong" ], "maintainers": [ "BochaoKong" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Analytic Functions of Several Complex Variables", "id": "https://doi.org/10.1090/chel/368", "authors": [ "Robert C. Gunning", "Hugo Rossi" ], "type": "book", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "Introduction to Complex Analytic Geometry", "id": "https://doi.org/10.1007/978-3-0348-7617-9", "authors": [ "Stanisław Łojasiewicz" ], "type": "book", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "Zum Eliminationsproblem der Potenzreihenideale", "id": "https://doi.org/10.1007/BF01448893", "authors": [ "Wolfgang Rückert" ], "type": "paper", "relationship": "background", "endorsement": "n/a" } ], "related": [ { "id": "https://github.com/BochaoKong/classical-complex-wpt/tree/b4a7273fe5c9752753c52e10494097569089642d", "relationship": "builds-on", "note": "Exact pinned analytic Weierstrass-preparation dependency. This project reuses its public convergence theorem and supplies its own germ division, geometry, Noetherianity, constant-rank, and Nullstellensatz developments." }, { "id": "https://github.com/leanprover-community/mathlib4/blob/81a5d257c8e410db227a6665ed08f64fea08e997/Mathlib/RingTheory/Nullstellensatz.lean", "relationship": "independent", "note": "Mathlib's result is Hilbert's polynomial Nullstellensatz; it is not the local analytic theorem proved here." } ], "classification": { "arxiv": [ "math.CV", "math.AG" ], "msc2020": [ "32A10", "32B05", "32C25", "13E05" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "OpenAI GPT-5-family Codex (exact backend identifier not exposed)" ], "framework": "Codex multi-agent v2" } ], "strongest": "agent", "models": [ "OpenAI GPT-5-family Codex (exact backend identifier not exposed)" ], "spend": "subscription-based; not tracked", "notes": "Codex performed implementation and automated audits under user direction. No external human mathematical or code review is claimed." }, "scope": "The holomorphic germ ring is the analytic subring of Mathlib's neighborhood function-germ ring at zero. Noetherianity holds in every finite dimension, including zero. Finite projection expresses rank-d finite freeness as unique quotient expansion in powers of the last-coordinate germ and couples it to an analytic representative with boundary control, finite fibers, surjectivity, and proper projection over an open base. Constant rank is stated at an arbitrary base point using four explicit analytic coordinate maps, their four local inverse identities, and the direct standard-rank formula; arbit", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "LocalComplexGeometry.holomorphicGerm_isNoetherian", "file": "LocalComplexGeometry/Palomar.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "LocalComplexGeometry.hypersurface_finiteProjection", "file": "LocalComplexGeometry/Palomar.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "LocalComplexGeometry.holomorphic_constantRank_normalForm", "file": "LocalComplexGeometry/Palomar.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "LocalComplexGeometry.localAnalyticNullstellensatz", "file": "LocalComplexGeometry/Palomar.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "LocalComplexGeometry.holomorphic_regularLevelSet_parametrization", "file": "LocalComplexGeometry/Analytic/LevelSet.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "LocalComplexGeometry.analyticNullstellensatz_ideal", "file": "LocalComplexGeometry/Nullstellensatz/Main.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The germ-ring, finite-projection, and Nullstellensatz results are based at the origin. The constant-rank theorem and its submersion, immersion, and regular-level-set corollaries permit an arbitrary base point. The compared finite-projection theorem expresses rank-d finite freeness as unique quotient expansion in powers of the last-coordinate germ and states all geometric coordinates directly; d bounds the number of distinct points in each fiber rather than separately exposing intersection multiplicities. The compared finite-family Nullstellensatz is representative-level and neighborhood-local, while the arbitrary-ideal radical equality is a proved but separately audited library theorem. The Challenge imports only Mathlib; ClassicalComplexWPT is used only by the proved Solution.", "alignment": true, "original": false, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [ "OpenAI Codex agents (automated; no external human reviewer)" ], "notes": "Separate agents audited theorem semantics, finite-projection topology, germ well-definedness, Nullstellensatz elimination, dependency discipline, and axiom closure. Palomar editorial review has not occurred." }, "canonical": { "repo": "bochaokong/nullstellensatz", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-30-000001", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-30-000001", "commit": "029697242294aea7989bf5d391199696a43490df", "trust": "high", "theorems": 4, "date": "2026-08-30" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "BochaoKong/oka-coherence/formalization.yaml", "repo": "BochaoKong/oka-coherence", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/BochaoKong/oka-coherence/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Oka Coherence in Finite-Dimensional Complex Coordinate Space", "description": "A Lean 4 formalization of the neighborhood-uniform finite-matrix kernel criterion for Oka coherence on every finite-dimensional complex coordinate space. For an analytic matrix germ at a point, the theorem constructs one finite family of analytic kernel sections on a common neighborhood whose germs generate every nearby kernel stalk.", "authors": [ "BochaoKong" ], "maintainers": [ "BochaoKong" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Sur les fonctions analytiques de plusieurs variables. VII. Sur quelques notions arithmétiques", "id": "https://doi.org/10.24033/bsmf.1408", "authors": [ "Kiyoshi Oka" ], "type": "paper", "relationship": "independently-proves", "endorsement": "" }, { "title": "Complex Analytic and Differential Geometry", "id": "https://www-fourier.univ-grenoble-alpes.fr/~demailly/manuscripts/agbook.pdf", "authors": [ "Jean-Pierre Demailly" ], "type": "book", "relationship": "adapts", "endorsement": "" }, { "title": "Analytic Functions of Several Complex Variables", "id": "https://doi.org/10.1090/chel/368", "authors": [ "Robert C. Gunning", "Hugo Rossi" ], "type": "book", "relationship": "background", "endorsement": "" } ], "related": [ { "id": "https://github.com/BochaoKong/classical-complex-wpt/tree/b4a7273fe5c9752753c52e10494097569089642d", "relationship": "builds-on", "note": "Exact pinned, public Solution-only dependency supplying complex-analytic Weierstrass preparation and division infrastructure. It is not in the Challenge source closure and does not prove Oka coherence." }, { "id": "https://github.com/BochaoKong/nullstellensatz/tree/029697242294aea7989bf5d391199696a43490df", "relationship": "builds-on", "note": "Exact pinned, public Solution-only LocalComplexGeometry dependency. Its analytic-germ and local-geometry results are reused by the proof; its Noetherian stalk theorem is not substituted for neighborhood-uniform coherence." } ], "classification": { "arxiv": [ "math.CV" ], "msc2020": [ "32C35", "32A10" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "OpenAI Codex, GPT-5 family (exact service identifier not recorded)" ], "framework": "Codex multi-agent coordination" } ], "strongest": "agent", "models": [ "OpenAI Codex, GPT-5 family (exact service identifier not recorded)" ], "spend": "subscription-based; not tracked", "notes": "Codex performed implementation and automated audits under user direction. No external human mathematical or code review is claimed." }, "scope": "The proved declaration OkaCoherence.okaCoherence is stated for every n, p, q : ℕ, every analytic q-by-p complex matrix germ A at x in ℂ^n, and all degenerate cases including n = 0, p = 0, and q = 0. It returns one open neighborhood U of x, one finite family of analytic kernel sections on U, and proves that the germs of that same family generate the kernel relation module at every y in U, with analytic coefficient germs and neighborhood-eventual equality. This is the explicit coordinate finite-matrix criterion for Oka coherence. The development does not construct the categorical holomorphic str", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "OkaCoherence.okaCoherence", "file": "OkaCoherence/Main.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The formal theorem is the explicit neighborhood-uniform finite-matrix kernel statement on ℂ^n. Conventional sheaf theory identifies it with coherence of the holomorphic structure sheaf, but this repository neither constructs that categorical sheaf nor proves the categorical equivalence in Lean. The theorem is local to finite-dimensional coordinate spaces and does not state the arbitrary-complex-manifold globalization. Allowing an arbitrary finite number q of matrix rows is the standard finite-free kernel strengthening of the single-row relation criterion, not a weaker stalkwise Noetherianity claim.", "alignment": true, "original": true, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [ "OpenAI Codex agents (automated; no external human reviewer)" ], "notes": "Independent automated audits checked the quantifier order, germ semantics, zero-rank cases, proof architecture, integration chain, prior art, Challenge dependency boundary, source scan, axiom closure, Comparator, and NanoDa replay. Palomar editorial review and external human mathematical review have not occurred." }, "canonical": { "repo": "bochaokong/oka-coherence", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "BochaoKong/permutohedron/formalization.yaml", "repo": "BochaoKong/permutohedron", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/BochaoKong/permutohedron/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "To cover a permutohedron", "description": "This Lean 4 formalization of Kong and Zeng's To cover a permutohedron (arXiv:2509.13877v3, 9 April 2026) determines the minimum number of affine hyperplanes needed to cover the permutation vertices of (1,...,n), excluding the hyperplane containing every vertex. The exact answer is n for odd n at least 3, and n-1 for even n at least 4. It also proves that an arbitrary injective real tuple of length n at least 2 has the sharp hyperplane intersection bound (n-1)! when its n-1 signed critical values are distinct. Genuine polynomial-root path lifting establishes local transpositions and the cycle a", "authors": [ "Bochao Kong" ], "maintainers": [ "Bochao Kong" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "To cover a permutohedron", "id": "https://arxiv.org/abs/2509.13877v3", "authors": [ "Bochao Kong", "Ji Zeng" ], "type": "paper", "relationship": "adapts", "endorsement": "other" }, { "title": "Covering the permutohedron by affine hyperplanes", "id": "https://arxiv.org/abs/2305.06202v3", "authors": [ "Gábor Hegedüs", "Gyula Károlyi" ], "type": "paper", "relationship": "adapts", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "52B05" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "GPT-6" ], "framework": "Codex desktop native multi-agent collaboration" } ], "strongest": "agent", "models": [ "GPT-6" ], "spend": "Not tracked", "notes": "AI-generated code and audits require the same kernel, statement-fidelity, dependency, and replay checks as other proof sources. Agent review is not represented as human mathematical review or peer review." }, "scope": "Theorem 1; both parity cases of Corollary 2; Theorem 3 and its covering consequence; repaired Claims 4 and 5; both sharp constructions; the generic-tuples and four-element consequences; exact convex-vertex, affine-span, arbitrary-hyperplane, and dimension-one bridges. Every selected declaration has a checked implementation and is stated in one independent Mathlib-only Challenge; comparator.json selects all eighteen. Their implementations were verified in the earlier split configuration; consolidation preserves their mathematical content. 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loop hypotheses: transported small meridians and a once-traversed large positive circle. Inversion orientation, adjacent-extrema indexing, compatible meridian access paths, and the cited even-facet typo are explicitly documented. The source’s Bézout argument is replaced by the checked symmetric-norm and vanishing-order proof; no general Bézout theorem is claimed. Dimension one is separately proved. The mathematical source-repair counterexamples are documented; their entire counterexample arguments are not separately advertised as formalized. The verified completion status is COMPLETE_WITH_DOCUMENTED_SOURCE_REPAIRS.", "alignment": true, "original": false, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [], "notes": "Independent AI agents audited source fidelity, statement meanings, hypotheses, geometric bridges, and proof dependencies. Human mathematical review, paper-author verification, and peer review are not claimed. 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It formalizes their star theorem for hereditary set families, sharp Boolean Fourier correlation inequality, antipodal corollary, sharpness results, and weighted strengthening. The formalization was completely prepared by GPT-6.", "authors": [ "Boon Suan Ho" ], "maintainers": [ "Boon Suan Ho" ], "license": "MIT", "role": "substantive-development", "substantive": "", "sources": [ { "title": "A proof of Chvátal's conjecture via a sharp correlation inequality", "id": "https://arxiv.org/abs/2609.19123v1", "authors": [ "Fan Chang", "Hong Liu", "Miao Liu" ], "type": "paper", "relationship": "formalizes", "endorsement": "other" } ], "related": [ { "id": "https://github.com/leanprover-community/mathlib4/tree/0df444a360eaa60ab8c11dca51a86af692955474", "relationship": "builds-on", "note": "Pinned mathematical library. The development reuses its finite-set, intersecting-family, finite-dimensional inner-product, and Gram-Schmidt results, with the normalizations and reductions needed for the paper proved locally." } ], "classification": { "arxiv": [ "math.CO", "math.PR" ], "msc2020": [ "05D05" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "GPT-6" ], "framework": "Codex" } ], "strongest": "agent", "models": [ "GPT-6" ], "spend": "", "notes": "GPT-6 generated the Lean formalization, scripts, and documentation, using parallel agents where useful. My involvement was prompting, setting requirements, and maintaining the repository. The original mathematical proof is Chang, Liu, and Liu's, not an AI-discovered result from this project. Compilation and kernel replay are mechanical checks, not human mathematical review." }, "scope": "The repository contains the full source-based development, including the numbered results and equations mapped in docs/PaperMap.md. Comparator's compact statement of record selects nine principal declarations: the expanded star theorem (1.1), sharp inequality (1.2), antipodal corollary (1.3), quadratic inequality (1.4), AND equality and coefficient optimality (5.1), numerical counterexample (5.2), and both the weighted bound and weighted maximum-attainment conclusion (5.3). The intermediate results remain proved and axiom-audited in the library; they are not separately selected Comparator clai", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "ChvatalSubmission.chvatal", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Chvatal.sharp_correlation", "file": "Chvatal/Main.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Chvatal.antipodal_correlation", "file": "Chvatal/Main.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Chvatal.two_spectral_le_quadratic_covariance", "file": "Chvatal/Correlation.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Chvatal.andFunction_sharp", "file": "Chvatal/Sharpness.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Chvatal.quarter_coefficient_optimal", "file": "Chvatal/Sharpness.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Chvatal.and_or_two_example", "file": "Chvatal/Sharpness.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Chvatal.kleitman_weighted_bound", "file": "Chvatal/Weighted.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Chvatal.exists_largest_weighted_intersecting_star", "file": "Chvatal/Weighted.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The paper's cube is represented by finite subsets of an arbitrary finite coordinate type; Fin n gives its n-dimensional case. Statements requiring a star center, a minimum coordinate, or the maximal-intersecting/antipodal characterization explicitly require a nonempty coordinate type. This states the paper's implicit n >= 1 convention. The main quadratic and harmonic correlation inequalities also hold for zero coordinates. Influence is mean squared flip difference, proved equal to change probability for Boolean functions. The empty-index maximum is zero and contributes nothing to the spectral sum. Lean's zero-denominator division agrees with Theorem 1.2's convention. Proposition 5.3 is first proved for any selector contained in each nonempty index and then specialized to a fixed total-order maximum. Its layer-cake step is a finite positive-level decomposition rather than an integral, with no restriction on real nonnegative decreasing weights. No weakening of the advertised principal statements was identified by the AI-assisted checks. These checks do not establish that every sentence of the informal paper is correct.", "alignment": true, "original": false, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [ "GPT-6 (Codex agents)" ], "notes": "Agents checked correspondence with the source, definitions, normalization, boundary cases, theorem statements, and axiom dependencies. No independent human expert review is claimed. No source-author endorsement or Palomar editorial review is claimed. Observed local verification outcomes and exact tool revisions are recorded in docs/Verification.md and logs/. 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Writing P_c(m, n) for the number of distinct m-by-n block patterns occurring in a configuration c, Nivat conjectured that if P_c(m, n) <= mn for some positive m and n, then c is periodic; that is, there exists nonzero h such that c(v+h) = c(v) for all v. The proof is due to GPT-6 Pro.", "authors": [ "Boon Suan Ho" ], "maintainers": [ "Boon Suan Ho" ], "license": "MIT", "role": "", "substantive": "", "sources": [ { "title": "Pattern complexity and Nivat's conjecture", "id": "paper/nivat.tex", "authors": [], "type": "paper", "relationship": "formalizes", "endorsement": "n/a" }, { "title": "An Algebraic Geometric Approach to Nivat's Conjecture", "id": "https://arxiv.org/abs/1605.05929v1", "authors": [ "Jarkko Kari", "Michal Szabados" ], "type": "paper", "relationship": "adapts", "endorsement": "" }, { "title": "Nonexpansive Z^2 subdynamics and Nivat's conjecture", "id": "https://arxiv.org/abs/1208.4090v2", "authors": [ "Van Cyr", "Bryna Kra" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Nivat's conjecture holds for sums of two periodic configurations", "id": "https://arxiv.org/abs/1710.05360v1", "authors": [ "Michal Szabados" ], "type": "paper", "relationship": "adapts", "endorsement": "" }, { "title": "Symbolic dynamics", "id": "https://doi.org/10.2307/2371264", "authors": [ "Marston Morse", "Gustav A. 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The mathematical proof and paper were not written by humans; the Lean code and documentation were generated by AI agents. Compilation and kernel checks are mechanical verification, not human mathematical review. Full cost and generation-run metadata were not tracked in this repository. The maintainer is not a specialist and does not plan to undertake mathematical digestion or expert exposition personally; work on these by experienced mathematicians is welcome." }, "scope": "The full finite-alphabet rectangular-complexity implication on Z^2, together with the intermediate results needed by the supplied proof, including its finite-rational two-factor theorem. Numbered examples and unnumbered illustrations in the manuscript are explanatory, not separate formal results. The compared declaration is NivatSubmission.nivat; Nivat.nivat is the corresponding reusable library endpoint.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "NivatSubmission.nivat", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "No weakening of the final theorem is known from the recorded automated statement audits. The Lean statement quantifies over all universes of finite alphabets and all Z^2 configurations. It counts restrictions over all translations of the original integer rectangle and yields one nonzero global period. Rational labeling is an internal proved reduction. Supporting interfaces use finite-pattern language inclusion instead of orbit-closure hypotheses where that is sufficient; local compactness lemmas discharge these hypotheses. Exact-line results package lattice coordinates as additive equivalences. The two-factor branch is proved independently of the descent. These representational choices are documented in docs/SOURCE_MAP.md and docs/READING_GUIDE.md. Formal checking of the Lean statements does not validate every sentence of the informal manuscript.", "alignment": true, "original": false, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [ "GPT-6 Astra (Codex agents)" ], "notes": "AI agents performed semantic checks of the final theorem and its expanded rectangle/pattern/period definitions, source-correspondence reviews, and dependency checks. No human expert review is claimed or documented. The paper and formalization have not been mathematically digested by humans. No human peer review, independent assessment of novelty or priority, or endorsement by the cited authors is claimed. Verification commands, tool versions, observed outcomes, and checker limitations are recorded in STATUS.md and docs/PALOMAR.md. This metadata records review before Palomar submission; it does not claim a Palomar review outcome or registration. An empty related_formalizations list means none are identified here, not that a comprehensive priority search was done." }, "canonical": { "repo": "boonsuan/nivat", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-14-000003", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-14-000003", "commit": "0d94547e934418c9d1b75fcbb7d6af493e86759d", "trust": "high", "theorems": 1, "date": "2026-09-14" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "BrandonMYates/allones-comparator/formalization.yaml", "repo": "BrandonMYates/allones-comparator", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/BrandonMYates/allones-comparator/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "All-ones polynomials over GF(2) under the substitution X ↦ X² + X + 1 (OEIS A071642, TheoremDB P12771)", "description": "Formalizes in Lean 4 the resolution of a 2019 OEIS conjecture of L. H. Gallardo (comment on A071642; TheoremDB P12771): for every n ≥ 2 the all-ones polynomial 1 + X + ⋯ + Xⁿ is irreducible over GF(2) if and only if 1 + P + ⋯ + Pⁿ is irreducible over GF(2), where P = X² + X + 1. The compared statements are the per-n equivalence, the literal set-level form of the OEIS comment (the terms ≥ 2 coincide), and the failure of the excluded case n = 1. The proof is elementary: a factorization of the all-ones polynomial composes to one of the substituted polynomial; conversely irreducibility forces n ev", "authors": [ "Brandon Yates" ], "maintainers": [ "Brandon Yates" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Proof of Gallardo's A071642 substitution conjecture via the Artin–Schreier trace criterion", "id": "", "authors": [], "type": "original-proof", "relationship": "other", "endorsement": "" }, { "title": "A071642: Numbers n such that x^n + x^(n-1) + ... + x + 1 is irreducible over GF(2)", "id": "https://oeis.org/A071642", "authors": [ "OEIS Foundation Inc." ], "type": "web post", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Does Substitution by x^2+x+1 Preserve All-Ones Irreducibility Exponents? (TheoremDB P12771)", "id": "https://theoremdb.org/statements/P12771/", "authors": [], "type": "web post", "relationship": "background", "endorsement": "n/a" }, { "title": "A Polynomial Variant of Perfect Numbers", "id": "https://cs.uwaterloo.ca/journals/JIS/VOL23/Gallardo/gallardo7.pdf", "authors": [ "Luis H. Gallardo" ], "type": "article", "relationship": "background", "endorsement": "not-contacted" }, { "title": "On the construction of irreducible self-reciprocal polynomials over finite fields", "id": "", "authors": [ "Helmut Meyn" ], "type": "article", "relationship": "background", "endorsement": "not-contacted" } ], "related": [], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11T06", "12E20" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "claude-opus-5", "claude-fable-5.1" ], "framework": "Claude Code" } ], "strongest": "agent", "models": [ "claude-fable-5.1", "claude-opus-5" ], "spend": "subscription-based", "notes": "Fully agent-written Lean under human direction; the human author audited the Challenge statements against the OEIS comment and the TheoremDB record and reviewed the informal proof." }, "scope": "Everything stated in the Challenge is proved: the equivalence for every n ≥ 2, the set-level OEIS form, and the n = 1 boundary. No hypothesis is weakened and nothing is omitted. The development is self-contained over Mathlib v4.32.0 (Lean 4.32.0).", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "none known; the Challenge transcribes the OEIS comment and the TheoremDB statement directly, adding only the explicit n = 1 boundary theorem.", "alignment": true, "original": true, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Brandon Yates" ], "notes": "Statement audit of Challenge.lean against OEIS A071642 and TheoremDB P12771 by the author. Mathematical review of the informal proof by Claude Fable 5.1, with three independent finite checkers (SymPy; two bit-polynomial Rabin tests) agreeing on n ≤ 129. Axiom audit by #print axioms on every compared declaration (main, A071642_terms_ge_two, boundary_one) and on every library lemma: each depends exactly on propext, Classical.choice and Quot.sound, the three standard Lean axioms and nothing else; no sorry, no native_decide, no user axioms. The same file was independently kernel-checked by TheoremDB's pinned-world verifier (Lean 4.33.0-rc1) with the identical axiom set." }, "canonical": { "repo": "brandonmyates/allones-comparator", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-11-000004", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-11-000004", "commit": "76f3a65407c45de21228824f7a41430acec473a4", "trust": "high", "theorems": 3, "date": "2026-09-11" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "BrandonMYates/csordas-comparator/formalization.yaml", "repo": "BrandonMYates/csordas-comparator", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/BrandonMYates/csordas-comparator/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "A counterexample to a log-concavity conjecture for the Riemann kernel", "description": "This project formalizes in Lean 4 a counterexample to a log-concavity conjecture of Coffey and Csordas (Math. Comp. 82 (2013) 2265-2272, Conjecture 2.5, where the quantity below is written S_n), later restated as Open Problem 4.13 of Csordas (Comput. Methods Funct. Theory 15 (2015) 373-391). Let Phi be the kernel of Riemann's integral representation, Phi(u) = sum_{n>=1} (4 pi^2 n^4 e^{9u/2} - 6 pi n^2 e^{5u/2}) e^{-pi n^2 e^{2u}}, and set J_n(t) = (Phi^(n)(t))^2 - Phi^(n-1)(t) Phi^(n+1)(t), the n-th Laguerre expression of Phi. The conjecture asserts J_n(t) > 0 for every real t and every n >= 1", "authors": [ "Brandon Yates" ], "maintainers": [ "Brandon Yates" ], "license": "Apache-2.0", "role": "", "substantive": "BrandonMYates/xilab", "sources": [ { "title": "Two concavity problems for the Riemann kernel", "id": "", "authors": [ "Brandon Yates" ], "type": "paper", "relationship": "formalizes", "endorsement": "" }, { "title": "On the log-concavity of a Jacobi theta function", "id": "https://doi.org/10.1090/S0025-5718-2013-02681-6", "authors": [ "Mark W. Coffey", "George Csordas" ], "type": "paper", "relationship": "background", "endorsement": "n/a" }, { "title": "Fourier transforms of positive definite kernels and the Riemann xi-function", "id": "https://doi.org/10.1007/s40315-014-0105-8", "authors": [ "George Csordas" ], "type": "paper", "relationship": "background", "endorsement": "n/a" }, { "title": "Conjectures and theorems in the theory of entire functions", "id": "", "authors": [ "George Csordas", "Dimitar K. Dimitrov" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "An Exact Level-Nine Counterexample to a Theta-Derivative Turan Conjecture", "id": "https://doi.org/10.5281/zenodo.21894000", "authors": [ "Ryan Kielhorn" ], "type": "paper", "relationship": "independently-proves", "endorsement": "not-contacted" }, { "title": "A Counterexample to Csordas' Open Problem 4.13 for the Riemann Xi Kernel", "id": "https://doi.org/10.5281/zenodo.21941139", "authors": [ "Akihiro Koide" ], "type": "paper", "relationship": "independently-proves", "endorsement": "not-contacted" }, { "title": "On a tempered xi function associated with the Riemann xi function", "id": "https://doi.org/10.1142/S0218348X23401163", "authors": [ "Xiao-Jun Yang" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" } ], "related": [ { "id": "https://github.com/leanprover-community/mathlib4", "relationship": "independent", "note": "Mathlib has no Riemann kernel Phi. The kernel, its evenness (proved from Mathlib's Gaussian Poisson summation), and the derivatives at the origin are all built in the supporting library." } ], "classification": { "arxiv": [ "math.CA", "math.CV", "math.NT" ], "msc2020": [ "30D15", "26A51", "30D10", "11M06" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "claude-opus-5" ], "framework": "Claude Code" } ], "strongest": "agent", "models": [ "claude-opus-5" ], "spend": "subscription-based", "notes": "The mathematics is the author's. The Lean formalization was produced by automation under the author's direction and verified mechanically." }, "scope": "Proved: J_9(0) < 0, and hence that the assertion \"J_n(t) > 0 for all real t and all n\" is false. The proof establishes, along the way, that Phi is even and that Phi^(8)(0) and Phi^(10)(0) are both positive. NOT proved and NOT claimed: the concavity theorem and double-Turan corollary of the source manuscript; the full sign sequence of Phi^(2j)(0) beyond the indices the counterexample needs; and any statement about the interval on which J_9 is negative.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The compared theorem is a statement about the kernel Phi as defined in the Challenge file. The source manuscript states the same counterexample in the Csordas-Dimitrov normalisation as well, where the origin value reads -7.8376e19; that restatement is not formalized, and the two differ by the positive factor recorded in the Csordas-Dimitrov source note. No other known divergence.", "alignment": true, "original": true, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [ "Claude Code agents (independent per-file rebuild and axiom audit)" ], "notes": "Each file of the supporting library was rebuilt from source and its main theorems audited with print-axioms by an agent other than its author. The numerical value of J_9(0) was independently recomputed from the definition of Phi at 60 decimal digits, by Cauchy-integral differentiation on a circle of radius 0.2, agreeing with the source manuscript to ten significant figures. No human expert has reviewed the mathematics, and the source manuscript has not been peer reviewed. Provenance chronology: the counterexample was found in this project on 3-4 August 2026. Subsequently, two independent LLM-assisted exact-certificate proofs were publicly deposited by Kielhorn on 11 August and Koide on 15 August 2026; both are cited in the source metadata, and both were audited here (their printed certific" }, "canonical": { "repo": "brandonmyates/csordas-comparator", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-27-000002", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-27-000002", "commit": "2b23c32b841044581b7fc7a2dca0df86255d490d", "trust": "high", "theorems": 1, "date": "2026-08-27" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "BrandonMYates/schwarzschild-comparator/formalization.yaml", "repo": "BrandonMYates/schwarzschild-comparator", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/BrandonMYates/schwarzschild-comparator/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Exact solutions of the vacuum and sourced Einstein field equations, verified in coordinates", "description": "Twenty-one theorems and thirty-three definitions, machine-checked in Lean 4 against Mathlib v4.32.0: the Schwarzschild, Schwarzschild-de Sitter (Kottler), Reissner-Nordstrom (with the Maxwell stress tensor defined and G = 8 pi T proved in all sixteen components), ingoing Vaidya (null dust of rank at most one, rank one wherever dm/dv ≠ 0, for a merely once-differentiable mass function), spatially flat FLRW (both Friedmann equations in Einstein-tensor form), static de Sitter, Minkowski-in-spherical and Kiselev metrics, together with Birkhoff uniqueness in the areal static gauge and the n != 2 eq", "authors": [ "Brandon Yates" ], "maintainers": [ "Brandon Yates" ], "license": "Apache-2.0", "role": "", "substantive": "BrandonMYates/grlab", "sources": [ { "title": "Uber das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie", "id": "", "authors": [ "Karl Schwarzschild" ], "type": "paper", "relationship": "independently-proves", "endorsement": "" }, { "title": "Uber die Eigengravitation des elektrischen Feldes nach der Einsteinschen Theorie", "id": "https://doi.org/10.1002/andp.19163550905", "authors": [ "Hans Reissner" ], "type": "paper", "relationship": "independently-proves", "endorsement": "" }, { "title": "On the Energy of the Gravitational Field in Einstein's Theory", "id": "", "authors": [ "Gunnar Nordstrom" ], "type": "paper", "relationship": "independently-proves", "endorsement": "" }, { "title": "Uber die physikalischen Grundlagen der Einsteinschen Gravitationstheorie", "id": "https://doi.org/10.1002/andp.19183611402", "authors": [ "Friedrich Kottler" ], "type": "paper", "relationship": "independently-proves", "endorsement": "" }, { "title": "Uber die allgemeinen kugelsymmetrischen Losungen der Einsteinschen Gravitationsgleichungen im Vakuum", "id": "https://doi.org/10.1007/s10714-005-0168-y", "authors": [ "Jorg Tofte Jebsen" ], "type": "paper", "relationship": "independently-proves", "endorsement": "" }, { "title": "Relativity and Modern Physics", "id": "https://doi.org/10.4159/harvard.9780674734487", "authors": [ "George David Birkhoff" ], "type": "book", "relationship": "independently-proves", "endorsement": "" }, { "title": "The gravitational field of a radiating star", "id": "https://doi.org/10.1007/bf03173260", "authors": [ "P. C. Vaidya" ], "type": "paper", "relationship": "independently-proves", "endorsement": "" }, { "title": "Uber die Krummung des Raumes", "id": "https://doi.org/10.1007/BF01332580", "authors": [ "Alexander Friedmann" ], "type": "paper", "relationship": "independently-proves", "endorsement": "" }, { "title": "Un Univers homogene de masse constante et de rayon croissant rendant compte de la vitesse radiale des nebuleuses extra-galactiques", "id": "https://doi.org/10.1093/mnras/91.5.483", "authors": [ "Georges Lemaitre" ], "type": "paper", "relationship": "independently-proves", "endorsement": "" }, { "title": "Quintessence and black holes", "id": "https://doi.org/10.1088/0264-9381/20/6/310", "authors": [ "V. V. Kiselev" ], "type": "paper", "relationship": "independently-proves", "endorsement": "" }, { "title": "General Relativity", "id": "https://doi.org/10.7208/chicago/9780226870373.001.0001", "authors": [ "Robert M. Wald" ], "type": "book", "relationship": "background", "endorsement": "" }, { "title": "A Relativist's Toolkit: The Mathematics of Black-Hole Mechanics", "id": "https://doi.org/10.1017/CBO9780511606601", "authors": [ "Eric Poisson" ], "type": "book", "relationship": "background", "endorsement": "" }, { "title": "Spacetime and Geometry: An Introduction to General Relativity", "id": "https://doi.org/10.1017/9781108770385", "authors": [ "Sean M. Carroll" ], "type": "book", "relationship": "background", "endorsement": "" } ], "related": [ { "id": "https://github.com/leanprover-community/mathlib4", "relationship": "independent", "note": "Supplies deriv/HasDerivAt, fderiv, rpow and the Matrix machinery. Contains NO curvature: ten names grepped at the pinned revision, zero hits; RiemannianMetric is positive-definite only." }, { "id": "https://arxiv.org/abs/2608.21502", "relationship": "independent", "note": "Chow, Liao and Qin, Hamilton's three-manifold theorem in Lean 4. Real curvature machinery, but positive-definite Riemannian for Ricci flow: no field equations, no exact solutions." }, { "id": "https://arxiv.org/abs/2405.08863", "relationship": "independent", "note": "HepLean/PhysLean. Its Relativity directory is special-relativistic only, and Cosmology/FLRW POSITS the Friedmann equations as a definition with no metric or curvature tensors; Part VIII derives them." }, { "id": "https://www.isa-afp.org/entries/No_FTL_observers_Gen_Rel.html", "relationship": "independent", "note": "Stannett et al., first-order axiomatic general relativity in Isabelle: no metric tensor, no curvature, no field equations. The search was targeted, not exhaustive; absence is not proof of novelty." } ], "classification": { "arxiv": [ "gr-qc", "math.DG", "cs.LO" ], "msc2020": [ "83C15", "83C05", "83C22", "83F05", "53B30", "68V20" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "claude-opus-5", "claude-fable-5" ], "framework": "Claude Code" } ], "strongest": "agent", "models": [ "claude-fable-5", "claude-opus-5" ], "spend": "subscription-based", "notes": "Roles, stated separately. Direction: the author set the target statements and their scope, chose what to certify and what to leave out, supplied the validation-gated computer-algebra suite used as one of the two oracles, and takes responsibility for the submission. Mathematical argument: classical, with no step originating here. The Lean proof architecture and the computations were developed by AI agents (Claude Code, claude-opus-5) under the author's direction, including the row-plus-trace split that keeps the Vaidya theorem C^1 and the machine-closed 304-entry table diff, both agent findings" }, "scope": "PROVED: the twenty-one compared theorems, unconditionally, under exactly the hypotheses shown: chart-local, on the stated open regions, Vaidya needing only a C^1 mass function. NOT PROVED and NOT CLAIMED: any manifold-level or coordinate-independent claim; any maximal extension; anything at a horizon or pole; Birkhoff's theorem itself (staticity is assumed); Ricci symmetry in general or the Bianchi identities; Maxwell's equations; any matter model, fluid or energy condition beyond the defined Maxwell tensor; non-flat FLRW; the outgoing Vaidya chart; every library result not in comparator.json.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "Chart-only: a metric is a matrix-valued function on an open set in R^4; with no atlas, no manifold and no coordinate-independence theorem, so nothing is invariant. Signature (-,+,+,+), Wald/Carroll convention: de Sitter has POSITIVE scalar curvature; the opposite sign is also current. Coordinate order differs by Part: Vaidya (v,r,theta,phi), FLRW comoving. Vaidya is INGOING with g_vr = +1 and needs m only C^1; FLRW is k = 0; Kiselev uses Real.rpow on r > 0. Vacuum means R_uv = 0 (III, V, X, w=0 of XI), R_uv = Lambda g_uv (IV, IX), or G = 8 pi T for the one defined Maxwell tensor (VI); VII and VIII define none. A library docstring calls the flat spherical chart eleven-symbol; it is nine.", "alignment": true, "original": true, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [ "Claude Code agents (clean rebuild, axiom audit over all 54 compared declarations, source-level signature diff, sorry scan, Crossref/OpenAlex verification of every source)" ], "notes": "Rebuilt from scratch against Mathlib v4.32.0 (8,673 jobs, clean). The Challenge produces exactly 21 sorry warnings, one per compared theorem, and nothing else; the Solution builds clean. All 54 compared declarations report exactly propext, Classical.choice and Quot.sound, and a source diff found all 54 signatures identical across library, Solution and Challenge. Only the author has reviewed it." }, "canonical": { "repo": "brandonmyates/schwarzschild-comparator", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-27-000020", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-27-000020", "commit": "d8b6acf8263d4924356ac8abb62668c137ab61cd", "trust": "high", "theorems": 21, "date": "2026-08-27" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "BrandonMYates/tempered-xi-comparator/formalization.yaml", "repo": "BrandonMYates/tempered-xi-comparator", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/BrandonMYates/tempered-xi-comparator/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Zeros of the odd channel of Riemann's kernel", "description": "This project formalizes in Lean 4 the zero geometry of the odd channel of Riemann's kernel, refuting conjectures of X.-J. Yang (Fractals 32 (2024) 2340116). Let Phi be the kernel of Riemann's integral representation, Phi(u) = sum_{n>=1} (4 pi^2 n^4 e^{9u/2} - 6 pi n^2 e^{5u/2}) e^{-pi n^2 e^{2u}}, the positive even function whose cosh transform is the completed zeta function, and let F(w) = 2 int_0^infty Phi(u) sinh(wu) du be the odd channel of the same kernel; this is Yang's \"tempered xi function\", in shifted coordinates. Six statements are compared: F is entire; F has infinitely many zeros; ", "authors": [ "Brandon Yates" ], "maintainers": [ "Brandon Yates" ], "license": "Apache-2.0", "role": "", "substantive": "BrandonMYates/xilab", "sources": [ { "title": "The zeros of the tempered xi function", "id": "", "authors": [ "Brandon Yates" ], "type": "paper", "relationship": "formalizes", "endorsement": "" }, { "title": "The integral representation for the Riemann Xi-function", "id": "https://doi.org/10.1016/0022-314X(71)90016-3", "authors": [ "Robert Spira" ], "type": "paper", "relationship": "independently-proves", "endorsement": "not-contacted" }, { "title": "On a tempered xi function associated with the Riemann xi function", "id": "https://doi.org/10.1142/S0218348X23401163", "authors": [ "Xiao-Jun Yang" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "A new odd entire function of order one arising in the heat problem", "id": "https://doi.org/10.2298/TSCI221105010Y", "authors": [ "Xiao-Jun Yang", "M. Abdel-Aty", "L.-L. Geng" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "The integral of the Riemann xi-function", "id": "https://arxiv.org/abs/1106.4348", "authors": [ "Jeffrey C. Lagarias", "David Montague" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Bemerkung ueber die Integraldarstellung der Riemannschen xi-Funktion", "id": "", "authors": [ "George Polya" ], "type": "paper", "relationship": "background", "endorsement": "n/a" }, { "title": "A note on the Riemann xi-function", "id": "", "authors": [ "Aurel Wintner" ], "type": "paper", "relationship": "background", "endorsement": "n/a" } ], "related": [ { "id": "https://github.com/leanprover-community/mathlib4", "relationship": "independent", "note": "Mathlib supplies completedRiemannZeta and Gaussian Poisson summation but has no Riemann kernel Phi and no odd channel; both are defined in the supporting library, which also builds the holomorphic logarithm of a nowhere-zero entire function and Hadamard-style order-one machinery that Mathlib lacks." } ], "classification": { "arxiv": [ "math.NT", "math.CV", "math.CA" ], "msc2020": [ "30D15", "30D10", "11M06", "44A15" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "claude-opus-5" ], "framework": "Claude Code" } ], "strongest": "agent", "models": [ "claude-opus-5" ], "spend": "subscription-based", "notes": "The mathematics is the author's. The Lean formalization was produced by automation under the author's direction and verified mechanically." }, "scope": "Proved: the six compared statements. F is entire; F has infinitely many zeros; every nonzero zero has nonzero real and imaginary parts; quartet symmetry of the zero set; the negation of the source paper's Conjecture 1; and the negation of the exactly formalizable content of its Conjecture 2. NOT proved in Lean, and not relied on by any compared statement: the identification of F with the odd part of the completed zeta function; the order bound on F (proved in the supporting library but not compared by this configuration); Conjecture 3 of the source paper, which is not formalized; and simplicit", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The compared theorems quantify over the integral F defined in the Challenge file, not over the completed zeta function. The source manuscript treats the two as identified; the formal development does not assert the identification, which was checked numerically only. The refutation of the source paper's Conjecture 1 is stated in the shifted coordinates recorded in the alignment notes. No other known divergence.", "alignment": true, "original": true, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [ "Claude Code agents (independent per-file rebuild and axiom audit)" ], "notes": "Each file of the supporting library was rebuilt from source and its main theorems audited with print-axioms by an agent other than its author. No human expert has reviewed the mathematics, and the draft manuscript has not been peer reviewed. Provenance chronology: the draft manuscript underlying this formalization and its accompanying literature sweep are externally dated 3 August 2026, in a write-once file; this is recorded for provenance, and no claim of first public presentation is made. A further literature search completed 26 August 2026 identified no refutation of the source conjectures and no other proof-assistant formalization of any compared statement; every recorded citation of the source paper found by that search is a self-citation, and the conjectures are restated as open in a" }, "canonical": { "repo": "brandonmyates/tempered-xi-comparator", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-27-000001", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-27-000001", "commit": "d226851cc05dee3cad2244ce2660ad72b898a636", "trust": "high", "theorems": 6, "date": "2026-08-27" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "BrandonMYates/yz-cubic-comparator/formalization.yaml", "repo": "BrandonMYates/yz-cubic-comparator", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/BrandonMYates/yz-cubic-comparator/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "No low-degree polynomial family for a cubic Diophantine equation", "description": "This project formalizes in Lean 4 the statement that the Diophantine equation y*z*(y+z) = x^3 + x^2 + 3x - 1 admits no polynomial parametrization over the rationals whose x-coordinate has degree between 1 and 4. Writing f = X^3 + X^2 + 3X - 1, the compared theorem is that there is no triple of rational polynomials (g, u, v) with g nonconstant of degree at most 4 satisfying the identity u*v*(u+v) = f composed with g. Exhibiting such a family is one of the two standard ways an equation of this shape is settled affirmatively -- the other being a sporadic solution -- and the theorem closes that ro", "authors": [ "Brandon Yates" ], "maintainers": [ "Brandon Yates" ], "license": "Apache-2.0", "role": "", "substantive": "BrandonMYates/yzcubic", "sources": [ { "title": "No polynomial family of degree at most four for yz(y+z) = x^3 + x^2 + 3x - 1", "id": "", "authors": [ "Brandon Yates" ], "type": "original-proof", "relationship": "other", "endorsement": "" }, { "title": "Is y^2 z + z^2 y = x^3 + x^2 + 3x - 1 solvable in integers?", "id": "https://mathoverflow.net/questions/509449", "authors": [], "type": "web discussion", "relationship": "background", "endorsement": "not-contacted" }, { "title": "On the shortest open cubic equations", "id": "https://arxiv.org/abs/2603.29831", "authors": [ "Bogdan Grechuk", "Ashleigh Ratcliffe" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "The new shortest open cubic equations", "id": "https://mathoverflow.net/questions/466803", "authors": [], "type": "web discussion", "relationship": "background", "endorsement": "not-contacted" }, { "title": "The Mordell-Schinzel conjecture for cubic diophantine equations", "id": "https://arxiv.org/abs/2412.12080", "authors": [ "Janos Kollar", "Jennifer Li" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "A uniform bound on the Brauer groups of certain log K3 surfaces", "id": "https://arxiv.org/abs/1705.04529", "authors": [ "Martin Bright", "Julian Lyczak" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" } ], "related": [ { "id": "https://github.com/leanprover-community/mathlib4", "relationship": "independent", "note": "Mathlib supplies everything the statement needs -- Polynomial, Polynomial.comp, natDegree -- so the Challenge introduces no definitions. The proof uses Mathlib's fermatLastTheoremThree (Fermat's Last Theorem for exponent three) for the degree-3 window, and otherwise only elementary polynomial algebra; no Groebner-basis or ideal-membership computation is replayed or relied upon." } ], "classification": { "arxiv": [ "math.NT", "math.AC" ], "msc2020": [ "11D25", "12E05", "13P15" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "claude-opus-5", "claude-fable-5" ], "framework": "Claude Code" } ], "strongest": "agent", "models": [ "claude-fable-5", "claude-opus-5" ], "spend": "subscription-based", "notes": "Roles, stated separately. Direction: the author selected the problem (the smallest open cubic on the MathOverflow list), set the target statement and its scope, directed the work at every stage, and takes responsibility for the submission. Mathematical argument: developed by AI agents (Claude Code, claude-opus-5) under that direction, not by the author -- the structural lemmas (degree divisibility, slot-degree windows, the square-completion exclusion) were identified in a July 2026 agent-run computational search, and the decisive reduction of the degree-3 window to Fermat's Last Theorem for ex" }, "scope": "PROVED: no triple of rational polynomials (g, u, v) with 1 <= deg g <= 4 satisfies u*v*(u+v) = f composed with g, where f = X^3 + X^2 + 3X - 1. NOT PROVED and NOT CLAIMED: any statement about the existence of integer or rational solutions of y*z*(y+z) = x^3 + x^2 + 3x - 1; the case deg g >= 5; and every other result in the author's working notes on this equation, none of which is advertised here.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The compared theorem quantifies over polynomials with rational coefficients. The underlying question is about integer solutions; the polynomial-family statement over the rationals is strictly stronger than the same statement over the integers, and no integrality hypothesis is imposed. The degree bound in the formal statement is exactly the bound proved: degree 5 is excluded from the statement rather than silently assumed away. No other known divergence.", "alignment": true, "original": true, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [ "Claude Code agents (independent clean rebuild, axiom audit, statement diff, rfl bridge)" ], "notes": "Reviewed internally, mechanically: the pinned library was rebuilt from a clean checkout against Mathlib v4.32.0; every public theorem was checked with #print axioms to depend on exactly propext, Classical.choice and Quot.sound; a comment-stripped scan found no sorry, native_decide or axiom declarations; the Challenge statement was diffed against the library theorem and the definitional bridge (X^3 + X^2 + 3X - 1 = YZCubic.fq, by rfl) was compiled in a separate file against the built artifacts. No human expert has reviewed the mathematics." }, "canonical": { "repo": "brandonmyates/yz-cubic-comparator", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-27-000010", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-27-000010", "commit": "70082c378b76b2eb9cec74de67506d1c0e36a514", "trust": "high", "theorems": 1, "date": "2026-08-27" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "bryangingechen/CombinatorialRigidity/formalization.yaml", "repo": "bryangingechen/CombinatorialRigidity", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/bryangingechen/CombinatorialRigidity/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "CombinatorialRigidity", "description": "", "authors": [ "Bryan Gin-ge Chen" ], "maintainers": [ "Bryan Gin-ge Chen" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "On graphs and rigidity of plane skeletal structures", "id": "doi:10.1007/BF01534980", "authors": [ "G. Laman" ], "type": "article", "relationship": "formalizes", "endorsement": "n/a" }, { "title": "On generic rigidity in the plane", "id": "doi:10.1137/0603009", "authors": [ "L. Lovász", "Y. Yemini" ], "type": "article", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "Pebble game algorithms and sparse graphs", "id": "doi:10.1016/j.disc.2007.07.104", "authors": [ "A. Lee", "I. Streinu" ], "type": "article", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "On the problem of decomposing a graph into n connected factors", "id": "doi:10.1112/jlms/s1-36.1.221", "authors": [ "W. T. Tutte" ], "type": "article", "relationship": "formalizes", "endorsement": "n/a" }, { "title": "Edge-disjoint spanning trees of finite graphs", "id": "doi:10.1112/jlms/s1-36.1.445", "authors": [ "C. St. J. A. Nash-Williams" ], "type": "article", "relationship": "formalizes", "endorsement": "n/a" }, { "title": "The union of matroids and the rigidity of frameworks", "id": "doi:10.1137/0401025", "authors": [ "W. Whiteley" ], "type": "article", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "Rigidity of multi-graphs. I. Linking rigid bodies in n-space", "id": "doi:10.1016/0095-8956(84)90016-9", "authors": [ "T.-S. Tay" ], "type": "article", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "Linking (n-2)-dimensional panels in n-space II: (n-2,2)-frameworks and body and hinge structures", "id": "doi:10.1007/BF01788678", "authors": [ "T.-S. Tay" ], "type": "article", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "The generic rank of body-bar-and-hinge frameworks", "id": "doi:10.1016/j.ejc.2009.03.030", "authors": [ "B. Jackson", "T. Jordán" ], "type": "article", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "A proof of the molecular conjecture", "id": "doi:10.1007/s00454-011-9348-6", "authors": [ "N. Katoh", "S. Tanigawa" ], "type": "article", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "On the rigidity of molecular graphs", "id": "doi:10.1007/s00493-008-2287-z", "authors": [ "B. Jackson", "T. Jordán" ], "type": "article", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "Generic rigidity in three-dimensional bond-bending networks", "id": "J. Phys. A: Math. Gen. 31 (1998), 6653-6668", "authors": [ "D. J. Jacobs" ], "type": "article", "relationship": "adapts", "endorsement": "not-contacted" }, { "title": "Generic rigidity of molecular graphs via ear decomposition", "id": "doi:10.1016/S0166-218X(99)00188-2", "authors": [ "D. S. Franzblau" ], "type": "article", "relationship": "adapts", "endorsement": "not-contacted" }, { "title": "Statics of frameworks and motions of panel structures: a projective geometric introduction", "id": "Structural Topology 6 (1982), 43-82", "authors": [ "H. Crapo", "W. Whiteley" ], "type": "article", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "Combinatorial Rigidity: Graphs and Matroids in the Theory of Rigid Frameworks", "id": "doi:10.2969/msjmemoirs/03401C020", "authors": [ "T. Jordán" ], "type": "book", "relationship": "background", "endorsement": "n/a" }, { "title": "Matroid Theory (2nd ed., Oxford Graduate Texts in Mathematics 21)", "id": "isbn:978-0-19-856694-6", "authors": [ "J. G. Oxley" ], "type": "book", "relationship": "background", "endorsement": "n/a" } ], "related": [ { "id": "https://github.com/apnelson1/Matroid", "relationship": "builds-on", "note": "Peter Nelson's Lean 4 matroid package (Apache-2.0). The matroid-union / Edmonds-partition subsystem under CombinatorialRigidity/Matroid/ is ported from it - rebased from its shelved WIP/{Submodular,Union}.lean onto the package's live FiniteCircuitMatroid constructor - and it also supplies Matroid.ofFun and Graph.cycleMatroid. Each vendored file carries an upstream copyright header with a provenanc" } ], "classification": { "arxiv": [ "math.CO", "math.MG", "cs.CG" ], "msc2020": [ "52C25", "05B35", "05C70", "15A75", "68R10", "92E10" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "claude-opus-4-7", "claude-opus-4-8", "claude-fable-5", "claude-sonnet-5", "claude-sonnet-4-6", "claude-haiku-4-5-20251001" ], "framework": "Claude Code" } ], "strongest": "agent", "models": [ "claude-fable-5", "claude-haiku-4-5-20251001", "claude-opus-4-7", "claude-opus-4-8", "claude-sonnet-4-6", "claude-sonnet-5" ], "spend": "subscription-based usage (Claude Code); no metered API spend", "notes": "Human-in-the-loop agentic formalization: the author directs phase scoping and reviews output; Claude Code performs the bulk of the proof engineering, blueprint authoring, and documentation. CI gates every push (full `lake build` green, blueprint web/PDF build, checkdecls verifying every blueprint \\lean{...} pointer, and an upstreaming dashboard)." }, "scope": "A Lean 4 / mathlib4 formalization of combinatorial rigidity theory, in four arcs. (1) Laman's theorem (1970): a graph on >= 2 vertices is generically rigid in the plane iff it contains a (2,3)-tight spanning subgraph, both directions via the Henneberg construction. (2) The planar rigidity matroid (Lovász-Yemini) in both its combinatorial (2,3)-count and linear-matroid forms, plus an executable, certificate-carrying (k,l)-sparsity decision procedure - the Lee-Streinu pebble game - with Decidable instances and a `lake exe pebble-game` CLI. (3) Matroid union / Edmonds partition (ported from apnel", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "SimpleGraph.isGenericallyRigid_two_iff_exists_isLaman_le", "file": "CombinatorialRigidity/LamanTheorem.lean", "description": "Laman's theorem (1970): for n >= 2, G is generically rigid in the plane iff it contains a Laman (= (2,3)-tight) spanning subgraph.", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "SimpleGraph.linearRigidityMatroid_eq_rigidityMatroid", "file": "CombinatorialRigidity/LinearRigidityMatroid.lean", "description": "Lovász-Yemini, linear-matroid form: the generic-placement linear rigidity matroid (via Matroid.ofFun) equals the combinatorial (2,3)-count rigidity matroid.", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "SimpleGraph.countMatroid_indep_iff_runPebbleGame", "file": "CombinatorialRigidity/PebbleGame/Correctness.lean", "description": "Lee-Streinu pebble-game correctness: in the matroidal regime l < 2k, edge-set independence in the (k,l)-count matroid is decided by the verdict-bearing pebble game. Powers the executable decision procedure `SimpleGraph.instDecidableIsLaman` / `instDecidableIsSparse` / `instDecidableIsTight` (PebbleGame/Exec.lean) and the `lake exe pebble-game` CLI.", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "Graph.tutte_nash_williams", "file": "CombinatorialRigidity/BodyBar/TreePacking.lean", "description": "Tutte-Nash-Williams tree-packing (1961): a multigraph is the edge-disjoint union of k forests iff it is (k,k)-sparse. Specialized from the local Edmonds matroid-partition machinery.", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "Graph.kFrameMatroid_eq_unionPow_cycleMatroid", "file": "CombinatorialRigidity/BodyBar/KFrame.lean", "description": "Whiteley 1988 Theorem 1: the generic k-frame matroid equals the k-fold union of the graphic (cycle) matroid, restricted to E(G).", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "Graph.BodyBarFramework.tay_witness", "file": "CombinatorialRigidity/BodyBar/TayTheorem.lean", "description": "Tay's body-bar theorem (Tay 1984, Whiteley 1988 Thm 8), existence-of-realization form: for d = n(n+1)/2, a multigraph carries an independent (resp. isostatic) body-bar framework in R^n iff it is (d,d)-sparse (resp. (d,d)-tight).", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "Graph.BodyHingeFramework.body_hinge_tay", "file": "CombinatorialRigidity/BodyBar/BodyHinge.lean", "description": "Body-hinge / panel-hinge Tay-Whiteley theorem (Tay 1989, Whiteley 1988), existence-of-realization form: a graph carries an independent (resp. isostatic) body-hinge framework in R^n iff its (d-1)-fold edge-multiplication is (d,d)-sparse (resp. tight), equivalently a packing of d forests.", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "CombinatorialRigidity.Molecular.PanelHingeFramework.theorem_55_minimalKDof_gen", "file": "CombinatorialRigidity/Molecular/AlgebraicInduction/Theorem55.lean", "description": "Katoh-Tanigawa 2011 Theorem 5.5 (the realization theorem, all three cases including the hardest, Case III): every minimal k-dof-graph on >= 2 vertices admits a nondegenerate-hinge panel realization whose rigidity matrix attains rank D(|V|-1) - k, at every dimension n >= 3 (D = (n+1 choose 2) >= 6) and over any infinite field of any characteristic (phase 33; KT's real statement is the K = R case); ", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "CombinatorialRigidity.Molecular.PanelHingeFramework.rankHypothesis_of_theorem_55_gen", "file": "CombinatorialRigidity/Molecular/AlgebraicInduction/Theorem55.lean", "description": "Katoh-Tanigawa 2011 Theorem 5.6: every simple spanning multigraph has a panel-hinge realization attaining the deficiency rank, rank R(G,p) = D(|V|-1) - def(G~), at every dimension n >= 3 and over any infinite field of any characteristic (phase 33). With the def = corank bridge this completes the Katoh-Tanigawa Proposition 1.1 reconciliation (CombinatorialRigidity.Molecular.rigidityMatrix_prop11).", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "CombinatorialRigidity.Molecular.PanelHingeFramework.molecular_conjecture", "file": "CombinatorialRigidity/Molecular/AlgebraicInduction/Theorem55.lean", "description": "The Molecular Conjecture (Tay-Whiteley 1984; Katoh-Tanigawa 2011 Conjecture 1.2): a simple spanning graph on >= 2 bodies has an infinitesimally rigid genuine body-hinge realization iff it has one as a panel-hinge framework, at every dimension n >= 3 and over any infinite field of any characteristic (phase 33).", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "SimpleGraph.molecule_rank_formula", "file": "CombinatorialRigidity/Molecular/Molecule/Application.lean", "description": "The molecule rank formula (Jackson-Jordán 2008; Katoh-Tanigawa 2011 Corollary 5.7): for a simple graph G of minimum degree >= 2, the rank of the square graph G^2 in the 3-D generic bar-joint rigidity matroid is r(G^2) = 3|V| - 6 - def(G~) - the combinatorial flexibility count for a molecule modelled on G.", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "SimpleGraph.jacobs", "file": "CombinatorialRigidity/JacobsTheorem.lean", "description": "Jacobs' conjecture (Jacobs 1998; Jackson-Jordán 2008 Conjecture 5.1 / Theorem 5.4, here unconditional): the square G^2 of a simple graph is independent in the 3-D generic bar-joint rigidity matroid iff G^2 satisfies the three-dimensional Laman counting condition |E(X)| <= 3|X| - 6 for all |X| >= 3.", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "SimpleGraph.degree_one_rank", "file": "CombinatorialRigidity/JacobsDegreeOne.lean", "description": "The degree-1 rank formula (Jackson-Jordán 2008 Lemma 4.2; the tree case SimpleGraph.degree_one_rank_tree is due to Franzblau 2000): for a connected non-tree G, r(G^2) = r((G^core)^2) + 2|V \\ V(G^core)| + |V_1(G)|, reducing the squared rank to the two-core (the maximal subgraph of minimum degree >= 2), where the molecule rank formula's hypothesis fails; for a tree, r(G^2) = 2|V| - 5 + |V_1(G)|.", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "SimpleGraph.molecule_generic_square_packing", "file": "CombinatorialRigidity/Molecular/Molecule/Application.lean", "description": "Generic rigidity of squares from six spanning trees (Jackson-Jordán 2010 p. 586, unconditional now that the molecular conjecture is a theorem; phase 34): if 5G contains six edge-disjoint spanning trees, then every bar-joint realization of G^2 in R^3 at a placement generic for row independence is infinitesimally rigid.", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "Graph.BodyBarFramework.isIndependent_and_isInfinitesimallyRigid_ofEndpoints_iff", "file": "CombinatorialRigidity/BodyBar/GenericLift.lean", "description": "Tay's theorem at every generic endpoint assignment (Tay 1984; Jackson-Jordán 2010 section 5; phase 34): the body-bar realization placing each bar on the segment between its assigned endpoints, at an endpoint assignment generic for row independence, is isostatic (independent and infinitesimally rigid) iff the multigraph is (d,d)-tight - the generic form of the body-bar existence theorem, quantifyin", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "CombinatorialRigidity.Molecular.BodyHingeFramework.isInfinitesimallyRigidOn_ofHinge_isGenericHingePoints_iff_spanningTrees", "file": "CombinatorialRigidity/Molecular/GenericLift/HingeGeneric.lean", "description": "Generic body-and-hinge rigidity from spanning trees (Tay 1989, Whiteley 1988; Jackson-Jordán 2010 Thm 6.1 / Cor 6.3, with the existential quantifier upgraded to every generic realization; phase 34): for a simple spanning multigraph on >= 2 bodies, at any dimension n >= 3 and over any infinite field, the affine-hinge framework at every hinge-point assignment generic for row independence is infinite", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "CombinatorialRigidity.Molecular.molecular_conjecture_multigraph", "file": "CombinatorialRigidity/Molecular/AlgebraicInduction/Theorem55.lean", "description": "The Molecular Conjecture at full multigraph strength (Katoh-Tanigawa 2011 Conjecture 1.2 for multigraphs; phase 35): a spanning multigraph on any nonempty body set - parallel edges and loops admitted - has an infinitesimally rigid genuine body-hinge realization iff it has an infinitesimally rigid hinge-coplanar panel realization (KT's containment model: each hinge lies in the panel of each endpoin", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] } ], "comparator": false, "literature_dependencies": 0, "divergences": "Most statements are more general than their classical sources - over mathlib's multigraph `Graph alpha beta`, at arbitrary dimension, and (for the Katoh-Tanigawa chain) over any infinite field rather than R, with the original real statements as the K = R case. Two scope restrictions run the other way: the (k,l)-count matroid and the pebble game are developed only in the matroidal regime l < 2k, which covers the (2,3) Laman case, and the molecular results only at dimension n >= 3. The body-bar and body-hinge realization theorems are formalized twice: in existence-of-realization form with an explicit witness, and in a generic form following Jackson-Jordán 2010's coordinate route, where \"almost all\" is delivered as one nonzero polynomial per framework class rather than as a measure-zero statement. One deliberate method divergence: where Katoh-Tanigawa fix coordinates algebraically independent over the rationals (their footnote 6), the formalization chooses each inductive seed off the zero locus of the finitely many polynomials that composition tests; the statements proved are the same. Nothing is cited in place of a proof - every result the arguments use is formalized, the vendored ma", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Bryan Gin-ge Chen" ], "notes": "No external human peer review. Confidence rests on: a fully green CI build on every push (Lean compile, blueprint web/PDF, and checkdecls cross-checking every blueprint \\lean{...} pointer against the Lean environment); per-result `#print axioms` checks (all seventeen main results use only propext / Classical.choice / Quot.sound); the lakefile's `warn.sorry` backstop; and continuous AI-agent review (between-phase cleanup rounds auditing Lean/blueprint divergence and long proofs, plus /code-review and friction-review passes), overseen by the author." }, "canonical": { "repo": "bryangingechen/combinatorialrigidity", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "carlok/sharp-symmetry-bounds-lean/formalization.yaml", "repo": "carlok/sharp-symmetry-bounds-lean", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/carlok/sharp-symmetry-bounds-lean/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Sharp symmetry bounds for real plane algebraic curves (Theorem 1)", "description": "Let C be an infinite real plane algebraic curve defined by a polynomial of degree d >= 2 that is irreducible over the complex numbers, and suppose C is not a circle. Its group of Euclidean symmetries is finite, the orientation-preserving part is cyclic of order at most max(d, 2d - 4), and the full group has order at most 2d. Both bounds are attained in every degree d >= 2. For d >= 5, curves attaining the rotation bound are exactly, up to orientation-preserving similarity, the curves Re(z^(d-2)(|z|^2 + a)) = 0 with |a| = 1 and a not real; conversely each such curve with d - 2 >= 3 attains the ", "authors": [ "Carlo Perassi" ], "maintainers": [ "Carlo Perassi" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Sharp symmetry bounds for real algebraic curves (working draft, 2026), Theorem 1", "id": "", "authors": [ "Carlo Perassi" ], "type": "unpublished working note", "relationship": "formalizes", "endorsement": "participated" }, { "title": "Rotations, translations and symmetry detection for complexified curves", "id": "doi:10.1016/j.cagd.2008.09.004", "authors": [ "Peter Lebmeir", "Jürgen Richter-Gebert" ], "type": "article", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Feature Detection for Real Plane Algebraic Curves", "id": "", "authors": [ "Peter M. Lebmeir" ], "type": "thesis", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Distinct distances on algebraic curves in the plane", "id": "arXiv:1308.0177v3", "authors": [ "János Pach", "Frank de Zeeuw" ], "type": "preprint", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Symmetries and similarities of planar algebraic curves using harmonic polynomials", "id": "arXiv:1801.09962v1", "authors": [ "Juan Gerardo Alcázar", "Miroslav Lávička", "Jan Vršek" ], "type": "preprint", "relationship": "background", "endorsement": "not-contacted" } ], "related": [], "classification": { "arxiv": [ "math.AG", "math.MG" ], "msc2020": [ "14H50", "14P05", "51N20" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Claude Opus 5 (claude-opus-5)", "Codex Sol" ], "framework": "" } ], "strongest": "agent", "models": [ "Claude Opus 5 (claude-opus-5)", "Codex Sol" ], "spend": "subscription-based", "notes": "The human author wrote the note and directed the formalization, approving scope and statement-fidelity decisions. AI agents produced the Lean code. Statement alignment between the Challenge and the note was checked by the agents and the author; no independent human mathematical review of the Lean development has taken place." }, "scope": "Formalized: all five clauses of Theorem 1 as stated in Challenge.lean (finiteness, cyclic orientation-preserving part, both upper bounds, both sharpness statements in every degree d >= 2, the d >= 5 equality classification, and the converse for the extremal family). Not formalized in this entry: Theorem 2 of the note (ambient Mobius group, parameter equivalence, genus of the extremal family), which is developed separately in the author's separate, still private development and is incomplete (genus not proved).", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "(1) The plane is identified with C via z = x + iy; a curve is the real zero set of a real polynomial f(x, y) that is irreducible after extending coefficients to C. (2) Orientation-preserving symmetries are stated as isometries of the form z -> az + b with |a| = 1 (equivalent to determinant +1 of the linear part; that equivalence is proved in the library, not in the Challenge). (3) Cyclicity is stated as: the orientation-preserving symmetries are exactly the integer powers of one isometry. (4) Cardinalities use Set.ncard, which is 0 for infinite sets; finiteness is stated separately in the bound theorem, and the exact counts in the other theorems are positive. (5) \"Not real\" is stated as im(a) != 0. (6) The converse is proved without the normalization |a| = 1, a slight strengthening. (7) A symmetry T of C is required to satisfy T(z) in C exactly when z in C, i.e. T maps C onto C. (8) The sign character and the reflection statement of Lemma 3 are proof ingredients (GeometricRotation, Reflection), not compared statements.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "none" ], "notes": "Kernel-checked Lean proofs with a namespace-wide axiom audit (only propext, Classical.choice, Quot.sound). Challenge and Solution statements and definitions were compared by an elaborated-term pre-check, and Palomar's reusable mechanical preflight (Comparator, NanoDa, profile palomar-standard-v1) passed on commit 4408003 of this repository. The informal account with a step-by-step map to the Lean declarations is docs/THEOREM1.md. No independent human review." }, "canonical": { "repo": "carlok/sharp-symmetry-bounds-lean", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-18-000007", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-18-000007", "commit": "ced9fe2d4d2aa42aa03bbc19b1b56cdcd18c9413", "trust": "high", "theorems": 5, "date": "2026-09-18" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "catskillsresearch/cardb/formalization.yaml", "repo": "catskillsresearch/cardb", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/catskillsresearch/cardb/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "The number of topological bases of a finite set", "description": "A Lean formalization of an exact enumeration identity and two consequences: for a finite type α, the number of families B ⊆ Set α that are topological bases equals the sum, over all topologies τ on α, of 2^(|τ| − |M_τ|), where M_τ is the canonical minimal basis of minimal open neighbourhoods (Mathlib's nhdsKer {x} at each point). Kernel-reducible enumeration gives the exact table #(0)=2, #(1)=2, #(2)=10, #(3)=142. A quantitative sandwich then shows that the discrete fiber 2^(2^N − N) dominates: for N ≥ 2 the remaining fibers contribute at most 2^(N(N−1) + 3·2^(N−2)), the ratio to the discrete ", "authors": [ "Lars Warren Ericson" ], "maintainers": [ "Lars Warren Ericson" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "On the Number of Distinct Topological Bases of a Finite Set of Size N", "id": "https://github.com/catskillsresearch/cardb/blob/main/CARDB.md", "authors": [ "Lars Warren Ericson" ], "type": "original-proof", "relationship": "other", "endorsement": "participated" }, { "title": "Diskrete Räume", "id": "", "authors": [ "P. Alexandrov" ], "type": "article", "relationship": "background", "endorsement": "" }, { "title": "Topology on finite sets", "id": "", "authors": [ "D. Stephen" ], "type": "article", "relationship": "background", "endorsement": "" }, { "title": "The number of topologies on a finite set", "id": "https://cs.uwaterloo.ca/journals/JIS/VOL9/Benoumhani/benoumhani6.pdf", "authors": [ "M. Benoumhani" ], "type": "article", "relationship": "background", "endorsement": "" } ], "related": [ { "id": "https://github.com/catskillsresearch/scott_models/tree/main/CARDB", "relationship": "supersedes", "note": "This repository is a split and migration of the CARDB directory that previously lived in catskillsresearch/scott_models. That parent repository is about Scott models; CARDB was parked there for convenience and is not part of that project's intent. The present repo is the same author's continuation of that directory (same Apache-2.0 development, first commit a4d3fc0), not a reimplementation of an i" } ], "classification": { "arxiv": [ "math.GN", "math.CO" ], "msc2020": [ "54A05", "05A15", "54A10" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Cursor Grok 4.6 High Fast (SpaceXAI / Cursor)" ], "framework": "Cursor" }, { "method": "agent", "models": [ "Claude Sonnet 5 (Anthropic)" ], "framework": "Cursor" }, { "method": "agent", "models": [ "Gemini 3.7 Flash (Google DeepMind)" ], "framework": "Cursor" }, { "method": "manual", "models": [], "framework": "" } ], "strongest": "agent", "models": [ "Claude Sonnet 5 (Anthropic)", "Cursor Grok 4.6 High Fast (SpaceXAI / Cursor)", "Gemini 3.7 Flash (Google DeepMind)" ], "spend": "", "notes": "All three named models were used in the Cursor agent environment. None has an official Hugging Face model card; citations are in CARDB.md. Palomar Challenge / Solution / comparator / formalization.yaml packaging was added later by an agent under the author's direction." }, "scope": "Complete, with no unproved step. The identity, the small-N table through N = 3, the sandwich bound for N ≥ 2, #(N) ∼ 2^(2^N−N), and discrete dominance for N ≥ 10 are stated and proved for every finite type α (the identity and the limit omit DecidableEq; the table and bounds keep it). The empty type is included. Nothing is assumed from the literature; the compared family is proved from Mathlib alone.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "card_valid_bases", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "card_valid_bases_small", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "card_valid_bases_bounds", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "card_valid_bases_dominated", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "card_valid_bases_asymptotic", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The informal note speaks of a finite set S with |S| = N and writes #(N) for the count. The Lean statement is for an arbitrary finite type α; N is Fintype.card α and is not an explicit parameter. Informal gen(B) is Mathlib's generateFrom B; informal |τ| is ncard (opensOf τ); informal U_x is nhdsKer {x} (minimalOpen). The note's N = 0 row is included: the empty type is not excluded. Lemma names in CARDB.lean follow the note's development rather than a published paper.", "alignment": true, "original": true, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Lars Warren Ericson" ], "notes": "The author directed and reviewed the development as it was produced. No external or independent review has been performed, and no mathematician other than the author has read the Lean source. Every step of card_valid_bases is checked by Lean against Mathlib." }, "canonical": { "repo": "catskillsresearch/cardb", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-20-000003", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-20-000003", "commit": "d8c1d63052a18db0c43d44c1c77fb10ca8902ed0", "trust": "high", "theorems": 5, "date": "2026-08-20" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "catskillsresearch/hybrid_logic_lean_revisited/formalization.yaml", "repo": "catskillsresearch/hybrid_logic_lean_revisited", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/catskillsresearch/hybrid_logic_lean_revisited/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Completing Hybrid Logic L(∀) in Lean 4", "description": "A Lean 4 / mathlib continuation of Alex Oltean's hybrid_logic_lean formalization of hybrid logic L(∀). Oltean formalized syntax, Kripke semantics, a Hilbert proof system, and soundness, but left completeness and supporting Henkin/Lindenbaum constructions unfinished. This project ports that development to Lean 4.33.1 and closes those proof holes for nonempty nominal types, culminating in CompletenessTheorem: semantic consequence implies derivability under the explicit N.Nonempty hypothesis. Original project: https://github.com/alexoltean61/hybrid_logic_lean", "authors": [ "Lars Warren Ericson" ], "maintainers": [ "Lars Warren Ericson" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "hybrid_logic_lean", "id": "https://github.com/alexoltean61/hybrid_logic_lean", "authors": [ "Alex Oltean" ], "type": "software", "relationship": "other", "endorsement": "not-consulted" }, { "title": "Hybrid Completeness", "id": "https://doi.org/10.1093/jigpal/6.4.625", "authors": [ "Patrick Blackburn", "Miroslava Tzakova" ], "type": "article", "relationship": "formalizes", "endorsement": "" }, { "title": "Finishing Oltean's Completeness Proof in Lean 4 for Hybrid Logic L(∀)", "id": "https://arxiv.org/abs/2606.19761", "authors": [ "Lars Warren Ericson" ], "type": "article", "relationship": "other", "endorsement": "participated" }, { "title": "An Isabelle/HOL Framework for Synthetic Completeness Proofs", "id": "", "authors": [ "Asta Halkjær From" ], "type": "article", "relationship": "background", "endorsement": "" }, { "title": "Synthetic Completeness for a Terminating Seligman-Style Tableau System", "id": "", "authors": [ "Asta Halkjær From" ], "type": "article", "relationship": "background", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "cs.LO", "math.LO" ], "msc2020": [ "03B45", "03B70" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Cursor GPT-5.6 Sol" ], "framework": "Cursor" }, { "method": "manual", "models": [], "framework": "" } ], "strongest": "agent", "models": [ "Cursor GPT-5.6 Sol" ], "spend": "", "notes": "" }, "scope": "One compared theorem records strong completeness for L(∀). The original syntax, semantics, proof system, and soundness are attributed to Alex Oltean's hybrid_logic_lean. The revisited project ports that code and discharges 41 active original sorry/admit declarations across nine modules, including the extended Lindenbaum construction, witnessed-successor existence construction, language conservativity, and final completeness result for nonempty nominal types. The original final hole stated completeness without that hypothesis; the completed construction exposes N.Nonempty explicitly, so the Pal", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "CompletenessTheorem", "file": "Hybrid/Completeness.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "CompletenessTheorem is a propositional wrapper around the proof-producing noncomputable definition Completeness. The wrapper uses Nonempty (Γ ⊢ φ) because derivability is represented by a Type-valued derivation, while preserving the mathematical statement Γ ⊨ φ implies Γ ⊢ φ. The proof repairs and extends Oltean's existing architecture rather than claiming an independent re-formalization from scratch.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Lars Warren Ericson" ], "notes": "No external formal review has been performed." }, "canonical": { "repo": "catskillsresearch/hybrid_logic_lean_revisited", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "catskillsresearch/scott1964/formalization.yaml", "repo": "catskillsresearch/scott1964", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/catskillsresearch/scott1964/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Formalization of Scott's Measurement Structures and Linear Inequalities in Lean 4", "description": "A Lean 4 / mathlib formalization of Dana Scott's 1964 paper *Measurement Structures and Linear Inequalities* (Journal of Mathematical Psychology 1 (1964), 233–247). The paper derives necessary and sufficient conditions for three finite representation problems — intransitive indifference, ordered differences, and subjective probability — from the general solvability criterion for finite systems of linear inequalities. The development also covers Scott's ordered-group and signed-charge remarks, literal vector cancellation, and the KPS obstruction to de Finetti's five axioms. Every compared equiv", "authors": [ "Lars Warren Ericson", "Dana S. Scott", "Vijay D'Silva", "Brian Milnes" ], "maintainers": [ "Lars Warren Ericson" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "Measurement Structures and Linear Inequalities", "id": "", "authors": [ "Dana S. Scott" ], "type": "paper", "relationship": "formalizes", "endorsement": "participated" }, { "title": "Measures on Boolean Algebras", "id": "", "authors": [ "J. L. Kelley" ], "type": "paper", "relationship": "adapts", "endorsement": "" }, { "title": "Foundations of Measurement, Volume I", "id": "", "authors": [ "D. H. Krantz", "R. D. Luce", "P. Suppes", "A. Tversky" ], "type": "book", "relationship": "background", "endorsement": "" }, { "title": "The Axioms of Subjective Probability", "id": "", "authors": [ "P. C. Fishburn" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Necessary and Sufficient Conditions for Existence and Uniqueness of Weak Qualitative Probability Structures", "id": "", "authors": [ "M. A. Cohen" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Scaled Boolean Algebras", "id": "", "authors": [ "M. Hardy" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "The Logic of Comparative Cardinality", "id": "", "authors": [ "Y. Ding", "M. Harrison-Trainor", "W. H. Holliday" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Modal Logic with “Most”", "id": "", "authors": [ "X. Fu", "Z. Zhao" ], "type": "paper", "relationship": "background", "endorsement": "" } ], "related": [ { "id": "https://github.com/catskillsresearch/scott1972", "relationship": "other", "note": "Same-author formalization of Scott Continuous Lattices (1972); not imported here." }, { "id": "https://github.com/catskillsresearch/scott1976", "relationship": "other", "note": "Same-author formalization of Scott Data Types as Lattices (1976); not imported here." }, { "id": "https://github.com/catskillsresearch/scott1980", "relationship": "other", "note": "Same-author formalization of Scott PRG-19 (1980); not imported here." }, { "id": "https://github.com/catskillsresearch/scott1982", "relationship": "other", "note": "Same-author formalization of Scott 1982 information systems; not imported here." } ], "classification": { "arxiv": [ "cs.LO", "math.LO", "econ.TH" ], "msc2020": [ "91C05", "15A39", "91B16", "68V20" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Cursor Grok 4.6 High Fast (SpaceXAI / Cursor)" ], "framework": "Cursor" }, { "method": "agent", "models": [ "Cursor Composer 2.5 Fast" ], "framework": "Cursor" }, { "method": "agent", "models": [ "Anthropic Claude Sonnet 5 (medium reasoning)" ], "framework": "Cursor" }, { "method": "agent", "models": [ "Anthropic Claude Opus 4.8 (high reasoning)" ], "framework": "Cursor" }, { "method": "agent", "models": [ "OpenAI GPT 5.6" ], "framework": "Cursor" }, { "method": "manual", "models": [], "framework": "" } ], "strongest": "agent", "models": [ "Anthropic Claude Opus 4.8 (high reasoning)", "Anthropic Claude Sonnet 5 (medium reasoning)", "Cursor Composer 2.5 Fast", "Cursor Grok 4.6 High Fast (SpaceXAI / Cursor)", "OpenAI GPT 5.6" ], "spend": "", "notes": "Palomar Challenge / Solution / comparator / formalization.yaml packaging follows the scott1972 / scott1976 / cardb template." }, "scope": "Complete sorry-free formalizations of Scott's eight numbered theorems 1.1–1.4, 2.1, 3.1, 3.2, and 4.1. Challenge.lean contains the deliberate Palomar proof holes, while Solution.lean re-exports all proofs. A ninth compared result is explicitly labelled as a modern reconstruction of the infinite probability theorem Scott announced but did not publish here. It is an iff characterization for arbitrary Boolean algebras, not merely an existence implication: finite-additive probability representation is equivalent to nontriviality, bottom-minimality, totality, and an explicit generalized Kelley cove", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "Working source: sources/ScottMeasurement1964.pdf (a vision transcription sources/ScottMeasurement1964_vision.md is produced by scripts/ocr_pdf_pipeline.sh). Section 4 of arxiv.md is an exhaustive paper-order concordance (verbatim Scott passage, exact Lean fragment or an explicit gap, and a Lean-driven mathematical reconstruction). Coverage, quotations, and declaration names are checked by scripts/check_concordance.py; cards are emitted by scripts/emit_concordance.py. Challenge.lean renders Scott's condition (4_B) with the atom-counting reading of the algebraic sum of characteristic functions, which Scott gives in words immediately after the theorem; theorem_4_1_vector separately proves the equivalent literal vector-sum form. Theorem 2.1 currently uses the direct Scott–Suppes staircase construction, while Scott's local cycle reductions are formalized separately. Exact divergences are listed in known_gaps and in arxiv.md. Its countable-cover and separator-combination argument adapts Kelley's 1959 measure-existence method, but its universal event-span formulation, weak-comparison cone, generalized Kelley condition, and iff statement are not claimed to occur in Kelley or Scott. Based o", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Lars Warren Ericson" ], "notes": "Lars Warren Ericson directed and reviewed the Lean development. Dana S. Scott, Vijay D'Silva, and Brian Milnes are co-authors of the report; this is not recorded as an independent external review of the Lean source. Every proof step is checked by the Lean kernel." }, "canonical": { "repo": "catskillsresearch/scott1964", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-25-000015", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-25-000015", "commit": "2c35b7d2b7f5d60c20381a2f497bb0ab7d4d0a1f", "trust": "high", "theorems": 9, "date": "2026-09-25" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "catskillsresearch/scott1972/formalization.yaml", "repo": "catskillsresearch/scott1972", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/catskillsresearch/scott1972/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Scott 1972 Continuous Lattices — Theorem 4.4", "description": "A Lean 4 / mathlib formalization of Dana Scott's 1972 paper *Continuous Lattices* (LNM 274), through the Milner correction (pp. 135–136). The Palomar compared result is Theorem 4.4 as worded in sources/ScottContinLatt1972.md: the inverse limit D_∞ of the recursively defined sequence ⟨D_n, j_n⟩ of function spaces is not only a continuous lattice, but it is also homeomorphic to its own function space [D_∞ → D_∞]. The compared Homeomorph explicitly uses the product/subspace topology on the inverse limit and the pointwise Pi topology on the function space. The full paper development (injectivity, ", "authors": [ "Lars Warren Ericson" ], "maintainers": [ "Lars Warren Ericson" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "Continuous Lattices", "id": "", "authors": [ "Dana S. Scott" ], "type": "article", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "A Lean 4 Formalization of Scott's Continuous Lattices (1972)", "id": "https://arxiv.org/abs/2606.30782", "authors": [ "Lars Warren Ericson" ], "type": "article", "relationship": "other", "endorsement": "participated" }, { "title": "Continuous Lattices (Milner correction)", "id": "", "authors": [ "Robin Milner" ], "type": "article", "relationship": "background", "endorsement": "not-contacted" } ], "related": [ { "id": "https://github.com/catskillsresearch/scott_models", "relationship": "other", "note": "Same-author sibling with cross-presentation equivalence theorems for Scott 1972 / 1980 / 1982. That Palomar entry was withdrawn after a registration glitch; this repository is an independent submission for the 1972 paper alone. See PROVENANCE.md." }, { "id": "https://github.com/catskillsresearch/scott1980", "relationship": "other", "note": "Same-author formalization of Scott PRG-19 (1980); not imported here." }, { "id": "https://github.com/catskillsresearch/scott1982", "relationship": "other", "note": "Same-author formalization of Scott 1982 information systems; not imported here." } ], "classification": { "arxiv": [ "math.LO", "math.GN" ], "msc2020": [ "06B35", "54H99", "03B40" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Cursor Grok 4.6 High Fast (SpaceXAI / Cursor)" ], "framework": "Cursor" }, { "method": "agent", "models": [ "Cursor Composer 2.5 Fast" ], "framework": "Cursor" }, { "method": "agent", "models": [ "Anthropic Claude Sonnet 5 (medium reasoning)" ], "framework": "Cursor" }, { "method": "agent", "models": [ "Anthropic Claude Opus 4.8 (high reasoning)" ], "framework": "Cursor" }, { "method": "manual", "models": [], "framework": "" } ], "strongest": "agent", "models": [ "Anthropic Claude Opus 4.8 (high reasoning)", "Anthropic Claude Sonnet 5 (medium reasoning)", "Cursor Composer 2.5 Fast", "Cursor Grok 4.6 High Fast (SpaceXAI / Cursor)" ], "spend": "", "notes": "Palomar Challenge / Solution / comparator / formalization.yaml packaging was added under the author's direction, following the cardb template." }, "scope": "Complete for the compared claim and the underlying Scott1972 library it uses. Theorem 4.4 is the compared declaration theorem_4_4: for a continuous lattice D₀ and projection j₀, the inverse limit D_∞ of the function-space tower is a continuous lattice homeomorphic to [D_∞ → D_∞]. The Homeomorph names Scott's product/subspace inverse-limit topology and Definition 3.1 pointwise function-space topology explicitly. Challenge.lean fixes the function-space lattice by pointwise order and pointwise suprema, the inverse-limit lattice by coordinatewise order and infima, and the projection tower by the r", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "Scott1972.ContinuousLattice.theorem_4_4", "file": "Scott1972/ContinuousLattice/FunctionSpaceTower.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "Working source: sources/ScottContinLatt1972.md. Theorem 4.4 there is quoted in Challenge.lean and as theorem_4_4. Lean DInf is the inverse limit of the recursively defined sequence ⟨D_n, j_n⟩ of function spaces (D_{n+1} = [D_n → D_n], j_{n+1} = [j_n → j_n], D_0 and j_0 given as in the paragraph above 4.4). Definition 3.1 takes [X → Y] with the product topology, represented by scottMapInducedPiTopology. The inverse limit uses inverseLimitTopology, the subspace topology induced from the Pi product of the stage Scott topologies. Proposition 4.1 follows Scott's route: coordinatewise maximal extensions from Proposition 3.8 are made compatible by Lemma 3.9, yielding injectivity, and Theorem 2.12 supplies both the continuous-lattice result and the topology identification. Lemma 4.5 follows Scott's induction after moving a projection through a directed supremum. The source-topology homeomorphism uses Scott's formulas i_∞(x) = ⨆ₙ (i_{n∞} ∘ x_{n+1} ∘ j_{∞n}) and j_∞(f) = ⨆ₙ i_{(n+1)∞}(j_{∞n} ∘ f ∘ i_{n∞}). Theorem 3.3 identifies the pointwise function topology with the lattice Scott topology. The abstract's extra “and isomorphic” is theorem_4_4_orderIso and is not the numbered theorem.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Lars Warren Ericson" ], "notes": "The author directed and reviewed the development. No external review of the Lean source, and no review or endorsement by Dana Scott. Every proof step is checked by the Lean kernel." }, "canonical": { "repo": "catskillsresearch/scott1972", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-24-000003", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-24-000003", "commit": "278ba1745b6fb023fc3e9ffe53b7106ce60222c1", "trust": "high", "theorems": 19, "date": "2026-08-24" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "catskillsresearch/scott1976/formalization.yaml", "repo": "catskillsresearch/scott1976", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/catskillsresearch/scott1976/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Scott 1976 Data Types as Lattices", "description": "A Lean 4 / mathlib formalization of Dana Scott's 1976 paper *Data Types as Lattices* (Technical Monograph PRG-5; SIAM J. Comput. 5 (1976), 522–587). The library target is a 1:1 onto translation of every numbered definition, theorem, example, and Tables 1–3. Palomar Challenge/Solution select the 63 theorem declarations and 70 locked definitions named in comparator.json. A Comparator match establishes those selected statements, not every informal clause of every numbered theorem. The locked coproduct `plusR` uses the doubly strict conditional of (4.4) rather than the ordinary conditional written", "authors": [ "Lars Warren Ericson" ], "maintainers": [ "Lars Warren Ericson" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "Data Types as Lattices", "id": "", "authors": [ "Dana S. Scott" ], "type": "article", "relationship": "formalizes", "endorsement": "other" } ], "related": [ { "id": "https://github.com/catskillsresearch/scott1972", "relationship": "other", "note": "Same-author formalization of Scott Continuous Lattices (1972); not imported here." }, { "id": "https://github.com/catskillsresearch/scott1980", "relationship": "other", "note": "Same-author formalization of Scott PRG-19 (1980); not imported here." }, { "id": "https://github.com/catskillsresearch/scott1982", "relationship": "other", "note": "Same-author formalization of Scott 1982 information systems; not imported here." }, { "id": "https://github.com/catskillsresearch/scott_models", "relationship": "other", "note": "Same-author sibling with cross-presentation equivalence theorems. This repository is an independent Palomar submission for the 1976 paper alone. See PROVENANCE.md." } ], "classification": { "arxiv": [ "math.LO", "cs.LO" ], "msc2020": [ "06B35", "68Q55", "03B70" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Cursor Grok 4.6 High Fast (SpaceXAI / Cursor)" ], "framework": "Cursor" }, { "method": "agent", "models": [ "Cursor Composer 2.5 Fast" ], "framework": "Cursor" }, { "method": "agent", "models": [ "Anthropic Claude Sonnet 5 (medium reasoning)" ], "framework": "Cursor" }, { "method": "agent", "models": [ "Anthropic Claude Opus 4.8 (high reasoning)" ], "framework": "Cursor" }, { "method": "manual", "models": [], "framework": "" } ], "strongest": "agent", "models": [ "Anthropic Claude Opus 4.8 (high reasoning)", "Anthropic Claude Sonnet 5 (medium reasoning)", "Cursor Composer 2.5 Fast", "Cursor Grok 4.6 High Fast (SpaceXAI / Cursor)" ], "spend": "", "notes": "Palomar Challenge / Solution / comparator / formalization.yaml packaging follows the scott1972 / cardb template." }, "scope": "The library is a 1:1 onto translation of the numbered items in Scott 1976, except that plusR uses the doubly strict conditional of (4.4) in place of the ordinary conditional written in displayed (4.10). Comparator selects the 63 theorem declarations and 70 definitions named in comparator.json. That match establishes those selected statements; further library companions are not all compared.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "Scott1976.DataTypesAsLattices.theorem_1_1", "file": "Scott1976/DataTypesAsLattices/Continuous.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Scott1976.DataTypesAsLattices.theorem_1_2", "file": "Scott1976/DataTypesAsLattices/Graph.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Scott1976.DataTypesAsLattices.theorem_1_3", "file": "Scott1976/DataTypesAsLattices/FixedPoint.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] 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"sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Scott1976.DataTypesAsLattices.theorem_6_6", "file": "Scott1976/DataTypesAsLattices/Classification.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Scott1976.DataTypesAsLattices.theorem_6_7", "file": "Scott1976/DataTypesAsLattices/Classification.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Scott1976.DataTypesAsLattices.theorem_7_1", "file": "Scott1976/DataTypesAsLattices/Functionality.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Scott1976.DataTypesAsLattices.theorem_7_2", "file": "Scott1976/DataTypesAsLattices/Functionality.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Scott1976.DataTypesAsLattices.theorem_7_3", "file": "Scott1976/DataTypesAsLattices/Functionality.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Scott1976.DataTypesAsLattices.theorem_7_4", "file": "Scott1976/DataTypesAsLattices/Functionality.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "Working source: sources/Data_Types_as_Lattices_vision.md (transcription of sources/Data_Types_as_Lattices.pdf). Compared claims are the spanning theorems named in comparator.json. Remaining source-level notes are listed in known_gaps and in arxiv.md.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Lars Warren Ericson" ], "notes": "The author directed and reviewed the development. No external review of the Lean source, and no review or endorsement by Dana Scott. Every proof step is checked by the Lean kernel." }, "canonical": { "repo": "catskillsresearch/scott1976", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-25-000004", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-25-000004", "commit": "2081973122a631d34dc74f83cfc3f7495991415b", "trust": "high", "theorems": 6, "date": "2026-08-25" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "catskillsresearch/scott1980/formalization.yaml", "repo": "catskillsresearch/scott1980", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/catskillsresearch/scott1980/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Scott 1981 — D ⊴ U (Theorem 8.8, first sentence + two corollaries)", "description": "A Lean 4 / mathlib formalization of Dana Scott's 1980 Oxford lectures, published May 1981 as PRG-19 Lectures on a Mathematical Theory of Computation. The principal sourced Palomar result is the first sentence of Theorem 8.8: the neighbourhood system U over [0,1) ⊆ ℚ is universal, in the sense that every countable neighbourhood system D embeds as a subdomain (D ⊴ U). Comparator also selects two new formal corollaries: P4_embeds for Scott's Example 1.5 and Ssys_embeds for Exercise 7.22. Scott defines those systems but does not separately state the embeddings. Challenge.lean fixes the neighbourho", "authors": [ "Lars Warren Ericson" ], "maintainers": [ "Lars Warren Ericson" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Lectures on a Mathematical Theory of Computation", "id": "", "authors": [ "Dana S. Scott" ], "type": "paper", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "A Lean 4 Certificate of Scott's Lectures on a Mathematical Theory of Computation (1981) generated by LLM and registered by Palomar", "id": "", "authors": [ "Lars Warren Ericson" ], "type": "other", "relationship": "other", "endorsement": "participated" } ], "related": [ { "id": "https://github.com/catskillsresearch/scott1972", "relationship": "other", "note": "Same-author formalization of Scott 1972 Continuous Lattices. Not imported here. Palomar packaging follows that repository's Challenge / Solution pattern." }, { "id": "https://github.com/catskillsresearch/scott1982", "relationship": "other", "note": "Same-author formalization of Scott 1982 information systems; not imported here." }, { "id": "https://github.com/catskillsresearch/scott_models", "relationship": "other", "note": "Same-author sibling with cross-presentation equivalence theorems for Scott 1972 / 1980 / 1982. Not imported here." }, { "id": "https://github.com/catskillsresearch/cardb", "relationship": "other", "note": "Same-author Palomar template source; not imported here." } ], "classification": { "arxiv": [ "cs.LO", "math.LO" ], "msc2020": [ "68Q55", "06B35", "03B70" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Cursor Grok 4.6 High Fast (SpaceXAI / Cursor)" ], "framework": "Cursor" }, { "method": "agent", "models": [ "Cursor Composer 2.5 Fast" ], "framework": "Cursor" }, { "method": "agent", "models": [ "Anthropic Claude Sonnet 5 (medium reasoning)" ], "framework": "Cursor" }, { "method": "agent", "models": [ "Anthropic Claude Opus 4.8 (high reasoning)" ], "framework": "Cursor" }, { "method": "agent", "models": [ "GPT-5.6 Sol" ], "framework": "Cursor" }, { "method": "manual", "models": [], "framework": "" } ], "strongest": "agent", "models": [ "Anthropic Claude Opus 4.8 (high reasoning)", "Anthropic Claude Sonnet 5 (medium reasoning)", "Cursor Composer 2.5 Fast", "Cursor Grok 4.6 High Fast (SpaceXAI / Cursor)", "GPT-5.6 Sol" ], "spend": "", "notes": "Palomar Challenge / Solution / comparator / formalization.yaml packaging was added under the author's direction, following the cardb and scott1972 templates." }, "scope": "Complete for the three compared embedding results. The principal sourced declaration is theorem_8_8: for every countable neighbourhood system D one has D ⊴ U, i.e. D ≅ D' for some subsystem D' ◁ U. This is Scott's first Theorem 8.8 sentence. The other compared declarations are two new formal corollaries obtained by specialization: Example 1.5 (`P4_embeds`) and Exercise 7.22's system S (`Ssys_embeds`). Scott defines those systems but does not separately state these embeddings. Challenge.lean also locks the filter-element order (element_le), the Definition 8.7 membership predicate UMem, the non-", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "Scott1980.Neighborhood.theorem_8_8", "file": "Scott1980/Neighborhood/Theorem88a.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Scott1980.Neighborhood.Example15.P4_embeds", "file": "Scott1980/Neighborhood/PalomarExamples.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Scott1980.Neighborhood.Exercise722.Ssys_embeds", "file": "Scott1980/Neighborhood/PalomarExamples.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "Working source: sources/PRG19.md. Theorem 8.8 there is quoted in Challenge.lean and as theorem_8_8. Scott writes D ⊴ U; Lean Trianglelefteq is the prose before Lemma 6.15 (D ≅ D' for some D' ◁ U). Scott's standing assumption immediately before Definition 1.1 that the master token set Δ is non-empty is enforced by NeighborhoodSystem.master_nonempty. Definition 8.7 takes U as the non-empty finite unions of rational [r,s) inside [0,1). Lean UMem presents those unions by List (ℚ × ℚ) without per-pair bound side-conditions; U_mem_iff_scott in Definition87.lean proves this is equivalent to Scott's literal per-pair 0 ≤ r < s ≤ 1 family. The principal compared theorem is the general (non-effective) first sentence of 8.8; P4_embeds and Ssys_embeds are new formal corollaries. The effective refinement theorem_8_8_b and the finitary-projection correspondence theorem_8_8_c are proved in the library but are not Comparator targets.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Lars Warren Ericson" ], "notes": "The author directed and reviewed the development. No external review of the Lean source, and no review or endorsement by Dana Scott. Every proof step is checked by the Lean kernel. Palomar's automated v1 review identified two non-blocking presentation/fidelity warnings; this version addresses both before a fresh v2 review." }, "canonical": { "repo": "catskillsresearch/scott1980", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-24-000001", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-24-000001", "commit": "ef6c05671582b7883ecd40b7081b642ce5351963", "trust": "high", "theorems": 19, "date": "2026-08-25" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "catskillsresearch/scott1982/formalization.yaml", "repo": "catskillsresearch/scott1982", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/catskillsresearch/scott1982/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Scott 1982 — Theorem 7.2 (function-space elements)", "description": "A Lean 4 / mathlib formalization of Dana Scott's 1982 ICALP paper *Domains for Denotational Semantics*. The Palomar compared result is only the first sentence of Theorem 7.2: if A and B are information systems, then so is A → B, and the approximable mappings f : A → B are exactly the elements f ∈ |A → B|. Challenge.lean imports Mathlib only and locks Definition 2.1, Definition 2.2, Definition 5.1, and Definition 7.1 (tokens, consistency, and the constructive-witness form of entailment). The remaining apply / curry clauses of the same numbered theorem live in Scott1982/Theorem72.lean and are no", "authors": [ "Lars Warren Ericson" ], "maintainers": [ "Lars Warren Ericson" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "Domains for Denotational Semantics", "id": "", "authors": [ "Dana S. Scott" ], "type": "paper", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "A Lean 4 Formalization of Scott's Domains for Denotational Semantics (1982)", "id": "", "authors": [ "Lars Warren Ericson" ], "type": "other", "relationship": "other", "endorsement": "participated" } ], "related": [ { "id": "https://github.com/catskillsresearch/scott1972", "relationship": "other", "note": "Same-author formalization of Scott 1972 Continuous Lattices. Not imported here. Palomar packaging follows that repository's Challenge / Solution pattern." }, { "id": "https://github.com/catskillsresearch/scott1980", "relationship": "other", "note": "Same-author formalization of Scott PRG-19 (1981); not imported here." }, { "id": "https://github.com/catskillsresearch/scott_models", "relationship": "other", "note": "Same-author sibling with cross-presentation equivalence theorems for Scott 1972 / 1980 / 1982. Not imported here." }, { "id": "https://github.com/catskillsresearch/cardb", "relationship": "other", "note": "Same-author Palomar template source; not imported here." } ], "classification": { "arxiv": [ "cs.LO", "cs.PL", "math.LO" ], "msc2020": [ "68Q55", "68V20", "06B35", "03B70", "18C50" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Cursor Grok 4.6 High Fast (SpaceXAI / Cursor)" ], "framework": "Cursor" }, { "method": "agent", "models": [ "Cursor Composer 2.5 Fast" ], "framework": "Cursor" }, { "method": "agent", "models": [ "Anthropic Claude Sonnet 5 (medium reasoning)" ], "framework": "Cursor" }, { "method": "agent", "models": [ "Anthropic Claude Opus 4.8 (high reasoning)" ], "framework": "Cursor" }, { "method": "manual", "models": [], "framework": "" } ], "strongest": "agent", "models": [ "Anthropic Claude Opus 4.8 (high reasoning)", "Anthropic Claude Sonnet 5 (medium reasoning)", "Cursor Composer 2.5 Fast", "Cursor Grok 4.6 High Fast (SpaceXAI / Cursor)" ], "spend": "", "notes": "Palomar Challenge / Solution / comparator / formalization.yaml packaging was added under the author's direction, following the cardb and scott1972 templates." }, "scope": "Complete for the compared claim. theorem_7_2 is the compared declaration: approximable maps A → B are exactly the elements of the Definition 7.1 function-space system |A → B|. That is Scott's first Theorem 7.2 sentence only. Challenge.lean locks the information-system axioms, the carrier-inclusion order on elements, set-level entailment, approximable maps, and the concrete Definition 7.1 data (FunToken, FunCon, FunEnt, functionSystem). The broader paper is formalized in the same library but is not part of the advertised Comparator result. Additional Comparator theorem targets are proof obligat", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "Scott1982.InfoSys.theorem_7_2", "file": "Scott1982/Theorem72.lean", "description": "", "axioms": [ "propext", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "Ground truth: sources/Domains_for_Denotational_Semantics.pdf. Theorem 7.2 is quoted in Challenge.lean and as theorem_7_2. Definition 7.1 tokens are consistent pairs (u, v). Consistency is Scott's 7.1(iii). Entailment FunEnt is the constructive witness form of 7.1(iv): some s ⊆ w whose inputs are entailed by u' and whose output-union entails v', avoiding an undecidable filter on EntSet. Finset unions in the locked bodies are the choice-free ∪' restated in Challenge.lean (and proved in Scott1982.Constructive). The apply / curry clauses of Theorem 7.2 are proved in the library and are not Comparator targets. The compared statements assume DecidableEq on token types so the Finset operations of Definitions 2.1, 5.1, and 7.1 can be formed.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Lars Warren Ericson" ], "notes": "The author directed and reviewed the development. No external review of the Lean source, and no review or endorsement by Dana Scott. Every proof step is checked by the Lean kernel." }, "canonical": { "repo": "catskillsresearch/scott1982", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-24-000005", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-24-000005", "commit": "d3221eec09505aa75e1b818aabc73f1f04339bdc", "trust": "high", "theorems": 28, "date": "2026-08-24" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "catskillsresearch/scott_models/formalization.yaml", "repo": "catskillsresearch/scott_models", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/catskillsresearch/scott_models/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Equivalence of Scott's three presentations of domain theory", "description": "Order isomorphisms relating Scott’s three presentations of domains — continuous lattices (1972), neighbourhood systems (PRG-19), and information systems (1982) — including the round identification D ≃o RoundFilter ≃o RoundInfoSysElement and the S-expression equation T ≅ A + (T × T) as |T| ≃o |A + (T × T)| (SexSys.Element ≃o SexRhs.Element). Not a formalization of the three papers. Account: view.pdf.", "authors": [ "Lars Warren Ericson" ], "maintainers": [ "Lars Warren Ericson" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "Equivalence of Scott's Three Presentations of Domain Theory", "id": "https://github.com/catskillsresearch/scott_models/blob/main/view.pdf", "authors": [ "Lars Warren Ericson" ], "type": "notes", "relationship": "other", "endorsement": "participated" }, { "title": "Continuous Lattices", "id": "", "authors": [ "Dana Scott" ], "type": "article", "relationship": "background", "endorsement": "" }, { "title": "Lectures on a Mathematical Theory of Computation", "id": "", "authors": [ "Dana Scott" ], "type": "article", "relationship": "background", "endorsement": "" }, { "title": "Domains for Denotational Semantics", "id": "", "authors": [ "Dana Scott" ], "type": "article", "relationship": "background", "endorsement": "" }, { "title": "Folklore equivalence of the three presentations", "id": "", "authors": [], "type": "folklore", "relationship": "independently-proves", "endorsement": "" } ], "related": [ { "id": "https://github.com/catskillsresearch/domain_theory", "relationship": "supersedes", "note": "Same-author combined formalization that preceded this work (Apache-2.0). It was archived as the per-paper split progressed (scott1972 / scott1980 / scott1982 / scott_models), because each paper’s Lean outgrew one tree. This package plus vendor/ is the continuation, not a reimplementation by someone else. The archive is historical and is not vendored." }, { "id": "https://github.com/catskillsresearch/scott1972/tree/278ba1745b6fb023fc3e9ffe53b7106ce60222c1", "relationship": "other", "note": "Same-author per-paper home for Scott 1972 continuous lattices. vendor/scott1972 is a copy of revision 278ba1745b6fb023fc3e9ffe53b7106ce60222c1 (ledger vendor/FROZEN.txt), not the earlier 36bf01f… pin. Local modifications: the sibling Palomar kit is omitted (Challenge.lean, Solution.lean, comparator.json, formalization.yaml, preflight scripts, docs, PROVENANCE.md); lakefile.toml and CI default to t" }, { "id": "https://github.com/catskillsresearch/scott1980/tree/ef6c05671582b7883ecd40b7081b642ce5351963", "relationship": "other", "note": "Same-author per-paper home for PRG-19 neighbourhood systems. vendor/scott1980 is a copy of revision ef6c05671582b7883ecd40b7081b642ce5351963 (ledger vendor/FROZEN.txt), not the earlier f6cbc2d… pin. Local modifications: the sibling Palomar kit is omitted, including Neighborhood/PalomarExamples.lean and its Scott1980.lean import; lakefile.toml and CI default to the Scott1980 library only; Neighborh" }, { "id": "https://github.com/catskillsresearch/scott1982/tree/d3221eec09505aa75e1b818aabc73f1f04339bdc", "relationship": "other", "note": "Same-author per-paper home for Scott 1982 information systems (through Factoid 8.4 / domain equations). vendor/scott1982 is a copy of revision d3221eec09505aa75e1b818aabc73f1f04339bdc (ledger vendor/FROZEN.txt), not the earlier 7ed95c1… pin. Local modifications beyond the Palomar-kit omission and paper-library lakefile/CI: (1) Factoid81 bridge patch required by ScottModels/SexDomainEquation — Tree" }, { "id": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/DomainTheory", "relationship": "other", "note": "Public downstream ingest of catskillsresearch/domain_theory (candidates/provenance.jsonl: repo catskillsresearch/domain_theory, authors Catskills Research Company). Apache-2.0 permitted the copy. Incomplete relative to this snapshot (ContinuousLattice + Neighborhood + a short InfoSys; no full scott1982, no bridges). Not prior work this formalization follows; a later copy of an earlier repo of the " } ], "classification": { "arxiv": [ "cs.LO", "math.LO" ], "msc2020": [ "06B35", "68Q55" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Cursor Grok 4.5 (SpaceXAI / Cursor)" ], "framework": "Cursor" }, { "method": "agent", "models": [ "Cursor Composer 2.5 Fast (Anysphere)" ], "framework": "Cursor" }, { "method": "agent", "models": [ "Cursor Grok 4.6 (SpaceXAI / Cursor)" ], "framework": "Cursor" }, { "method": "agent", "models": [ "Claude Sonnet 5 (Anthropic)", "Claude Opus 4.8 (Anthropic)", "Gemini 3.5 Flash (Google DeepMind)" ], "framework": "Cursor" }, { "method": "manual", "models": [], "framework": "" } ], "strongest": "agent", "models": [ "Claude Opus 4.8 (Anthropic)", "Claude Sonnet 5 (Anthropic)", "Cursor Composer 2.5 Fast (Anysphere)", "Cursor Grok 4.5 (SpaceXAI / Cursor)", "Cursor Grok 4.6 (SpaceXAI / Cursor)", "Gemini 3.5 Flash (Google DeepMind)" ], "spend": "", "notes": "The compared isos were developed in earlier Cursor agent sessions and are already sorry-free in ScottModels/. Palomar Challenge / Solution / comparator / formalization.yaml packaging was added later by an agent under the author's direction. related_formalizations records the three per-paper siblings and the archived combined tree; see limitations for the separate Lean-search remark." }, "scope": "Complete for the compared family, with no unproved step in the ScottModels proof modules. The 1980 ↔ 1982 coding, the round continuous-lattice identification D ≃o RoundInfoSysElement, and the S-expression instance (sexNeighborhoodIso, sexIdealIso, sexDomainEquationIso) are stated and proved. The rest of the ScottModels library (general constructions, ScottMap conjugation, LRSOD) is out of the compared family. Nothing in the compared statements is imported from a prior Palomar entry.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "ScottModels.neighborhoodSystem_to_infoSys", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "ScottModels.infoSys_to_neighborhoodSystem", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "ScottModels.InfoSysToNeighborhood.domainOrderIso", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "ScottModels.presentation_domains_equiv", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "ScottModels.sexNeighborhoodIso", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "ScottModels.sexIdealIso", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "ScottModels.sexDomainEquationIso", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The informal note speaks of three presentations determining the same domains up to order isomorphism. The compared Lean statements are the named bridges, not a 2-categorical equivalence of the full categories with all morphisms. The 1972 corner is D ≃o RoundFilter ≃o RoundInfoSysElement, not raw |𝒟| ≃o D. Informal [u] is InfoSys.basicOpen; informal ↟a is wayBelowUp. DecidableEq on tokens (and on D for the ↟-coding) is required so InfoSys can use the choice-free Finset prelude. The sibling packages remain the per-paper remotes; this snapshot vendors modified copies in vendor/ (revisions and local patches: vendor/FROZEN.txt) and proves the maps between them.", "alignment": true, "original": true, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Lars Warren Ericson" ], "notes": "The author directed and reviewed the development as it was produced. No external or independent review has been performed, and no mathematician other than the author has read the Lean source. Every compared declaration is checked by Lean against Mathlib and the pinned sibling packages." }, "canonical": { "repo": "catskillsresearch/scott_models", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-25-000003", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-25-000003", "commit": "f0d6af25a06008457d399562ef6e26c11616ac51", "trust": "high", "theorems": 4, "date": "2026-08-25" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "CBirkbeck/uniform-sheafy-tate-domains-lean/formalization.yaml", "repo": "CBirkbeck/uniform-sheafy-tate-domains-lean", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/CBirkbeck/uniform-sheafy-tate-domains-lean/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Uniform sheafy Tate rings that are not stably uniform", "description": "A Lean 4 formalisation of *Uniform sheafy Tate rings that are not stably uniform* (Birkbeck–Torzewski), answering Question 7 of Kedlaya's *Nonarchimedean Scottish Book* in the negative: a sheafy uniform Huber pair need not be stably uniform. Buzzard–Verberkmoes and Mihara proved that stably uniform Tate Huber rings are sheafy; whether the converse holds was open. The paper gives two complete Tate algebras over a complete discretely valued field `k` that are uniform, nonnoetherian integral domains with `A° = A₀`, strongly sheafy (every Tate algebra `A⟨X₁, …, Xₙ⟩` is sheafy), and yet not stably ", "authors": [ "Christopher Birkbeck", "Alex Torzewski" ], "maintainers": [ "Christopher Birkbeck", "Alex Torzewski" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Uniform sheafy Tate rings that are not stably uniform", "id": "https://cbirkbeck.github.io/uniform-sheafy-tate-domains/", "authors": [ "Christopher Birkbeck", "Alex Torzewski" ], "type": "paper", "relationship": "formalizes", "endorsement": "participated" }, { "title": "Adic Spaces (lecture notes)", "id": "arXiv:1910.05934", "authors": [ "Torsten Wedhorn" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "The Nonarchimedean Scottish Book", "id": "https://scripts.mit.edu/~kedlaya/wiki/index.php?title=The_Nonarchimedean_Scottish_Book", "authors": [], "type": "web discussion", "relationship": "background", "endorsement": "not-contacted" } ], "related": [], "classification": { "arxiv": [ "math.AG", "math.NT" ], "msc2020": [ "14G22", "12J25", "13J10", "16W80", "14G45" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Claude Fable 5" ], "framework": "Claude Code" }, { "method": "other", "models": [ "ChatGPT 5.6 Sol" ], "framework": "" } ], "strongest": "agent", "models": [ "ChatGPT 5.6 Sol", "Claude Fable 5" ], "spend": "4 months Claude Code Max 20x + 1 month ChatGPT Pro", "notes": "COST CAVEAT: both figures are flat-rate subscriptions, not per-token metering, and each subscription covered all of the holder's work over that period rather than this project alone. So the subscription cost is an UPPER BOUND on what is attributable here, not a measurement of it, and no USD figure is quoted for that reason. Statement-level certification is available via leanprover/comparator; see `scripts/certify.sh` and the configs referenced under `status.main_results`. This repository pins Lean v4.33.0 (stable) + mathlib v4.33.0 — the first stable release line carrying the fix for kernel so" }, "scope": "Huber/Tate rings, adic spectra, rational localizations and the structure presheaf (Wedhorn); the finite-jet ring (Theorem 1.1) and its chart; the weighted-parity algebra (Theorem 8.1) — both uniform, strongly sheafy, non-noetherian domains with `A° = A₀` that are not stably uniform, the first with a non-reduced bad chart and the second with a bad chart that is an integral domain; and formal statements of the Nonarchimedean Scottish Book problem list. Both theorems are certified over an abstract complete ultrametric nonarchimedean field whose valuation ring is a DVR, not over a fixed witness fi", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "Palomar.fjp_1_1_isSheafy", "file": "Challenge.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Palomar.fjp_1_1_isUniform", "file": "Challenge.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Palomar.fjp_1_1_isDomain", "file": "Challenge.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Palomar.fjp_1_1_not_isNoetherianRing", "file": "Challenge.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Palomar.fjp_1_1_powerBounded_eq_unitBall", "file": "Challenge.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Palomar.fjp_1_1_stronglySheafy", "file": "Challenge.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Palomar.fjp_1_1_not_isStablyUniform", "file": "Challenge.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "FiniteJetOver.finiteJet_isSheafyComplete_of_dvr", "file": "Adic spaces/FJP/Over/SheafyEndpoints.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "FiniteJetOver.finiteJet_not_stablyUniform_of_dvr", "file": "Adic spaces/FJP/Over/SheafyEndpoints.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "WeightedParity.weightedParity_isSheafyComplete_of_dvr", "file": "Adic spaces/WP/Main.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "WeightedParity.weightedParity_not_stablyUniform_of_dvr", "file": "Adic spaces/WP/Main.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "(1) SCOPE OF THE FJP BASE. The paper states its main theorem over a complete discretely valued nonarchimedean field k. The certified statements (`FiniteJetMain.lean`) instantiate k at the restricted Laurent-series field over an arbitrary coefficient field F — the base of the revision the Lean docstrings cite; the `FiniteJetOver` layer additionally states the uniform / domain / nonnoetherian / not-stably-uniform endpoints over an abstract such k (any complete ultrametric nontrivially-normed field, with the uniformizer explicit or chosen from a DVR hypothesis). (2) THE WP BASE. Theorem 8.1 is certified over an abstract complete ultrametric nontrivially-normed field K with `IsDiscreteValuationRing 𝒪[K]` — the paper's \"complete discretely valued nonarchimedean field\", with \"discretely valued\" rendered as the DVR property of the valuation ring. (3) STRONGLY SHEAFY. The paper's \"(strongly) sheafy\" (every 𝒜⟨V₁,…,Vₛ⟩ is sheafy) is rendered by the shifted-weight algebras (`shiftWeight`): the certificate pins sheafiness of `WPA K (shiftWeight id s)` for every s, and sheafiness itself is \"every valid ring of integral elements gives a sheaf\" (`IsSheafyComplete`, the complete-ring form of Wedho", "alignment": false, "original": false, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [], "notes": "No human peer review, and no external reviewer. The review recorded here is automated: distinct AI passes examined the development after it was written, separately from the agent that produced it. Independently of review, the certified statements are checked mechanically: leanprover/comparator pins each statement in a challenge module that cannot see its own proof, replays the proof through the Lean kernel, and confirms the axiom set is within {propext, Quot.sound, Classical.choice}. 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Complex analytic spaces are defined as locally ringed spaces locally cut out in open subsets of ℂⁿ, and their structure sheaves are proved coherent (Oka). Schemes locally of finite type over ℂ are analytified, and for a proper scheme X over ℂ the three GAGA theorems are proved: the comparison maps Hᵠ(X, F) → Hᵠ(X^an, F^an) are bijective for coherent F, analytification is fully faithful on coherent sheaves, and every coherent analytic sheaf on X^an is algebraic; equivalently analytification is an equivalence Coh(X)", "authors": [ "Claude (Anthropic)", "Yuichiro Hoshi", "Junnosuke Koizumi", "Christian Merten" ], "maintainers": [ "Christian Merten" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Lecture notes on complex analysis in several variables", "id": "", "authors": [ "Eberhard Freitag" ], "type": "lecture notes", "relationship": "adapts", "endorsement": "not-contacted" }, { "title": "Géométrie algébrique et géométrie analytique", "id": "doi:10.5802/aif.59", "authors": [ "Jean-Pierre Serre" ], "type": "article", "relationship": "adapts", "endorsement": "n/a" }, { "title": "Revêtements étales et groupe fondamental (SGA 1), Exposé XII", "id": "arXiv:math/0206203", "authors": [ "Alexander Grothendieck", "Michèle Raynaud" ], "type": "book", "relationship": "adapts", "endorsement": "n/a" } ], "related": [], "classification": { "arxiv": [ "math.CV", "math.AG" ], "msc2020": [ "32C35", "32J25", "32B05", "14F06" ] }, "automation": { "methods": [ { "method": "manual", "models": [], "framework": "" }, { "method": "agent", "models": [ "claude-fable-5", "claude-opus-5", "claude-opus-5-5" ], "framework": "Claude Code" } ], "strongest": "agent", "models": [ "claude-fable-5", "claude-opus-5", "claude-opus-5-5" ], "spend": "", "notes": "Everything except the statement and proof of Oka's coherence lemma was formalised by Claude." }, "scope": "Oka's coherence theorem for arbitrary complex analytic spaces, and Serre's GAGA (all three comparison theorems and the equivalence of categories of coherent sheaves) for proper schemes over ℂ, proved from Lean's three standard axioms. Partial results towards the Riemann existence theorem (analytification of finite étale morphisms) are included.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "oka", "file": "Oka/Statement.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "ComplexAnalytic.AnalyticSpace.isCoherentStructureSheaf", "file": "Oka/AnalyticSpace/Coherent.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "ComplexAnalytic.gagaFull_proper", "file": "Oka/Analytification/GAGA/Proper/Equivalence.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "ComplexAnalytic.gagaEquivalenceOfIsProperℂ", "file": "Oka/Analytification/GAGA/Proper/Equivalence.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "ComplexAnalytic.gagaFull", "file": "Oka/Analytification/GAGA/ProjectiveGAGA3.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "LocalOkaRing.mem_radical_span_of_eventually", "file": "Oka/Nullstellensatz/Germ.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "AlgebraicGeometry.exists_chow_of_irreducibleSpace", "file": "Oka/AlgebraicGeometry/Chow.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] } ], "comparator": false, "literature_dependencies": 0, "divergences": "Analytic spaces are locally ringed spaces with a ℂ-algebra structure, locally isomorphic to subspaces cut out by finitely many holomorphic functions in open subsets of ℂⁿ; no separation axiom is imposed. Coherence is formulated for sheaves of modules on a site. For GAGA on proper schemes the proof differs from SGA 1 XII in two places: the relative comparison for the Chow morphism is proved directly over boxes using Cartan's Theorem B for 𝒪 on product domains and algebraic syzygies (no Grauert direct image theorem), and essential surjectivity avoids the comparison of Ext¹ groups by an Artin–Rees embedding argument combined with GAGA-2 and Rückert's Nullstellensatz. The Nullstellensatz is proved for the zero locus of a finite generating family of an ideal.", "alignment": true, "original": false, "review": { "status": "unchecked", "bucket": "unchecked", "reviewers": [], "notes": "All results are machine-checked by the Lean kernel; the main theorems depend only on propext, Classical.choice and Quot.sound. The formal statements have not been independently reviewed by a human beyond the hand-written statement of Oka's coherence lemma." }, "canonical": { "repo": "chrisflav/oka", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "ckkogler/odlyzko-poonen-lean/formalization.yaml", "repo": "ckkogler/odlyzko-poonen-lean", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/ckkogler/odlyzko-poonen-lean/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "The Odlyzko–Poonen conjecture", "description": "Irreducibility and sharp reducibility asymptotics for uniformly random monic integer polynomials with constant coefficient one and intermediate coefficients zero or one. Rational irreducibility has probability tending to one; reducibility equals the minus-one root probability, and sqrt(2/(pi*n)), each up to O(1/n). The development also proves the exact modulo-four autocorrelation companion bound, independent factor-pair estimate, reciprocal-factor bounds, uniform noncyclotomic fixed-factor estimate, cyclotomic bound and exact binomial identities needed for these conclusions. Signed difference ", "authors": [ "Constantin Kogler" ], "maintainers": [ "Constantin Kogler" ], "license": "0BSD", "role": "", "substantive": "", "sources": [ { "title": "The Odlyzko–Poonen conjecture on irreducibility of random polynomials", "id": "https://arxiv.org/abs/2609.26771", "authors": [ "Constantin Kogler" ], "type": "paper", "relationship": "formalizes", "endorsement": "" }, { "title": "Zeros of polynomials with 0,1 coefficients", "id": "https://www.cecm.sfu.ca/organics/papers/odlyzko/paper/html/paper.html", "authors": [ "A. M. Odlyzko", "B. Poonen" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Irreducibility of random polynomials: general measures", "id": "https://arxiv.org/abs/2007.14567v3", "authors": [ "Lior Bary-Soroker", "Dimitris Koukoulopoulos", "Gady Kozma" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Irreducibility of random polynomials of large degree", "id": "https://arxiv.org/abs/1810.13360v2", "authors": [ "Emmanuel Breuillard", "Péter P. Varjú" ], "type": "paper", "relationship": "independently-proves", "endorsement": "" }, { "title": "An effective lower bound for the height of algebraic numbers", "id": "https://arxiv.org/abs/1211.3110", "authors": [ "Paul Voutier" ], "type": "paper", "relationship": "background", "endorsement": "" } ], "related": [ { "id": "lp-densities-lean, commit 1ede5c88a2f7fcbb43907447f2363db89fbde0a2", "relationship": "builds-on", "note": "One contained polynomial-root-power proof module and reusable verification scripts were copied from the author's earlier local development under 0BSD. Exact source paths, revision and hashes are in provenance/. No public URL is asserted and no sibling project is a build dependency." } ], "classification": { "arxiv": [ "math.NT", "math.PR" ], "msc2020": [ "11C08", "60C05" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "GPT-6 (Codex)" ], "framework": "Codex" } ], "strongest": "agent", "models": [ "GPT-6 (Codex)" ], "spend": "", "notes": "Constantin Kogler supplied project direction and the mathematical source. Codex authored and revised proofs and inspected their statements and axiom lists. Model identifiers across earlier context segments were not exhaustively recorded. Costs and total model-inference time were not reliably aggregated." }, "scope": "The proved development includes irreducibility, sharp reducibility estimates, cyclotomic factorization and degree bounds, a strengthened reciprocal-divisor estimate, signed difference-multiset counts with exponential error, an explicit three-term expansion, and periodic half-power expansions to every fixed order. All selected mathematical targets are proved; the phase-retrieval application is outside the selected scope. Challenge states three main theorems and three preliminary model lemmas. The complete default build, all statement and axiom checks, strict comparison of six claims, NanoDa and", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The explicit three-term and arbitrary-order periodic half-power expansions are proved and included in the completed local verification. Proved statements preserve their mathematical hypotheses and constants. The phase-retrieval application is outside the selected scope; polynomial autocorrelation and signed difference-multiset consequences are included. The parameter m of binaryProbability counts internal coefficients, so degree n uses m=n-1, with a proved uniform-counting equivalence. The value at n=0 is a total extension irrelevant to the asymptotic statements. An algebraic proof of the factor constants replaces the positivity argument; that positivity observation is separately proved. The arithmetic inputs are proved through controlled separating primes, confluent determinants and sparse coefficients. The strengthened reciprocal-divisor proof averages selected-bit weights exactly and uses a prime interval with absolute upper factor sixteen; these alternatives preserve the stated final probability bounds. Separate concentration results use the interval from T to 2T and the actual word law. No general Breuillard–Varju theorem, sharper external Mahler bound, historical result menti", "alignment": true, "original": true, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [], "notes": "The developing agent inspected definitions, finite-model indexing, quantifier order, constants, floors, real exponents and uniformity of asymptotic errors. The comparison now selects three main theorems and three preliminary model lemmas. Their actual full types and explicit definitions were reviewed, and the strict Comparator and both required kernel replays accepted all six. The fresh authored checkout reproduced the full statement output. Current evidence identifies exact checked sources; historical receipts retain their own scope. Recorded GitHub verification covers the compiler-portability update and the same mathematical sources. There is no separate human review or registry acceptance." }, "canonical": { "repo": "ckkogler/odlyzko-poonen-lean", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "consensusframework/yang-mills-mass-gap/formalization.yaml", "repo": "consensusframework/yang-mills-mass-gap", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/consensusframework/yang-mills-mass-gap/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Finite-Volume Lattice Gauge Theory in Lean 4 — A Formalization Program Around the Yang–Mills Mass Gap", "description": "Exploratory, multi-phase Lean 4 / Mathlib formalization of finite-volume lattice gauge theory. The active library is Phase 3, namespace `LatticeGauge` (Phase3/LatticeGauge): Wilson action, Gibbs measure, gauge invariance of the Gibbs expectation, Haar probability measure on U(n), translation invariance, the β = 0 product state and its factorization, and a polymer / cluster expansion of Kotecký–Preiss type in the small-β (strong-coupling, Wilson convention) regime. Principal results: Version 49 — finite-volume cluster-expansion identity for the log partition function; Version 50 — finite-volume", "authors": [ "Jucelha Carvalho (Smart Tour Brasil; ORCID 0009-0004-6047-2306)", "GPT-5.6 \"Sol\" (AI model)", "GPT Astra (AI model)", "Claude Fable 5 (AI model)", "Claude Fable 5.1 (AI model)", "Kimi 3 (AI model)", "Manus AI 1.6 (AI model)", "Codex v2 (AI model)", "Claude Opus 4.5 (AI model)", "Claude Opus 4.6 (AI model)", "Claude Opus 4.7 (AI model)", "Claude Opus 5 (AI model)", "GPT-5.2 (AI model)", "Gemini 3 Pro (AI model)", "Grok 4.5 (AI model)", "Grok 4.6 (AI model)" ], "maintainers": [ "Jucelha Carvalho (Smart Tour Brasil; ORCID 0009-0004-6047-2306)" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Finite-Volume Lattice Gauge Theory in Lean 4 — the complete project through Version 52 (verified Phase 3 library LatticeGauge; Phases 1–2 archived at Version 51)", "id": "https://doi.org/10.5281/zenodo.17397622", "authors": [ "Jucelha Carvalho (Smart Tour Brasil; ORCID 0009-0004-6047-2306)", "GPT-5.6 \"Sol\" (AI model)", "GPT Astra (AI model)", "Claude Fable 5 (AI model)", "Claude Fable 5.1 (AI model)", "Kimi 3 (AI model)", "Manus AI 1.6 (AI model)", "Codex v2 (AI model)", "Claude Opus 4.5 (AI model)", "Claude Opus 4.6 (AI model)", "Claude Opus 4.7 (AI model)", "Claude Opus 5 (AI model)", "GPT-5.2 (AI model)", "Gemini 3 Pro (AI model)", "Grok 4.5 (AI model)", "Grok 4.6 (AI model)" ], "type": "formalization", "relationship": "other", "endorsement": "participated" }, { "title": "Confinement of quarks", "id": "https://doi.org/10.1103/PhysRevD.10.2445", "authors": [ "Kenneth G. Wilson" ], "type": "article", "relationship": "background", "endorsement": "n/a" }, { "title": "Cluster expansion for abstract polymer models", "id": "https://doi.org/10.1007/BF01211762", "authors": [ "Roman Kotecký", "David Preiss" ], "type": "article", "relationship": "adapts", "endorsement": "not-contacted" }, { "title": "Statistical Mechanics of Lattice Systems: A Concrete Mathematical Introduction", "id": "ISBN 978-1-107-18482-4", "authors": [ "Sacha Friedli", "Yvan Velenik" ], "type": "book", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Quantum Yang–Mills Theory (Clay Mathematics Institute Millennium Problem description)", "id": "https://www.claymath.org/millennium/yang-mills-the-maths-gap/", "authors": [ "Arthur Jaffe", "Edward Witten" ], "type": "web post", "relationship": "background", "endorsement": "n/a" } ], "related": [], "classification": { "arxiv": [ "math-ph", "hep-lat" ], "msc2020": [ "81T13", "81T25", "82B20", "03B35" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "GPT-5.6 \"Sol\"" ], "framework": "ChatGPT" }, { "method": "agent", "models": [ "GPT Astra" ], "framework": "ChatGPT" }, { "method": "agent", "models": [ "Claude Fable 5", "Claude Fable 5.1" ], "framework": "Claude (Cowork / Claude Code)" }, { "method": "agent", "models": [ "Codex v2" ], "framework": "Codex" }, { "method": "agent", "models": [ "Kimi 3" ], "framework": "" }, { "method": "agent", "models": [ "Manus AI 1.6" ], "framework": "" }, { "method": "agent", "models": [ "Claude Opus 4.5", "Claude Opus 4.6", "Claude Opus 4.7", "Claude Opus 5", "GPT-5.2", "Gemini 3 Pro", "Grok 4.5", "Grok 4.6" ], "framework": "" }, { "method": "manual", "models": [], "framework": "" } ], "strongest": "agent", "models": [ "Claude Fable 5", "Claude Fable 5.1", "Claude Opus 4.5", "Claude Opus 4.6", "Claude Opus 4.7", "Claude Opus 5", "Codex v2", "GPT Astra", "GPT-5.2", "GPT-5.6 \"Sol\"", "Gemini 3 Pro", "Grok 4.5", "Grok 4.6", "Kimi 3", "Manus AI 1.6" ], "spend": "subscription-based usage across several model providers; not itemised", "notes": "Human–AI collaborative development coordinated by the human coordinator under the Consensus Framework, a multi-agent coordination method in which distinct AI models take separate roles (architecture and specification, implementation, reproduction, adversarial review, custody audit) and the human coordinator holds custody, integration and release; products and interfaces named under `framework` are technical environments only, never authorship or affiliation. For Versions 49–51, every accepted gate went through: written specification → Lean implementation → verifiable git bundle → reproduction " }, "scope": "Verified scope (Phase 3, Phase3/LatticeGauge): finite periodic 4-dimensional lattice, compact gauge group with a probability base measure and a bounded measurable character χ with |χ| ≤ 1; Gibbs measure and expectation; gauge and translation invariance; Haar probability on U(n) with unconditional capstones |⟨W⟩| ≤ 1; β = 0 factorization; polymer gas and Kotecký–Preiss type cluster expansion with explicit constants in the small-β (strong-coupling) regime; exponential covariance decay (Version 50); exponential stability under remote polymer-activity restriction (Version 51); exponential stabilit", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "LatticeGauge.abs_profileExpectation_sub_profileExpectation_le_local_exp_decay", "file": "Phase3/LatticeGauge/ActivityProfileDampingStability.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "LatticeGauge.abs_activityDampedExpectation_sub_activityDampedExpectation_le_local_exp_decay_refined", "file": "Phase3/LatticeGauge/ActivityDampingLipschitzRefined.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "LatticeGauge.abs_activityDampedExpectation_sub_activityDampedExpectation_le_local_exp_decay", "file": "Phase3/LatticeGauge/ActivityDampingLipschitz.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "LatticeGauge.abs_gibbsExpectation_sub_activityDampedExpectation_le_local_exp_decay", "file": "Phase3/LatticeGauge/ActivityDampingStability.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "LatticeGauge.abs_gibbsExpectation_sub_activityRestrictedExpectation_le_local_exp_decay", "file": "Phase3/LatticeGauge/ActivityRestrictionStability.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "LatticeGauge.abs_gibbsCovariance_le_local_exp_decay", "file": "Phase3/LatticeGauge/CovarianceDecay.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "LatticeGauge.logPartition_eq_tsum_unrooted", "file": "Phase3/LatticeGauge/KPLogPartition.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] } ], "comparator": false, "literature_dependencies": 0, "divergences": "The Lean statements were developed within the project (see the project source entry), so there is no single source text to diverge from. Relative to the informal cluster-expansion literature: everything is finite volume; the polymer gas, Ursell coefficients, Penrose tree bound and Kotecký–Preiss induction are formalized in a finite, explicit-constant form (2/113 per link, exponential rate 1/2, β ≤ 1/40000) rather than in the general abstract-polymer-model generality; \"distance\" is walk separation in the plaquette graph, not Euclidean distance; activityRestrictedExpectation is a normalized polymer functional obtained by suppressing activities, not a second Gibbs measure or a boundary condition. Earlier project texts used the phrase \"small coupling\"; in Wilson's convention small β is strong coupling, and the wording was corrected in the repository from commit 624af23a on.", "alignment": true, "original": false, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [ "GPT-5.6 \"Sol\" (AI model) — architect's review of every module", "Kimi 3 (AI model) — external adversarial mathematical review (Stone 51); final adversarial mathematical and code review of the integrated Stone 52 at 00600e03e36f5fdfdfda1983a17929c3efa6b44f, with errata C1 and addendum C2, reading only, no Lean execution by the reviewer; final adversarial review of the Stone 53 candidate abad16674ab9b713c7ef9d0344c8f5cc02b09739 with errata C1, reading only, no Lean execution, not blind; adversarial review of the Stone 54 scientific merge 9c0f6fdecee5c8628a2434d032e421edc78bb722, reading and mathematical analysis, no Lean execution, not blind; adversarial review of the Stone 55 candidate 0c50b6d5e23915f13d6f36d560a58ba6f24ae97b and scientific merge d6ce3d7232f08b5f975dfa8c6f43ec5031937e10 with errata C1 on the attribution of the tests, reading and mathematical analysis, no Lean execution, not blind", "Codex v2 (AI model) — custody, reproduction and reading audit (Stone 51), independent from the Lean implementer but not independent from the architect's model family; not a human peer review", "Manus AI 1.6 (AI model) — reproduction of gate 51-A", "GPT Astra (AI model) — review of the Stone 52 integration and custody checks on GitHub; architecture and review of the documentary consolidation; scientific architecture of Stones 53, 54 and 55 and review of their sources, Git objects, evidence and publication metadata, without a Lean build of its own", "Claude Fable 5.1 (AI model), separate auditing instance — seven stage audits of Stone 52 (gates A0, A, A/C1, B, C, D, E) on its own pinned bench; same model as the implementer, prior exposure declared from 52-B on, explicitly not blind for 52-E; reports and evidence in docs/audits/stone52/; for Stone 53, one reproduction by execution of the candidate abad16674ab9b713c7ef9d0344c8f5cc02b09739 on its own bench (113 modules rebuilt, own certificates and tests), same model as the constructor, not blind; report and evidence in docs/audits/stone53/; for Stone 54, one reproduction by execution of the candidate 410bde8146e8f14b698a03387da395a79f81f9eb (114 modules rebuilt, 0 replayed, own certificates and tests), same model, not blind; report and evidence in docs/audits/stone54/; for Stone 55, one reproduction by execution of the candidate 0c50b6d5e23915f13d6f36d560a58ba6f24ae97b (116 modules rebuilt from the committed manifest, 0 replayed, 68 own certificates, own tests W0–W5x), same model, not blind; report and evidence in docs/audits/stone55/" ], "notes": "Machine verification: Lean 4 kernel and GitHub Actions CI, green on main for the integrated Versions 49–53 (Version 52 commit f4015c5c…, run 482; Version 53 commit 571b83aa…, run 486), for Stone 54 (merge commit 9c0f6fde…, run 488), for the Version 54 commit 9358faa2… (run 490) and for Stone 55 (merge commit d6ce3d72…, run 494, id 35263497257). Human review: coordinator's process audit. No peer review by human mathematicians has taken place; public discussions of the project in the Lean community produced comments acknowledged in README.md, without endorsement of the results." }, "canonical": { "repo": "consensusframework/yang-mills-mass-gap", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "CoolRmal/Besicovitchs-1-2/formalization.yaml", "repo": "CoolRmal/Besicovitchs-1-2", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/CoolRmal/Besicovitchs-1-2/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "A machine-checked bound of 0.6934 for Besicovitch's 1/2-problem", "description": "Besicovitch's 1/2-conjecture asserts that a Borel set in the plane with finite one-dimensional Hausdorff measure whose lower density exceeds 1/2 almost everywhere is countably 1-rectifiable. Writing sigma_1(R^2) for the least threshold that forces countable 1-rectifiability, the conjecture is sigma_1(R^2) = 1/2. The previously published record is sigma_1(R^2) <= 7/10, obtained by De Lellis, Glaudo, Massaccesi and Vittone. This development proves, with a complete Lean 4 proof term, sigma_1(H) <= 6934/10000 = 0.6934 for every real Hilbert space H, without finite-dimensionality or separability as", "authors": [ "Yongxi Lin" ], "maintainers": [ "Yongxi Lin" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "The bound sigma_1(R^2) <= 6934/10000 for Besicovitch's 1/2-problem, via thirty rational Gram certificates for the two-colour six-centre problem", "id": "", "authors": [], "type": "original-proof", "relationship": "other", "endorsement": "" }, { "title": "Besicovitch's 1/2 problem and linear programming", "id": "arXiv:2404.17536", "authors": [ "Camillo De Lellis", "Federico Glaudo", "Annalisa Massaccesi", "Davide Vittone" ], "type": "article", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Besicovitch's example in higher dimensions: a purely unrectifiable set with large lower density", "id": "arXiv:2607.05206", "authors": [ "Jaume Capdevila" ], "type": "article", "relationship": "other", "endorsement": "not-contacted" }, { "title": "Study of extreme cases with respect to the densities of irregular linearly measurable plane sets of points", "id": "Math. Ann. 116 (1939), 358-373", "authors": [ "D. B. Dickinson" ], "type": "article", "relationship": "other", "endorsement": "n/a" }, { "title": "On Besicovitch's 1/2-problem", "id": "J. London Math. Soc. (2) 45 (1992), 279-287", "authors": [ "David Preiss", "Jaroslav Tiser" ], "type": "article", "relationship": "background", "endorsement": "not-contacted" }, { "title": "On the fundamental geometrical properties of linearly measurable plane sets of points, parts I and II", "id": "Math. Ann. 98 (1928), 422-464; Math. Ann. 115 (1938), 296-329", "authors": [ "A. S. Besicovitch" ], "type": "article", "relationship": "background", "endorsement": "n/a" } ], "related": [], "classification": { "arxiv": [ "math.CA", "math.MG" ], "msc2020": [ "28A75", "28A78", "49Q15", "68V20", "90C05" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "OpenAI ChatGPT project sessions; exact model versions not recorded locally" ], "framework": "ChatGPT project workspace with local file mirror" }, { "method": "agent", "models": [ "claude-opus-5", "claude-fable-5-1" ], "framework": "Claude Code" }, { "method": "manual", "models": [], "framework": "n/a" } ], "strongest": "agent", "models": [ "OpenAI ChatGPT project sessions; exact model versions not recorded locally", "claude-fable-5-1", "claude-opus-5" ], "spend": "subscription-based; not separately tracked", "notes": "Two stages with a clean handover. The mathematics was produced first in a ChatGPT project and delivered as a written argument together with thirty exact rational certificates and two independent verifiers. The Lean formalization was then produced by an agent in Claude Code. Before any formalization the certificate data was re-derived from the definitions in exact rational arithmetic by a third independent transcription, which reproduced all thirty published upper bounds exactly; that agreement also guards the transcription into Lean, since one mistyped integer changes the computed bound. All f" }, "scope": "Formalized in full: sigma_1(E) <= 6934/10000 for every real Hilbert space E, without a separability assumption, in particular for every R^n, stated as Besicovitch.sigma_one_le_6934_div_10000; its planar case is Besicovitch.sigma_one_plane_le_6934_div_10000 over EuclideanSpace R (Fin 2). sigma_1 is defined inside Challenge.lean as the infimum of the thresholds that force countable 1-rectifiability. The proof term is complete and uses only propext, Classical.choice and Quot.sound. The supporting chain is formalized too: the geometric measure theory layer, the Besicovitch pair condition, the six-", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "None from a cited source, since no source result is being formalized: the background references supply framework and history only, and the bound proved here is stated for the first time by this project. Two qualifications a reader should carry, besides the standing one that sigma_1 is an infimum whose attainment is not claimed. First, the theorem is weaker than the optimum of the finite problem it solves, by about 9.36e-5; the reason is recorded under scope. Second, the thirty certificates were discovered by an unverified numerical search. That search is not part of the proof: Lean recomputes each certificate's exact rational upper bound from the stored integers and rechecks every sign condition, and the positive semidefiniteness on which the argument rests is proved identically from a rank-one decomposition, with no eigenvalue computation or numerical test anywhere.", "alignment": true, "original": true, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "none" ], "notes": "No external mathematical review. Checks performed by the authors. The statement was audited by computing the transitive definition closure of the submitted theorem and confirming it is identical between Challenge.lean and Solution.lean: five constants, sigmaOne, ForcesOneRectifiability, lowerOneDensity, IsCountablyOneRectifiable and one proof term. Axioms of the submitted declaration were printed and are exactly propext, Classical.choice and Quot.sound. The repository contains no native_decide, no Lean.ofReduceBool, no admit and no axiom declaration. The certificate data was verified three times independently before formalization, as described under automation. Literature search (performed 2026-09-02) Scope and method. The search asks two questions. (a) Does any published or publicly avail" }, "canonical": { "repo": "coolrmal/besicovitchs-1-2", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-02-000011", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-02-000011", "commit": "8b0e98aa712374c96da7e619624f5b36ed8d5fd3", "trust": "high", "theorems": 5, "date": "2026-09-20" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "CoolRmal/centered-maximal-constant/formalization.yaml", "repo": "CoolRmal/centered-maximal-constant", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/CoolRmal/centered-maximal-constant/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Bounds 1.6855 and 3.879 for the planar centred Hardy-Littlewood maximal constant over squares", "description": "Let c_d be the weak type (1,1) constant of the centred Hardy-Littlewood maximal operator over axis-parallel cubes in R^d: the least C such that alpha |{Mf > alpha}| <= C ||f||_1 for every integrable f and every level alpha. Its value is unknown for every d >= 2; in the plane the previously published lower bound is c_2 >= 3/4 - sqrt(2)/4 + sqrt(6)/2 = 1.62119... (Aldaz 2000) and the best known upper bound is 4. This development proves c_2 >= Phi = 1.68550999335552518..., an explicit algebraic number of degree 16, from a lattice with unequal masses: columns at spacing h = (5 + sqrt 22)/6 carryin", "authors": [ "Yongxi Lin" ], "maintainers": [ "Yongxi Lin" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "The lower bound c_2 >= 1.68550999... for the weak type (1,1) constant of the centred Hardy-Littlewood maximal operator over squares, from a lattice with alternating masses 1 and (17 + 4 sqrt 22)/9", "id": "", "authors": [], "type": "original-proof", "relationship": "other", "endorsement": "" }, { "title": "Kernel comparison for the centred maximal operator: an unpublished manuscript proving c_2 < 3.615749", "id": "", "authors": [], "type": "other", "relationship": "other", "endorsement": "not-contacted" }, { "title": "A remark on the centered n-dimensional Hardy-Littlewood maximal function", "id": "Czechoslovak Math. J. 50 (2000), no. 1, 103-112", "authors": [ "J. M. Aldaz" ], "type": "paper", "relationship": "other", "endorsement": "not-contacted" }, { "title": "Improving the planar centered maximal constant (research brief)", "id": "", "authors": [], "type": "other", "relationship": "other", "endorsement": "" }, { "title": "245A, Notes 5: Differentiation theorems (Exercise 42)", "id": "", "authors": [ "Terence Tao" ], "type": "web discussion", "relationship": "other", "endorsement": "not-contacted" }, { "title": "The best constant for the centered Hardy-Littlewood maximal inequality", "id": "Ann. of Math. (2) 157 (2003), 647-688", "authors": [ "A. D. Melas" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Weak type (1, 1) inequalities of maximal convolution operators", "id": "Rend. Circ. Mat. Palermo (2) 41 (1992), 342-352, doi:10.1007/BF02848939", "authors": [ "M. Trinidad Menarguez", "F. Soria" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Some remarks on dimension-free estimates for the discrete Hardy-Littlewood maximal functions", "id": "arXiv:2010.07379", "authors": [ "D. Kosz", "M. Mirek", "P. Plewa", "B. Wrobel" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Centered Hardy-Littlewood maximal constant in dimension 2 (constant 47a)", "id": "", "authors": [], "type": "other", "relationship": "background", "endorsement": "" } ], "related": [ { "id": "https://github.com/fpvandoorn/carleson", "relationship": "independent", "note": "Formalizes an uncentred maximal function on doubling metric measure spaces with a non-sharp weak type bound. Not used here." }, { "id": "https://github.com/TauCetiProject/TauCeti", "relationship": "independent", "note": "Contains a centred maximal function over balls with the weak type bound 4^n. Not used here; the challenge imports only Mathlib." } ], "classification": { "arxiv": [ "math.CA", "math.FA" ], "msc2020": [ "42B25", "26D15", "35R11" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "claude-fable-5-1" ], "framework": "Claude Code" }, { "method": "agent", "models": [ "claude-opus-5" ], "framework": "Claude Code with the mathlib-quality plugin" }, { "method": "agent", "models": [ "claude-opus-5" ], "framework": "Claude Code" }, { "method": "manual", "models": [], "framework": "n/a" } ], "strongest": "agent", "models": [ "claude-fable-5-1", "claude-opus-5" ], "spend": "subscription-based; not separately tracked", "notes": "The mathematics and the Lean formalization were produced by AI agents under the maintainer's direction, in two stages with an explicit handover summarised in docs/HISTORY.md. No AI system is listed as an author." }, "scope": "Formalized in full, using only propext, Classical.choice and Quot.sound: c_2 >= Phi (CenteredMaximal.ofReal_phi_le_weakTypeConstant_two), c_d <= 2^d for every d including d = 0 (CenteredMaximal.weakTypeConstant_le_two_pow), 1.685 < Phi < 1.686 (CenteredMaximal.lt_phi, CenteredMaximal.phi_lt), and c_2 <= 3.879 (CenteredMaximal.weakTypeConstant_two_le_upper), which is below the classical 4. The definitions live in Challenge.lean with Mathlib imports only; cubes are Metric.closedBall in Fin d -> R with its sup norm, the maximal function takes values in [0, infinity], and level sets are measured w", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "No cited source result is formalized for the principal theorem, which is stated here for the first time. The upper bound 3.879 DIVERGES from its source manuscript: the manuscript proves 3.615749 with a fractional generator of order 6/5 and a 1201-coefficient spline kernel, whereas the formalization carries the same comparison method through the elementary case alpha = 1 with a two-parameter kernel, giving a weaker constant; the manuscript's existence proof for the obstacle problem, which uses a fractional Sobolev inequality, is replaced by Gagliardo averaging inequalities. The upper bound follows Tao's Exercise 42 except that Mathlib's Vitali lemma needs an enlargement tau > 1, so the proof gives (1 + tau)^d for each tau > 1 and passes to the limit. The smearing of point masses follows Aldaz's Lemma 1.1, adapted to atoms of unequal mass.", "alignment": true, "original": true, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [ "six independent AI referee agents (Claude Code), for the exploratory certificate; no human review" ], "notes": "No external human review. The exploratory certificate was attacked by six independent referee agents and all concluded that the claim holds. The Lean statement was checked by reading Challenge.lean against the informal definitions; the compared theorems' axioms were printed and are exactly propext, Classical.choice and Quot.sound." }, "canonical": { "repo": "coolrmal/centered-maximal-constant", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-19-000002", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-19-000002", "commit": "8e1ee3de55681cd852cf9db16384484036273a4d", "trust": "high", "theorems": 5, "date": "2026-09-20" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "CoolRmal/FavardLength/formalization.yaml", "repo": "CoolRmal/FavardLength", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/CoolRmal/FavardLength/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "The Favard-length decay exponent of the four-corner Cantor set is at least 1/4", "description": "For the four-corner Cantor set approximants K_n = C_n × C_n, where C_n is the union of the 2^n intervals of length 4^{-n} with left endpoints sum_{j<=n} w_j 4^{-j}, w_j in {0,3}, let α_Fav be the supremum of the a >= 0 for which the Favard length Fav(K_n) = (1/π) ∫_0^π |π_θ(K_n)| dθ is at most C n^{-a} for all n >= 1. This project proves in Lean that every a < 1/4 is admissible, so 1/4 <= α_Fav, together with the classical bound α_Fav <= 1. The previous best lower bounds were 1/6 (Nazarov-Peres-Volberg, 2010) and 1/5 (Marshall, arXiv:2509.02882 v2, 2026, whose v1 claim of 1/4 was withdrawn). T", "authors": [ "Yongxi Lin" ], "maintainers": [ "Yongxi Lin" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Fav(K_n) <= C_a n^{-a} for every a < 1/4 for the four-corner Cantor set, hence 1/4 <= α_Fav, via a joint negative-moment estimate for the low-frequency product", "id": "", "authors": [], "type": "original-proof", "relationship": "other", "endorsement": "" }, { "title": "The power law for the Buffon needle probability of the four-corner Cantor set", "id": "arXiv:0801.2942", "authors": [ "Fedor Nazarov", "Yuval Peres", "Alexander Volberg" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Improved Power Laws for the Favard Length Problem in All Dimensions", "id": "arXiv:2509.02882", "authors": [ "Caleb Marshall" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "An estimate from below for the Buffon needle probability of the four-corner Cantor set", "id": "arXiv:0807.2953", "authors": [ "Michael Bateman", "Alexander Volberg" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "How likely is Buffon's needle to fall near a planar Cantor set?", "id": "doi:10.2140/pjm.2002.204.473", "authors": [ "Yuval Peres", "Boris Solomyak" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Optimization constants: the Favard-length decay exponent of the four-corner Cantor set (constant 60a)", "id": "https://teorth.github.io/optimizationproblems/constants/60a.html", "authors": [], "type": "web discussion", "relationship": "background", "endorsement": "not-contacted" } ], "related": [], "classification": { "arxiv": [ "math.CA", "math.MG" ], "msc2020": [ "28A80", "28A75", "42A38" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "OpenAI ChatGPT project sessions; exact model versions not recorded locally" ], "framework": "ChatGPT project workspace with a local file mirror" }, { "method": "agent", "models": [ "claude-opus-5-5" ], "framework": "Claude Code (desktop app), with its Workflow multi-agent orchestration" }, { "method": "manual", "models": [], "framework": "n/a" } ], "strongest": "agent", "models": [ "OpenAI ChatGPT project sessions; exact model versions not recorded locally", "claude-opus-5-5" ], "spend": "subscription-based; not separately tracked", "notes": "Two stages. First, the mathematics was produced in a ChatGPT project as a written argument with internal reviews and a formalization plan. Second, Claude Code agents produced the Lean formalization, splitting it into independently checked parts joined by frozen Prop contracts. The Lean kernel confirms the stated theorems, and so does the toolchain's lake comparator, run in CI under bubblewrap with the bundled NanoDa and con-ron kernels. The figures above come from the workflow runtime's usage reports and git timestamps." }, "scope": "Formalized in full, with only the axioms propext, Classical.choice and Quot.sound. The definitions are literal, in Challenge.lean: the Cantor approximants C_n (digits {0,3}, base 4), K_n = C_n × C_n, the projection x cos θ + y sin θ, Lebesgue measure of the projection, Fav(K_n) as (1/π) times the interval integral over [0, π], the admissible exponents, and α_Fav as their real supremum. Four compared theorems: every a in [0, 1/4) is admissible (favard_le_rpow_of_lt_quarter); 1/4 <= α_Fav (one_quarter_le_decayExponent); every admissible exponent is at most 1 (le_one_of_mem_admissibleExponents); ", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The Lean proof establishes the manuscript's qualitative conclusion, α_Fav >= 1/4, by routes that sometimes differ from the manuscript; docs/proof-account.md lists them. For each target exponent a < 1/4 the moment parameter s is fixed, instead of choosing s by depth. Hence the (2 + log n)^{7/2} bound is not claimed. The exceptional-direction estimate J9 is proved in an integrated form rather than through a Markov-selected subset. The mean-one property of the lacunary cosine products uses an antiperiodicity induction rather than Fourier expansion. The derivative bound in J4 avoids Parseval. The Plancherel step is replaced by the Fourier inversion formula for the triangle function. The elementary lower bound is proved with constant 1/320 rather than 1/(4n + sqrt 2). The Challenge requires n >= 1, where the registry's definition reads \"all n in N\", because n^{-a} is undefined at n = 0. No known divergence affects the compared statements.", "alignment": true, "original": true, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [ "Claude Code audit agents (claude-opus-5-5), run separately from the proving agents" ], "notes": "No human mathematical review of the manuscript or the Lean proof. The Lean kernel and lake comparator (sandboxed, with the NanoDa and con-ron kernels) check the compared theorems. Two independent agent audits reviewed the statement surface. The first checked every Challenge definition for fidelity, coercions and junk values, with compiled sanity checks: C_0, C_1 and depth-2 membership, interval integrability and positivity of the Favard integral, and non-vacuity of the real supremum. The second checked the Comparator setup: kernel-term identity of the Challenge and Solution definitions, import separation, the configuration against the Palomar policy, and the absence of sorry, axioms, native_decide and kernel-bypassing options. Both found the statements faithful. Their one material comment," }, "canonical": { "repo": "coolrmal/favardlength", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-24-000001", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-24-000001", "commit": "65f19d4c36d4e576417da893f54cf351aef9d672", "trust": "high", "theorems": 4, "date": "2026-09-24" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "CoolRmal/NKBesicovitch/formalization.yaml", "repo": "CoolRmal/NKBesicovitch", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/CoolRmal/NKBesicovitch/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "NKBesicovitch", "description": "Lean proofs of Hausdorff dimension at least n-(n-k)/p_c^k for all (n,k)-Besicovitch sets with 1<=k<=n, and positive Lebesgue measure for Lebesgue measurable such sets when p_c^(k-1)+k>n. Here a Besicovitch set contains a translated unit k-disk in every k-dimensional direction, and p_c is the unique root in (2,3) of p^3-2p^2-2p+2=0. Building on Bourgain's arithmetic-combinatorial approach, Nets Hawk Katz and Terence Tao's New bounds for Kakeya problems (2002; preprint 2001) established the Minkowski-dimension bound 1+(n-1)/beta_c, where beta_c=p_c/(p_c-1)=1.67513... satisfies beta_c^3-4*beta_c+", "authors": [ "Yongxi Lin" ], "maintainers": [ "Yongxi Lin" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "New bounds for Kakeya problems", "id": "arXiv:math/0102135", "authors": [ "Nets Hawk Katz", "Terence Tao" ], "type": "article", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Improved bounds for the Kakeya maximal conjecture in higher dimensions", "id": "arXiv:1908.05589", "authors": [ "Jonathan Hickman", "Keith M. Rogers", "Ruixiang Zhang" ], "type": "article", "relationship": "background", "endorsement": "not-contacted" }, { "title": "The projection estimate near 1.675", "id": "", "authors": [], "type": "supplied manuscript", "relationship": "formalizes", "endorsement": "" }, { "title": "From Projection Inequalities to Mixed-Norm Local X-Ray Estimates", "id": "", "authors": [], "type": "supplied proof notes", "relationship": "formalizes", "endorsement": "" }, { "title": "Two bounds for the X-ray transform", "id": "doi:10.1007/s00209-009-0589-5", "authors": [ "Richard Oberlin" ], "type": "article", "relationship": "adapts", "endorsement": "not-contacted" }, { "title": "Bounds for Kakeya-type maximal operators associated with k-planes", "id": "arXiv:math/0512377", "authors": [ "Richard Oberlin" ], "type": "article", "relationship": "adapts", "endorsement": "not-contacted" } ], "related": [], "classification": { "arxiv": [ "math.CA" ], "msc2020": [ "42B25", "28A75" ] }, "automation": { "methods": [ { "method": "autonomous", "models": [ "GPT-6" ], "framework": "OpenAI Codex" } ], "strongest": "autonomous", "models": [ "GPT-6" ], "spend": "not tracked", "notes": "AI-generated Lean development directed by the project owner. Existing supplied research drafts and prose audits are used as references only. No human review of the new Lean statements has been recorded." }, "scope": "Completed Lean proofs, with final external verification pending. The exact critical exponent, strict parameter margins, full-dimensional boundary case, normalized disk averages, canonical Grassmannian probability and its uniqueness, measurability of plate maximal functions on all Borel nonnegative inputs, the exact two-slice projection seed, intrinsic pair-incidence mass, dual-height code algebra, double-projection density, low-density deletion, the pointwise code marginal, pair-fiber projection bound, simultaneous finite density refinement and code-fiber selection, the code-density bound by e", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The statements include the elementary k=n boundary and require 1<=k<=n. The disk property uses closed radius-one disks and has no measurable-center selection hypothesis. Lebesgue measurability uses NullMeasurableSet. Both geometric targets and numerical exponent bounds are proved as stated. Hausdorff dimension is Mathlib's dimH, valued in the extended nonnegative reals; ENNReal.ofReal embeds the stated real lower bound. Its set has no measurability or boundedness hypothesis, and the endpoint follows by continuity from subcritical estimates, not an endpoint operator bound. Helper analytic implications explicitly display their assumptions.", "alignment": true, "original": false, "review": { "status": "self-assessed, incomplete", "bucket": "self-assessed", "reviewers": [], "notes": "Source definitions and the exact exponent were checked against supplied notes and Oberlin's statement. Challenge imports only Mathlib. Comparator is configured to compare definitions, the numerical critical-exponent bounds, and both geometric theorems, with no permission for sorryAx. All targets and the reusable Hausdorff transfer pass the standard-axiom audit. A fresh Comparator macOS development run on 2026-09-09 accepted all three current targets, compared fixed definition bodies, and replayed their exports in Lean. This run had no Linux sandbox and no independent kernel. Full verification remains pending; see verification/hausdorff-development.md. Independent mathematical review and novelty assessment remain pending. The fixed definitions are not listed as editable definition holes." }, "canonical": { "repo": "coolrmal/nkbesicovitch", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-19-000001", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-19-000001", "commit": "e12e5981d840eb5ebdc25403691b3c63a9c7aff8", "trust": "high", "theorems": 3, "date": "2026-09-19" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "corun1024/4ct/formalization.yaml", "repo": "corun1024/4ct", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/corun1024/4ct/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "The Four Colour Theorem in Lean 4", "description": "The Four Colour Theorem, formalized in Lean 4 over Mathlib, following the architecture of Gonthier and Werner's Coq proof. The statement is Gonthier's, clause for clause, over Mathlib's reals: every simple map of the real plane - a partial equivalence relation on points of R^2 whose classes are open and connected - admits a colouring with at most four regions in which adjacent regions differ. The development carries the whole proof: hypermaps, the Euler formula, planarity implying the Jordan curve property, Kempe chain surgery, the reduction from the plane to finite maps by compactness, reduci", "authors": [ "Chris Emery" ], "maintainers": [ "Chris Emery" ], "license": "MIT", "role": "substantive-development", "substantive": "", "sources": [ { "title": "Formal Proof - The Four-Color Theorem", "id": "Notices of the AMS 55 (11), 2008, 1382-1393", "authors": [ "Georges Gonthier" ], "type": "paper", "relationship": "adapts", "endorsement": "" }, { "title": "The four-colour theorem", "id": "J. Combin. Theory Ser. B 70 (1997), 2-44; doi:10.1006/jctb.1997.1750", "authors": [ "Neil Robertson", "Daniel Sanders", "Paul Seymour", "Robin Thomas" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Mathlib", "id": "https://github.com/leanprover-community/mathlib4", "authors": [ "The mathlib community" ], "type": "other", "relationship": "background", "endorsement": "" } ], "related": [ { "id": "https://github.com/rocq-community/fourcolor", "relationship": "adapts", "note": "The development this one follows. Three kinds of data are mechanical translations of it: FourColor/Configurations.lean from configurations.v, the presentation scripts FourColor/Present*.lean from present*.v, and the quiz data computed from the configurations. No Coq source text is copied. Distributed under CeCILL-B, whose article 5.3.2 permits a derived work to carry another licence provided the a" } ], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05C15", "68V20" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Claude Fable 5.1" ], "framework": "Claude Code" }, { "method": "agent", "models": [ "Claude Opus 5" ], "framework": "Claude Code" } ], "strongest": "agent", "models": [ "Claude Fable 5.1", "Claude Opus 5" ], "spend": "Subscription-based rather than metered: under $200 to the maintainer on a Claude Max subscription. Estimated at roughly $5,000 had the same token volume been billed at API list prices; that estimate is not measured.", "notes": "Interactive agent-directed development throughout. The maintainer set the targets and directed the work; the models wrote the Lean." }, "scope": "Complete. FourColor.fourColorTheorem proves FourColor.FourColorTheorem, the general (not merely finite) statement over the real plane. Both computational halves are discharged in the development rather than assumed: reducibility of all 633 configurations, and the seven unavoidability presentations. Every computation is `decide +kernel`; there is no `native_decide`, no `axiom`, no `sorry`, and no `partial`, `unsafe` or `opaque` definition. Two results of the reference are deliberately not ported because nothing uses them - Jordan_planar and Jordan_WalkupE, the converse direction of the planarit", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "FourColor.fourColorTheorem", "file": "FourColor/Complete.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] } ], "comparator": false, "literature_dependencies": 0, "divergences": "The statement is a faithful transcription of the reference's realplane.v and fourcolor.v, definition by definition, with one difference: Coq states the theorem over an arbitrary Real.model, whose points are denotations and whose regions are not required to be extensional, because Coq's logic has no quotient types. Here everything is stated over Mathlib's R, the extensional case. The reference's own header notes that value equality on points is not needed to state or prove the theorem, and realcategorical.v proves all models isomorphic; the reduction between the two forms is not formally verified here. The mathematics is expressed in Mathlib's structures rather than transliterated from MathComp: the four colours are an AddCommGroup (the Klein four-group), a hypermap is three Equiv.Perms with its cycle relations as Equiv.Perm.SameCycle, orbit counts are Nat.card of a quotient, and arities are Function.minimalPeriod. Several proofs are shorter in consequence. The largest divergence is in how reducibility is checked. Coq runs a Kempe closure program inside its kernel on a bytecode virtual machine added for that proof. Lean's kernel has no equivalent and `native_decide` was ruled out, s", "alignment": true, "original": false, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [], "notes": "Five independent adversarial audits were run over the development, by agents, covering the faithfulness of the statement, soundness escapes, the composition of the logical chain, the generated reducibility certificates, and the build and verification machinery. They found no defect in the mathematics. The certificate audit corrupted certificate data and confirmed the checks reject it, including a cross-module test in which an over-claiming chord was rejected by the kernel; the chain audit compared all 633 configurations against the reference's configurations.v with no mismatches. Several real defects were found in the verification harness and fixed: freshness is now decided by content fingerprint rather than file timestamps, and the checker now tests exit codes and requires the theorem to " }, "canonical": { "repo": "corun1024/4ct", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "dcposch/albertson-berman-lean/formalization.yaml", "repo": "dcposch/albertson-berman-lean", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/dcposch/albertson-berman-lean/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "albertson-berman-palomar", "description": "Lean 4 / Mathlib certificate that the Albertson–Berman conjecture is false: Jung’s 31-vertex maximal planar seed T is a spherical triangulation with 87 edges and 58 faces, Euler characteristic 2, and induced-forest number a(T)=15 < 16. Planarity is a finite sphere certificate; a(T)≤15 is Jung’s selected-edge transfer on two icosahedral gadgets in a pentagonal bipyramid, not a 2^31 enumeration. The compared statement does not claim that 29 is the minimum order of a counterexample.", "authors": [ "Dan Clemens Posch" ], "maintainers": [ "Dan Clemens Posch" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "A 15/31 Counterexample Family to the Albertson–Berman Conjecture", "id": "arXiv:2608.17350", "authors": [ "Heejae Jung" ], "type": "article", "relationship": "independently-proves", "endorsement": "not-contacted" }, { "title": "A conjecture on planar graphs", "id": "", "authors": [ "Michael O. Albertson", "David M. Berman" ], "type": "article", "relationship": "background", "endorsement": "n/a" } ], "related": [], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05C10", "05C69" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "grok-4.6" ], "framework": "Grok Build TUI" } ], "strongest": "agent", "models": [ "grok-4.6" ], "spend": "not tracked", "notes": "Coordinator scaffolded the Palomar project; this lane closed the remaining Solution.lean sorries. Scratch files verified with scripts/lean_verify.sh --file FILE --receipt-only, patterned on jc2-lean/max11-partial-y/scripts/box_lean_verify.sh." }, "scope": "Compared: AlbertsonBerman.counterexample, the conjunction of the 31-vertex seed’s edge/face counts, connectedness, triangular faces, 2-face incidence, edge-set equality, Euler characteristic 2, a 15-vertex induced tree, and the universal bound that no 16-vertex induced subgraph is acyclic. Not compared: the infinite 15k/31k annular family, minimum counterexample order, or degree-multiset claims.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "Challenge/Solution state the 31-vertex seed certificate rather than the full 15k/31k family. Upper bound a(T)≤15 uses the gadget profile and bipyramid score β=3, transported by injective embeddings, instead of enumerating 2^31 subsets. Does not formalize minimum order 29.", "alignment": true, "original": true, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "none" ], "notes": "Statement surface follows Jung’s seed proposition: planarity plus a(T)=15." }, "canonical": { "repo": "dcposch/albertson-berman-lean", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "dcposch/austin-pair-lean/formalization.yaml", "repo": "dcposch/austin-pair-lean", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/dcposch/austin-pair-lean/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "austin-pair", "description": "Thin Palomar wrapper of an Austin pair from the Equational Theories Project: magma laws Equation 3588 (x*y = z*((x*y)*z)) and Equation 3994 (x*y = (z*(x*y))*z) are equivalent on every finite magma, but neither implies the other for magmas in general. The compared theorems exact InfModel.Finite.Equation3994_implies_Equation3588 together with its dual and the two infinite countermodels in teorth/equational_theories.", "authors": [ "Dan Clemens Posch" ], "maintainers": [ "Dan Clemens Posch" ], "license": "Apache-2.0", "role": "", "substantive": "teorth/equational_theories", "sources": [ { "title": "The Equational Theories Project: Advancing Collaborative Mathematical Research at Scale", "id": "arXiv:2512.07087", "authors": [ "Matthew Bolan", "Joachim Breitner", "Jose Brox", "Nicholas Carlini", "Mario Carneiro", "Floris van Doorn", "Martin Dvorak", "Andrés Goens", "Aaron Hill", "Harald Husum", "Hernán Ibarra Mejia", "Zoltan A. Kocsis", "Bruno Le Floch", "Amir Livne Bar-on", "Lorenzo Luccioli", "Douglas McNeil", "Alex Meiburg", "Pietro Monticone", "Pace P. Nielsen", "Emmanuel Osalotioman Osazuwa", "Giovanni Paolini", "Marco Petracci", "Bernhard Reinke", "David Renshaw", "Marcus Rossel", "Cody Roux", "Jérémy Scanvic", "Shreyas Srinivas", "Anand Rao Tadipatri", "Terence Tao", "Vlad Tsyrklevich", "Fernando Vaquerizo-Villar", "Daniel Weber", "Fan Zheng" ], "type": "paper", "relationship": "formalizes", "endorsement": "not-contacted" } ], "related": [ { "id": "https://github.com/teorth/equational_theories", "relationship": "builds-on", "note": "Substantive development. Compared proofs exact InfModel.Finite.Equation3994_implies_Equation3588, InfModel.Equation3994_not_implies_Equation3588, and InfModel.Equation3588_not_implies_Equation3994 at commit e5a88a1479011ece4aad8e3c2e7e5c0ebc0a5b2a." } ], "classification": { "arxiv": [ "math.RA", "math.LO" ], "msc2020": [ "08A05", "03C05" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "grok-4.6" ], "framework": "Grok Build TUI" } ], "strongest": "agent", "models": [ "grok-4.6" ], "spend": "not tracked", "notes": "Wrapper only. No new mathematical proof. Lean 4.29.1 matches the upstream equational_theories toolchain." }, "scope": "Compared: finite equivalence of Equation 3994 and Equation 3588, and the two non-implications in general. Not compared: other Austin pairs (206/1648), Austin laws (Kisielewicz 28770 and 374794), or the rest of the ETP implication graph.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "Magma operation is Mathlib Mul rather than equational_theories.Magma (same binary operation, different typeclass, to keep Challenge Mathlib-only). The wiki's \"Equation 3944\" is Equation 3994 in Lean. The compared existential does not include an Infinite hypothesis; the ETP models are on ℕ.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "none" ], "notes": "Statement surface follows InfModel.lean Equations 3588 and 3994." }, "canonical": { "repo": "dcposch/austin-pair-lean", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "dcposch/jc2-lean/gcd3-69-composition/formalization.yaml", "repo": "dcposch/jc2-lean", "path": "gcd3-69-composition/formalization.yaml", "directory": "gcd3-69-composition", "origins": [ "search" ], "url": "https://github.com/dcposch/jc2-lean/blob/HEAD/gcd3-69-composition/formalization.yaml", "version": "v0.4", "missing": [], "name": "jc2-gcd3-69-common-cubic-divisible-source-exclusion", "description": "Lean 4 / Mathlib proof of the exhaustive divisible-core normalized common-cubic exclusion at actual partial degrees (6,9). For literal bivariate source polynomials with leading coefficients H^2 and H^3, nonzero constant coefficientwise-inner Jacobian, and 3 dividing deg H, the formalization splits H into its polynomial-cube and noncube cases and derives a contradiction in both. It does not derive the normalization from an arbitrary Keller pair or establish maximum partial degree eleven.", "authors": [ "Dan Clemens Posch" ], "maintainers": [ "Dan Clemens Posch" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Exhaustive divisible-core common-cubic source composition at (6,9)", "id": "", "authors": [ "Dan Clemens Posch" ], "type": "original-proof", "relationship": "other", "endorsement": "participated" } ], "related": [ { "id": "PALOMAR-2026-08-25-000002", "relationship": "other", "note": "Version 2 is the registered source-facing noncube exclusion used on the branch where H is not a polynomial cube." }, { "id": "PALOMAR-2026-08-29-000002", "relationship": "other", "note": "The registered cube-branch entry (project gcd3-69-cube in this repository). This composition consumes the locally strengthened variant of that exclusion, including constant cube roots (GCD369PolynomialCubeSourceExclusionV2), which extends the registered theorem within the same project." } ], "classification": { "arxiv": [ "math.AG", "math.AC" ], "msc2020": [ "14R15", "13P10" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "OpenAI Codex (GPT-5 family)" ], "framework": "Codex" }, { "method": "manual", "models": [], "framework": "n/a" } ], "strongest": "agent", "models": [ "OpenAI Codex (GPT-5 family)" ], "spend": "subscription-based", "notes": "The submitted composition invokes no external computer algebra system or trusted certificate. Its two substantive branches are imported from the pinned sibling projects and the final case split is checked by Lean's kernel." }, "scope": "Fully proved over an algebraically closed characteristic-zero field: a literal normalized source p,q in k[x][y] of outer degrees 6 and 9, with leading coefficients H^2 and H^3, nonzero constant coefficientwise-inner Jacobian, and 3 dividing the polynomial degree of H, cannot exist. The proof performs the exhaustive split according to whether H is a polynomial cube. The cube branch uses the complete local cube exclusion; the noncube branch uses PALOMAR-2026-08-25-000002 v2. Not proved here: normalization of arbitrary Keller pairs, the nondivisible-core history route, scalar-extension descent, t", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "GCD369PolynomialCommonCubicSourceExclusionV2", "file": "GCD369DivisibleSourceExclusion.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The headline begins with the literal normalized common-cubic source and the historical divisibility condition 3 | deg H. It does not claim the preceding global normalization, the nondivisible-core route, or the later maximum-eleven composition. No other divergence is known.", "alignment": true, "original": true, "review": { "status": "self-assessed; imported branch formalizations have their own recorded audits; this composition awaits external human review", "bucket": "self-assessed", "reviewers": [ "Dan Clemens Posch" ], "notes": "The project build and axiom script check the exact headline declaration. No external peer review of this composition has yet occurred." }, "canonical": { "repo": "dcposch/jc2-lean", "directory": "gcd3-69-composition" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "dcposch/jc2-lean/gcd3-69-core/formalization.yaml", "repo": "dcposch/jc2-lean", "path": "gcd3-69-core/formalization.yaml", "directory": "gcd3-69-core", "origins": [ "search" ], "url": "https://github.com/dcposch/jc2-lean/blob/HEAD/gcd3-69-core/formalization.yaml", "version": "v0.4", "missing": [], "name": "jc2-gcd3-69-core", "description": "Lean 4 / Mathlib formalization of six reusable algebraic statements from the first common-cubic gate at actual partial degrees (6,9): the exact top-row identity, nontrivial Kummer alignment, simultaneous depression, invariance of a simple generated field under a nonzero base-field scalar, an explicit order-three Davenport--Stothers certificate, and its general differential identity. This project intentionally does not formalize the later (6,9), 3|H exclusion or the maximum-eleven automorphism theorem.", "authors": [ "Dan Clemens Posch" ], "maintainers": [ "Dan Clemens Posch" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "GCD3 (6,9) common-cubic first gate", "id": "", "authors": [ "Dan Clemens Posch" ], "type": "working campaign report", "relationship": "formalizes", "endorsement": "participated" }, { "title": "Hostile different-model review: GCD3 (6,9) common-cubic first gate", "id": "", "authors": [ "Grok review agent under the JC2 campaign" ], "type": "campaign review report", "relationship": "background", "endorsement": "not-contacted" }, { "title": "GCD3 (6,9) mixed-cube-factor adversarial control", "id": "", "authors": [ "Dan Clemens Posch" ], "type": "working campaign report", "relationship": "background", "endorsement": "participated" } ], "related": [ { "id": "PALOMAR-2026-08-19-000005", "relationship": "sibling project from the same campaign and repository", "note": "Theorem A, the weighted polynomial ODE-rigidity theorem." }, { "id": "PALOMAR-2026-08-20-000001", "relationship": "sibling project from the same campaign and repository", "note": "The normalized (2,2) vertex-gap obstruction." } ], "classification": { "arxiv": [ "math.AG", "math.AC" ], "msc2020": [ "14R15" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "OpenAI Codex (GPT-5 family)" ], "framework": "Codex" }, { "method": "manual", "models": [], "framework": "n/a" } ], "strongest": "agent", "models": [ "OpenAI Codex (GPT-5 family)" ], "spend": "subscription-based", "notes": "The Lean formalization is machine-checked. The source mathematics had campaign-internal adversarial model review, but this project has not yet received external human peer review." }, "scope": "Fully formalized and proved: GCD369TopRowIdentity, GCD369KummerAlignment, GCD369SimultaneousDepression, GCD369CubeFactorNeutral, GCD369DavenportStothersCertificate, and GCD369DavenportStothersDerivativeIdentity. Not formalized: derivation of the Kummer model from an arbitrary Keller pair; the eight high source rows; lower Pfaffian and terminal rows; target-translation corrections; cube-mismatch and cube-trajectory closures; the coverage composition; the (6,9), 3|H exclusion; the maximum-eleven theorem; or JC2.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "(1) GCD369KummerAlignment abstracts the campaign situation to a characteristic-zero domain with hypotheses saying that differential constants are sigma-fixed and A,B have common weight omega; it does not construct the Kummer extension or assert sigma(s)=omega*s. These omitted facts are not used by the alignment argument once the stated hypotheses are available. (2) GCD369CubeFactorNeutral is the general elementary simple-field statement underlying the mixed-cube-factor control, not its full divisor analysis. (3) The Davenport--Stothers certificate is written in the denominator-free parameter lambda=2*mu; over a characteristic-zero field this parametrizes the same family as the source's lambda. (4) The formalization stops at first-gate algebra and makes no claim that a Keller trajectory enters or persists on the Davenport--Stothers component. No other divergences known.", "alignment": true, "original": false, "review": { "status": "self-assessed; informal source passed campaign-internal different-model review; lean statements and source alignment await human audit; no external peer review", "bucket": "self-assessed", "reviewers": [ "Dan Clemens Posch" ], "notes": "Kernel checks show no sorryAx or custom axioms in Solution.lean. The cited hostile review is evidence about the informal source and its scope, not an independent review of this Lean extraction." }, "canonical": { "repo": "dcposch/jc2-lean", "directory": "gcd3-69-core" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "dcposch/jc2-lean/gcd3-69-cube/formalization.yaml", "repo": "dcposch/jc2-lean", "path": "gcd3-69-cube/formalization.yaml", "directory": "gcd3-69-cube", "origins": [ "palomar", "search" ], "url": "https://github.com/dcposch/jc2-lean/blob/HEAD/gcd3-69-cube/formalization.yaml", "version": "v0.4", "missing": [], "name": "jc2-gcd3-69-source-cube-exclusion", "description": "Lean 4 / Mathlib proof of the source-facing polynomial-cube exclusion at actual partial degrees (6,9). From literal bivariate polynomials of outer degrees six and nine, leading coefficients s^6 and s^9 for a common polynomial s, and nonzero constant Keller bracket, the formalization proves that no such source exists, including when s is constant. It constructs the finite-pole or infinity expansion, exhausts every timing of the eight Faber loads through rho2, and excludes the remaining elliptic, cusp, zero-invariant, and constant-core terminal fibres. It does not claim the reduction from an arb", "authors": [ "Dan Clemens Posch" ], "maintainers": [ "Dan Clemens Posch" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "GCD3 (6,9) cube-core mismatch gate", "id": "", "authors": [ "Dan Clemens Posch" ], "type": "other", "relationship": "formalizes", "endorsement": "participated" }, { "title": "Hostile review: GCD3 (6,9) cube-core mismatch gate", "id": "", "authors": [ "Grok 4.6 review agent under the JC2 campaign" ], "type": "other", "relationship": "background", "endorsement": "n/a" }, { "title": "GCD3 (6,9) cube-core rational trajectory gate", "id": "", "authors": [ "Dan Clemens Posch" ], "type": "other", "relationship": "formalizes", "endorsement": "participated" }, { "title": "Hostile review: GCD3 (6,9) cube-core rational trajectory gate", "id": "", "authors": [ "Grok 4.6 review agent under the JC2 campaign" ], "type": "other", "relationship": "background", "endorsement": "n/a" }, { "title": "On the Jacobian conjecture and the configurations of roots", "id": "doi:10.1515/crll.1983.340.140", "authors": [ "Tzuong-Tsieng Moh" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "On the shape of possible counterexamples to the Jacobian Conjecture", "id": "doi:10.1016/j.jalgebra.2016.08.039", "authors": [ "Christian Valqui", "Jorge A. Guccione", "Juan J. Guccione" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" } ], "related": [ { "id": "PALOMAR-2026-08-25-000002", "relationship": "other", "note": "Version 2 proves the source-facing noncube exclusion at the same actual partial degrees (6,9). The present entry proves the polynomial-cube companion; neither entry alone contains their global Keller-pair partition and degree-recursion handoff." } ], "classification": { "arxiv": [ "math.AG", "math.AC" ], "msc2020": [ "14R15", "13P10" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "OpenAI Codex (GPT-5 family)", "Grok 4.6" ], "framework": "Codex with delegated Grok CLI sessions" }, { "method": "manual", "models": [], "framework": "n/a" } ], "strongest": "agent", "models": [ "Grok 4.6", "OpenAI Codex (GPT-5 family)" ], "spend": "subscription-based", "notes": "Informal source identities used symbolic computation during discovery and campaign review. The submitted Lean proof invokes no external CAS or trusted certificate. Every accepted identity is checked by Lean's kernel. No external human review of this formalization has yet occurred." }, "scope": "Fully proved over an algebraically closed characteristic-zero field: a literal polynomial source p,q in k[x][y] with outer degrees 6 and 9, leading coefficients s^6 and s^9, nonzero s, and nonzero constant coefficientwise-inner Jacobian cannot exist. For positive-degree s, the proof constructs a finite normalized-coefficient pole, rules out the exact cubic-square normal fallback, builds the first transverse Hahn scale, and excludes the later fibres. For constant s, it expands at infinity, exhausts all timing branches for d,c7,c5,c4,c2,c1,rho1,rho2, and excludes the terminal constant-core norma", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "GCD369PolynomialCubeSourceExclusion", "file": "GCD369Cube.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "(1) The public theorem begins with the literal normalized cube source whose leading coefficients are s^6 and s^9; it does not derive this normalization from an arbitrary plane Keller pair. (2) The informal reports organize the proof by weighted trajectory strata. Lean refines these into explicit strict, equality, zero-load, simple-root, double-root, and endpoint lemmas before composing them. (3) The final target translation and zero-invariant boundary are performed on the original literal polynomial source, avoiding any caller-supplied boundary witness. No other substantive divergence is known.", "alignment": true, "original": false, "review": { "status": "self-assessed; informal sources passed campaign-internal different-model hostile review; lean statements await external human audit; no external peer review", "bucket": "self-assessed", "reviewers": [ "Dan Clemens Posch" ], "notes": "The independent Grok reports review the informal mathematics, not this Lean code. Full builds and the project axiom script find no sorryAx, custom axioms, or trust escapes in the solution theorem." }, "canonical": { "repo": "dcposch/jc2-lean", "directory": "gcd3-69-cube" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-29-000002", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-29-000002", "commit": "ba44127093e65cbd862604a16e3fe79fb63e93cf", "trust": "high", "theorems": 1, "date": "2026-08-29" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "dcposch/jc2-lean/gcd3-69-noncube/formalization.yaml", "repo": "dcposch/jc2-lean", "path": "gcd3-69-noncube/formalization.yaml", "directory": "gcd3-69-noncube", "origins": [ "palomar", "search" ], "url": "https://github.com/dcposch/jc2-lean/blob/HEAD/gcd3-69-noncube/formalization.yaml", "version": "v0.5", "missing": [], "name": "jc2-gcd3-69-source-noncube-exclusion", "description": "Lean 4 / Mathlib proof of the source-facing noncube exclusion at actual partial degrees (6,9). From literal bivariate polynomials of outer degrees six and nine, leading coefficients H^2 and H^3, and nonzero constant Keller bracket, the formalization constructs the differential cubic-Kummer extension, affine alignment, coefficient weights, high-row normal form, and reduced rational presentations. It derives the invariant two-sheet split and excludes the zero, elliptic, and shifted Davenport--Stothers sheets. It does not claim the cube branch, all (6,9), maximum eleven, or the plane Jacobian con", "authors": [ "Dan Clemens Posch" ], "maintainers": [ "Dan Clemens Posch" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "GCD3 (6,9) first common-cubic gate", "id": "", "authors": [ "Dan Clemens Posch" ], "type": "other", "relationship": "formalizes", "endorsement": "participated" }, { "title": "Hostile different-model review: GCD3 (6,9) first common-cubic gate", "id": "", "authors": [ "Grok 4.6 review agent under the JC2 campaign" ], "type": "other", "relationship": "background", "endorsement": "n/a" }, { "title": "GCD3 (6,9) lower-Pfaffian successor gate", "id": "", "authors": [ "Dan Clemens Posch" ], "type": "other", "relationship": "formalizes", "endorsement": "participated" }, { "title": "Hostile different-model review: GCD3 (6,9) lower-Pfaffian successor", "id": "", "authors": [ "Grok 4.6 review agent under the JC2 campaign" ], "type": "other", "relationship": "background", "endorsement": "n/a" }, { "title": "On the Jacobian conjecture and the configurations of roots", "id": "doi:10.1515/crll.1983.340.140", "authors": [ "Tzuong-Tsieng Moh" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "On the shape of possible counterexamples to the Jacobian Conjecture", "id": "doi:10.1016/j.jalgebra.2016.08.039", "authors": [ "Christian Valqui", "Jorge A. Guccione", "Juan J. Guccione" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Elliptic Surfaces and Davenport-Stothers Triples", "id": "doi:10.14992/00008689", "authors": [ "Tetsuji Shioda" ], "type": "paper", "relationship": "adapts", "endorsement": "not-contacted" }, { "title": "On Davenport's bound for the degree of f^3 - g^2 and Riemann's Existence Theorem", "id": "doi:10.4064/AA-71-2-107-137", "authors": [ "Umberto Zannier" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" } ], "related": [], "classification": { "arxiv": [ "math.AG", "math.AC" ], "msc2020": [ "14R15", "13P10" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "OpenAI Codex (GPT-5 family)" ], "framework": "Codex" }, { "method": "manual", "models": [], "framework": "n/a" } ], "strongest": "agent", "models": [ "OpenAI Codex (GPT-5 family)" ], "spend": "subscription-based", "notes": "The Singular decomposition and SymPy replay in the source were used to discover and independently review the informal identities. The submitted Lean proofs do not invoke either system: all algebraic certificates are checked by Lean's kernel. No external human review of the Lean project has yet occurred." }, "scope": "Fully proved for a literal normalized source p,q in k[x][y] over an algebraically closed characteristic-zero field: outer degrees 6 and 9, leading coefficients H^2 and H^3, nonzero constant coefficientwise-inner Keller bracket, noncube H, and 3 dividing deg H are contradictory. The proof internally constructs a primitive cubic root, the irreducible differential Kummer extension and deck action, constant-field descent, simultaneous affine depression, deck coefficient weights, triangular integration of all eight high rows, the lower invariant-fibre split, and canonical reduced rational presentat", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "GCD369PolynomialNoncubeSourceExclusion", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "GCD369KummerHighRowsNormalize", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "GCD369RatFuncConstants", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "GCD369PolynomialNoncubeDifferentialKummerPackage", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "GCD369RatFuncDerivative", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "GCD369RatFuncStandardDifferential", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "GCD369RatFuncConstantsOfPolynomialDerivative", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "GCD369BaseFixingAutomorphismFixesConstants", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "GCD369KummerAlignmentFromBaseConstants", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "GCD369AlgebraicDifferentialConstantsDescend", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "GCD369KummerDeckCommutesWithDerivative", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "GCD369PolynomialNoncubeKummerExtension", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "GCD369NoncubeCubicKummerExtension", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "GCD369KummerAlignmentFromFirstRow", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "GCD369AlignedNoncubeExclusion", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "(1) The headline begins with a normalized common-cubic source whose leading coefficients are explicitly H^2 and H^3; it does not derive this global reduction from every arbitrary plane Keller pair. The affine change of variable, Kummer weights, function-field descent, and rational presentations are derived rather than assumed. (2) Rational functions are represented internally by reduced numerator/denominator pairs over an algebraically closed constant field. The nonzero elliptic exclusion uses direct evaluation at a forbidden value rather than the source's equivalent local-valuation phrasing; constant trajectories are handled separately from the uncleared Keller row. (3) The shifted Davenport--Stothers bracket doubles the degree-nine polynomial to remove the coefficient 63/2, so its bracket constant is 1134 rather than 567. (4) The valuation theorem explicitly assumes the effective-domain condition `B != 0`. Pointwise reducedness means every zero of `N` is not a zero of `B`; the theorem first proves `N` constant and then works entirely with polynomial root multiplicities. This is the global polynomial form of the source's finite valuations. No other divergences are known.", "alignment": true, "original": false, "review": { "status": "self-assessed; informal source passed campaign-internal different-model hostile review; lean statements and source alignment await human audit; no external peer review", "bucket": "self-assessed", "reviewers": [ "Dan Clemens Posch" ], "notes": "The copied hostile review independently reconstructs the informal mathematics but predates and does not inspect the Lean code. Kernel axiom reports and the project audit script find no sorryAx or non-permitted axioms in Solution.lean." }, "canonical": { "repo": "dcposch/jc2-lean", "directory": "gcd3-69-noncube" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-25-000002", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-25-000002", "commit": "b80667d80987ca48cb98e1eb31e1324251b7928a", "trust": "high", "theorems": 18, "date": "2026-08-26" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "dcposch/jc2-lean/max11-package/formalization.yaml", "repo": "dcposch/jc2-lean", "path": "max11-package/formalization.yaml", "directory": "max11-package", "origins": [ "search" ], "url": "https://github.com/dcposch/jc2-lean/blob/HEAD/max11-package/formalization.yaml", "version": "v0.4", "missing": [], "name": "jc2-max11-partial-y-degree-closure", "description": "Lean 4 / Mathlib proof that every plane Keller pair with maximum actual partial y-degree at most eleven generates the polynomial ring, over an algebraically closed field K of characteristic zero, modulo exactly two cited classical theorems stated as explicit hypotheses: Nagata's prime-total-degree-gcd theorem and the Guccione--Guccione--Valqui standard-pair endpoint obstruction. 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Every Lean theorem here additionally assumes that the field is algebraically closed (`[IsAlgClosed K]`); the low-scale leaves use this hypothesis, and no descent from the algebraically closed case to arbitrary characteristic-zero fields is formalized. This is the one substantive narrowing of the source statement. The Lean development uses the source's exact polynomial/Jacobian/ring-generation predicate (MvPolynomial.pderiv, degreeOf 1, Algebra.adjoin). It does not formalize the two cited classical theorems; they enter only as the explicitly typed hypotheses `PlaneKellerPrimeTotalDegreeGCDRoute` (Nagata 1989, Theorem 7.3, in coordinate-free total-degree form) and `PlaneKellerStandardEndpointGCDObstruction` (Guccione--Guccione--Valqui 2017, Corollary 7.9). GGV's statement about the globally minimal counterexample gcd is not strengthened to an arbitrary-pair twice-prime theorem. The finite composition follows the report's routing rules, but the source-shear bridge, the gcd-two normalization, and all five low-scale leaf exclusions are original to this formalization ", "alignment": true, "original": true, "review": { "status": "self-assessed; informal source passed campaign-internal different-model hostile review; lean statements and source alignment await human audit; no external peer review", "bucket": "self-assessed", "reviewers": [ "Dan Clemens Posch" ], "notes": "The copied hostile review predates and does not inspect the Lean code. This repository carries seven Comparator configurations at one commit: `comparator.json` (unconditional headline, solution module `Max11Solution`), `comparator-conditional.json` (the headline with the five leaf routes as hypotheses, solution module `Max11Assembly`), and `comparator-leaf{46,410,68,610,810}.json` (one per proved leaf route; solution modules `LowScale46ScaleTwo`, `Grok410ChainPromotionScratch`, `Grok68TerminalZeroMeetingCellScratch`, `Grok610ScaleZeroCubicLoadWallsScratch`, `Grok810RouteClosureScratch`). Every entry states its hypotheses, including the algebraically-closed characteristic-zero field hypothesis, in the Lean statement itself; nothing is assumed silently. Kernel axiom reports for all seven com" }, "canonical": { "repo": "dcposch/jc2-lean", "directory": "max11-package" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "dcposch/jc2-lean/max11-partial-y/formalization.yaml", "repo": "dcposch/jc2-lean", "path": "max11-partial-y/formalization.yaml", "directory": "max11-partial-y", "origins": [ "search" ], "url": "https://github.com/dcposch/jc2-lean/blob/HEAD/max11-partial-y/formalization.yaml", "version": "v0.4", "missing": [], "name": "jc2-max11-partial-y-degree-closure", "description": "Lean 4 / Mathlib proof that every plane Keller pair with maximum actual partial y-degree at most eleven generates the polynomial ring, over an algebraically closed field K of characteristic zero, modulo exactly two cited classical theorems stated as explicit hypotheses: Nagata's prime-total-degree-gcd theorem and the Guccione--Guccione--Valqui standard-pair endpoint obstruction. The proof is an original finite degree-routing composition: the zero-degree, equal-degree, divisibility, coprime, and odd-common-scale routes are proved from the cited theorems; a finite classifier reduces partial-degr", "authors": [ "Dan Clemens Posch" ], "maintainers": [ "Dan Clemens Posch" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "GCD3 (6,9) coverage composition and the max-(11) threshold", "id": "", "authors": [ "Dan Clemens Posch" ], "type": "other", "relationship": "other", "endorsement": "participated" }, { "title": "Hostile different-model review: GCD3 (6,9) coverage composition", "id": "", "authors": [ "Claude review agent under the JC2 campaign" ], "type": "other", "relationship": "background", "endorsement": "n/a" }, { "title": "Source-shear, common-core normalization, and low-scale Max-11 development", "id": "", "authors": [ "Dan Clemens Posch" ], "type": "original-proof", "relationship": "other", "endorsement": "participated" }, { "title": "Two-dimensional Jacobian Conjecture", "id": "", "authors": [ "Masayoshi Nagata" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "On the shape of possible counterexamples to the Jacobian Conjecture", "id": "doi:10.1016/j.jalgebra.2016.08.039", "authors": [ "Jorge A. Guccione", "Juan J. Guccione", "Christian Valqui" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" } ], "related": [ { "id": "PALOMAR-2026-08-25-000002", "relationship": "other", "note": "Version 2 is the registered source-facing `(6,9)` noncube exclusion used by the imported divisible-core assembly in this project." }, { "id": "local:gcd3-69-cube", "relationship": "other", "note": "The sibling project proves the complete source-facing polynomial-cube exclusion, including the constant-core branch, imported by this project." }, { "id": "local:gcd3-69-composition", "relationship": "other", "note": "The sibling composition performs the exhaustive cube/noncube dispatch at `(6,9)` and supplies the divisible common-cubic exclusion imported here." } ], "classification": { "arxiv": [ "math.AG", "math.AC", "math.CO" ], "msc2020": [ "14R15", "05A99" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "OpenAI Codex (gpt-6-astra; earlier Codex agent sessions, model not recorded)", "xAI Grok 4.6", "Claude Opus 5", "Claude Fable 5.1" ], "framework": "OpenAI Codex CLI, grok CLI, Claude Code" }, { "method": "manual", "models": [], "framework": "n/a" } ], "strongest": "agent", "models": [ "Claude Fable 5.1", "Claude Opus 5", "OpenAI Codex (gpt-6-astra; earlier Codex agent sessions, model not recorded)", "xAI Grok 4.6" ], "spend": "subscription plans plus prepaid API credits; not itemized", "notes": "No external human review of the Lean project has yet occurred. The submitted proofs use only kernel-checked Mathlib tactics; the axiom report of every comparator theorem is [propext, Classical.choice, Quot.sound]." }, "scope": "Field hypothesis. Every theorem in this project is stated for an arbitrary algebraically closed field K of characteristic zero (Lean binders `{K : Type*} [Field K] [CharZero K] [IsAlgClosed K]`). Algebraic closure is used in the low-scale leaves (for example to write a nonzero common core as H = h^2 at (4,6)); no result is claimed for non-algebraically-closed fields of characteristic zero, and scalar-extension descent is not formalized. 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"declaration": "Max11DegreeRoutes.planeKellerPair_69_alignmentDiscriminator_constant", "file": "DegreeRoutes.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Max11DegreeRoutes.planeKellerPair_69_cubeDiscriminatorNumerator_eq_zero", "file": "DegreeRoutes.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Max11DegreeRoutes.planeKellerPair_69_commonCore", "file": "DegreeRoutes.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Max11DegreeRoutes.Max11PlaneKellerGenerationWithElementaryRoutes", "file": "DegreeRoutes.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Max11ClassicalRoutes.Max11PlaneKellerGenerationWithZeroRoute", "file": "ClassicalRoutes.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Max11PlaneKellerGeneration", "file": "Max11Core.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "MaxPartialDegreeElevenClosure", "file": "Max11Core.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The informal source (the frozen campaign report) states its final theorem for characteristic-zero bivariate Keller pairs. Every Lean theorem here additionally assumes that the field is algebraically closed (`[IsAlgClosed K]`); the low-scale leaves use this hypothesis, and no descent from the algebraically closed case to arbitrary characteristic-zero fields is formalized. This is the one substantive narrowing of the source statement. The Lean development uses the source's exact polynomial/Jacobian/ring-generation predicate (MvPolynomial.pderiv, degreeOf 1, Algebra.adjoin). It does not formalize the two cited classical theorems; they enter only as the explicitly typed hypotheses `PlaneKellerPrimeTotalDegreeGCDRoute` (Nagata 1989, Theorem 7.3, in coordinate-free total-degree form) and `PlaneKellerStandardEndpointGCDObstruction` (Guccione--Guccione--Valqui 2017, Corollary 7.9). GGV's statement about the globally minimal counterexample gcd is not strengthened to an arbitrary-pair twice-prime theorem. The finite composition follows the report's routing rules, but the source-shear bridge, the gcd-two normalization, and all five low-scale leaf exclusions are original to this formalization ", "alignment": true, "original": true, "review": { "status": "self-assessed; informal source passed campaign-internal different-model hostile review; lean statements and source alignment await human audit; no external peer review", "bucket": "self-assessed", "reviewers": [ "Dan Clemens Posch" ], "notes": "The copied hostile review predates and does not inspect the Lean code. This repository carries seven Comparator configurations at one commit: `comparator.json` (unconditional headline, solution module `Max11Solution`), `comparator-conditional.json` (the headline with the five leaf routes as hypotheses, solution module `Max11Assembly`), and `comparator-leaf{46,410,68,610,810}.json` (one per proved leaf route; solution modules `LowScale46ScaleTwo`, `Grok410ChainPromotionScratch`, `Grok68TerminalZeroMeetingCellScratch`, `Grok610ScaleZeroCubicLoadWallsScratch`, `Grok810RouteClosureScratch`). Every entry states its hypotheses, including the algebraically-closed characteristic-zero field hypothesis, in the Lean statement itself; nothing is assumed silently. Kernel axiom reports for all seven com" }, "canonical": { "repo": "dcposch/jc2-lean", "directory": "max11-partial-y" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "dcposch/jc2-lean/strip-block/formalization.yaml", "repo": "dcposch/jc2-lean", "path": "strip-block/formalization.yaml", "directory": "strip-block", "origins": [ "search" ], "url": "https://github.com/dcposch/jc2-lean/blob/HEAD/strip-block/formalization.yaml", "version": "v0.4", "missing": [], "name": "jc2-strip-block-core", "description": "Lean 4 / Mathlib formalization of a supporting algebraic core for the uniform depth-two log-residue result (thm:R, Theorem 6.5 in the current campaign-paper compilation). It proves an exact finite-coefficient encoding of the weighted ODE (BlockToODEBridge), the resulting one-way characteristic-zero rigidity statement (InnerBlockRigidity), an exact binomial-transform coefficient identity for the displayed residue functional (OuterResidueIdentity), and the final variety intersection conditional on an explicit interface supplying the substantive inner and outer elimination equivalences (LogResidu", "authors": [ "Dan Clemens Posch" ], "maintainers": [ "Dan Clemens Posch" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Sheet-number obstruction theory for the plane Jacobian Conjecture: campaign theory bundle v1", "id": "doi:10.5281/zenodo.22002825", "authors": [ "Dan Clemens Posch" ], "type": "working paper (archived campaign theory bundle)", "relationship": "partially formalizes", "endorsement": "participated" }, { "title": "A Vertex-Gap Obstruction for Low-Degree Strip Pairs in the Plane Jacobian Conjecture", "id": "doi:10.5281/zenodo.21894922", "authors": [ "Dan Clemens Posch" ], "type": "working paper and reproduction artifact", "relationship": "background", "endorsement": "participated" } ], "related": [ { "id": "PALOMAR-2026-08-19-000005", "relationship": "reuses a local standalone copy of the sibling theorem", "note": "Theorem A (weighted-ODE rigidity) is already fully formalized and registered in theorem-a/. This package copies the proof locally so the project remains Mathlib-only and standalone." }, { "id": "PALOMAR-2026-08-20-000001", "relationship": "same campaign; complementary concrete block theorem", "note": "vertex-gap/ formalizes the actual polynomial/support equations and elimination for the concrete (k,d2)=(2,2) vertex-gap obstruction." } ], "classification": { "arxiv": [ "math.AG", "math.AC" ], "msc2020": [ "14R15" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "OpenAI Codex (GPT-5)" ], "framework": "Codex" }, { "method": "manual", "models": [], "framework": "n/a" } ], "strongest": "agent", "models": [ "OpenAI Codex (GPT-5)" ], "spend": "subscription-based", "notes": "Solution.lean and Rigidity.lean are kernel-checked. The scope boundary is part of the formalization: the two source-specific elimination inputs are named fields of DepthTwoBlockData, not hidden axioms or informal claims." }, "scope": "Fully proved in this package: (1) BlockToODEBridge, saying that the explicitly defined finite coefficient condition innerBlockKeysVanish is equivalent to A*C' - k*A'*C = 1 under the displayed degree bounds; (2) InnerBlockRigidity, the implication from that condition to deg A <= 1 in characteristic zero; (3) OuterResidueIdentity, identifying the displayed R_{k,d} with coefficient k+2 of an explicit finite binomial transform; and (4) LogResidueBlockVariety, the final elementary intersection once the actual inner- and outer-extra equivalences are supplied as fields of DepthTwoBlockData. NOT forma", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "This is a partial supporting formalization, not a declaration-equivalent translation of thm:R. First, innerBlockKeysVanish is a purpose-built finite coefficient encoding that already quantifies the polynomial C; the source instead defines actual extra keys after a triangular solve and proves their equivalence to existence of C. Second, OuterResidueIdentity proves only the elementary binomial-transform coefficient formula for R_{k,d}; it does not derive the surviving outer extra from the solved outer column or certify the scalar displayed in lem:outer. Third, LogResidueBlockVariety assumes both missing locus identifications through DepthTwoBlockData and proves only the final logical assembly. Fourth, the assembly theorem is stated over any field because characteristic zero is needed in the missing/rigidity input, not in the final zero-product argument itself. The finite encoding and ODE rigidity are stated over commutative integral domains where possible, strengthening the field assumptions for those isolated pieces.", "alignment": true, "original": false, "review": { "status": "self-assessed partial formalization; exact scope audited against the source during package preparation; not externally peer-reviewed and not yet recommended as a full-theorem palomar submission", "bucket": "self-assessed", "reviewers": [ "Dan Clemens Posch" ], "notes": "Any later submission should retain the explicit partial-scope warning. To claim a full formalization of thm:R, it must first replace the two DepthTwoBlockData equivalence fields with proofs from definitions of the actual block equations." }, "canonical": { "repo": "dcposch/jc2-lean", "directory": "strip-block" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "dcposch/jc2-lean/theorem-a/formalization.yaml", "repo": "dcposch/jc2-lean", "path": "theorem-a/formalization.yaml", "directory": "theorem-a", "origins": [ "palomar", "search" ], "url": "https://github.com/dcposch/jc2-lean/blob/HEAD/theorem-a/formalization.yaml", "version": "v0.4", "missing": [], "name": "jc72108-theorem-a", "description": "Lean 4 / Mathlib formalization of Theorem A (ODE rigidity, all weights; label thm:ode, Theorem 6.1 in the current compilation of the campaign paper): if polynomials A, C over an integral domain of characteristic zero and an integer weight nu >= 1 satisfy A*C' - nu*A'*C = c for a nonzero constant c, then deg A <= 1 (TheoremA). Also a standalone positive-characteristic strengthening extracted from the same proof (TheoremA_charP): the same conclusion over a domain of characteristic p under the degree bounds nu*deg A < p and deg C < p. The theorem is the rigidity engine of the depth-two block anal", "authors": [ "Dan Clemens Posch" ], "maintainers": [ "Dan Clemens Posch" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Sheet-number obstruction theory for the plane Jacobian Conjecture: campaign theory bundle v1", "id": "doi:10.5281/zenodo.22002825", "authors": [ "Dan Clemens Posch" ], "type": "working paper (archived campaign theory bundle)", "relationship": "formalizes", "endorsement": "participated" }, { "title": "Artifact for: A Vertex-Gap Obstruction for Low-Degree Strip Pairs in the Plane Jacobian Conjecture", "id": "doi:10.5281/zenodo.21894922", "authors": [ "Dan Clemens Posch" ], "type": "software", "relationship": "background", "endorsement": "participated" }, { "title": "An application of Newton-Puiseux charts to the Jacobian problem", "id": "", "authors": [ "Henryk Zoladek" ], "type": "article (topology 47 (2008), no. 6, 431-469)", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Nonzero constant Wronskians of polynomials and Laurent polynomials, and geometric consequences", "id": "arXiv:2410.18867", "authors": [ "Carlos Hermoso", "Juan Gerardo Alcázar" ], "type": "article (arxiv preprint, 2024)", "relationship": "independently-proves", "endorsement": "not-contacted" } ], "related": [], "classification": { "arxiv": [ "math.AG", "math.AC" ], "msc2020": [ "14R15" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Claude Fable 5 (Anthropic)" ], "framework": "Claude Code (Claude Agent SDK)" }, { "method": "manual", "models": [], "framework": "n/a" } ], "strongest": "agent", "models": [ "Claude Fable 5 (Anthropic)" ], "spend": "subscription-based", "notes": "Proofs were drafted by Claude (Fable) agents in a human-directed research campaign; the informal theorem and proof had previously passed campaign-internal adversarial review, including independent AI re-derivations and exact-arithmetic machine checks of the proof's two computational identities. This formalization is machine-checked; a campaign-internal semantic-fidelity review by an independent AI reviewer compared both Lean statements with the source (verdict: faithful) and its attribution corrections are incorporated. The human author reviewed the statements and takes responsibility." }, "scope": "Fully formalized and proved: TheoremA (characteristic zero, over any integral domain) and TheoremA_charP (characteristic p under nu*deg A < p and deg C < p, a standalone strengthening extracted from the source's proof and characteristic remark). Not formalized: everything else in the campaign theory - the strip-reduction combinatorics, the block-to-ODE bridge (label lem:bridge, Lemma 6.3 in the current compilation), the log-residue functional and the block-variety theorem (label thm:R, Theorem 6.5 in the current compilation), and the (72,108) application. Degrees are stated with Polynomial.nat", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "(1) TheoremA is stated over an arbitrary integral domain of characteristic zero; the source states a field. The formalized statement is stronger and specializes to the source's. (2) The kernel subtraction clears denominators (D = lc(A)^nu * C - lc(C) * A^nu instead of the source's C - (lc(C)/lc(A)^nu) * A^nu), which is why no field structure is needed. (3) TheoremA_charP is a standalone strengthening extracted from the Section 6 proof and positive-characteristic remark; the source records only the conservative block-level bound p > (k+1)*d2 and does not display the standalone statement. In the source's application (nu = k, deg A <= d2, deg C <= k*d2) that bound implies the two formalized hypotheses nu*deg A < p and deg C < p. No other divergences known.", "alignment": true, "original": true, "review": { "status": "self-assessed plus campaign-internal ai fidelity review; statements human-audited against the source; not yet externally peer-reviewed", "bucket": "self-assessed", "reviewers": [ "Dan Clemens Posch" ], "notes": "The human author audited Challenge.lean against Theorem A (label thm:ode, \"ODE rigidity, all weights\") of the campaign paper and MATHIEU.md Section 5.1 of the archived bundle. A campaign-internal semantic-fidelity review by an independent AI reviewer compared both Lean statements with the source, returned verdict \"faithful\" (including the integral-domain strengthening and the extracted characteristic-p statement), and flagged attribution defects that are corrected in this version. The informal proof was adversarially reviewed inside the campaign by independent AI systems with independently written verification code; the Lean proofs here were produced afterwards and are kernel-checked. No external human peer review yet." }, "canonical": { "repo": "dcposch/jc2-lean", "directory": "theorem-a" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-19-000005", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-19-000005", "commit": "6c0f56309226432afb90b6213638ec987d46f4a3", "trust": "high", "theorems": 2, "date": "2026-08-19" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "dcposch/jc2-lean/vertex-gap/formalization.yaml", "repo": "dcposch/jc2-lean", "path": "vertex-gap/formalization.yaml", "directory": "vertex-gap", "origins": [ "palomar", "search" ], "url": "https://github.com/dcposch/jc2-lean/blob/HEAD/vertex-gap/formalization.yaml", "version": "v0.4", "missing": [], "name": "jc2-vertex-gap", "description": "Lean 4 / Mathlib formalization of the vertex-gap obstruction at (k,d2) = (2,2) (label thm:22, Theorem 3.4 in the current compilation) from the campaign paper \"A Vertex-Gap Obstruction for Low-Degree Strip Pairs in the Plane Jacobian Conjecture\": for polynomials P, Q in K[x,y] over a field K of characteristic not in {2,3,5}, supported in the normalized type-(2,2) Newton strips along (1,2) with [P,Q] = x^2, the block coefficients satisfy a3 = 0 and a2^2*a6 = 0 - so the generic chart a2*a6 != 0 of the strip family is empty (VertexGap22, plus the side-symmetric companion VertexGap22_swapped in the", "authors": [ "Dan Clemens Posch" ], "maintainers": [ "Dan Clemens Posch" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "A Vertex-Gap Obstruction for Low-Degree Strip Pairs in the Plane Jacobian Conjecture (campaign paper, archived with its reproduction artifact \"Artifact for: A Vertex-Gap Obstruction for Low-Degree Strip Pairs in the Plane Jacobian Conjecture\")", "id": "doi:10.5281/zenodo.21894922", "authors": [ "Dan Clemens Posch" ], "type": "working paper (archived with the campaign reproduction artifact)", "relationship": "formalizes", "endorsement": "participated" }, { "title": "Sheet-number obstruction theory for the plane Jacobian Conjecture: campaign theory bundle v1", "id": "doi:10.5281/zenodo.22002825", "authors": [ "Dan Clemens Posch" ], "type": "working paper (archived campaign theory bundle)", "relationship": "background", "endorsement": "participated" }, { "title": "Increasing the degree of a possible counterexample to the Jacobian conjecture from 100 to 108", "id": "arXiv:2204.14178v1", "authors": [ "J. A. Guccione", "J. J. Guccione", "R. Horruitiner", "C. Valqui" ], "type": "article (arxiv preprint, 2022, unrefereed; version pinned at v1)", "relationship": "background", "endorsement": "not-contacted" } ], "related": [ { "id": "PALOMAR-2026-08-19-000005", "relationship": "sibling project from the same campaign and repository", "note": "Sibling Palomar project (theorem-a/ in the same repository): Theorem A (ODE rigidity), the rigidity engine of the depth-two block analysis of the same campaign, pinned to the same Lean/Mathlib toolchain." } ], "classification": { "arxiv": [ "math.AG", "math.AC" ], "msc2020": [ "14R15" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Claude Fable 5 (Anthropic)" ], "framework": "Claude Code (Claude Agent SDK)" }, { "method": "manual", "models": [], "framework": "n/a" } ], "strongest": "agent", "models": [ "Claude Fable 5 (Anthropic)" ], "spend": "subscription-based", "notes": "Proofs were drafted by a Claude (Fable) agent in a human-directed research campaign; the informal theorem and proof had previously passed campaign-internal adversarial review (independent AI re-derivations, exact-arithmetic replays, and the two corrections - side-symmetry and radical membership - now incorporated in the paper). This formalization is machine-checked; the human author reviewed the statements and takes responsibility." }, "scope": "Fully formalized and proved: the (2,2) vertex-gap obstruction (VertexGap22 and its side-symmetric companion VertexGap22_swapped, both at polynomial level with the Jacobian bracket hypothesis [P,Q] = x^2 stated via MvPolynomial.pderiv), the corner-enumeration lemma (CornerEnumeration = lem:enum), and the side-symmetric gap condition (GapConditionSideSymm = hypothesis (ii) per the erratum). Inside the proof, the vertex equation, the gap kill, and all nine block keys of the (2,2) cell are derived from the bracket, and the paper's elimination cascade (cancellations C1/C2, the rigidity event a3 = 0", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "(1) Labels: the paper states Theorem 3.4 after normalizing the labels of the pair (Lemma 2.2 plus the swap (P,Q) -> (Q,-P)); VertexGap22 is the normalized-label statement and VertexGap22_swapped the mirrored one, so the pair of theorems is the side-symmetric content of the printed theorem. (2) The Newton-polygon hypotheses (S1)-(S3) and (i) of the source are rendered as explicit lattice-support inequalities: P in the strip 0 <= 2i-j <= 2, Q in the strip 0 <= 2i-j <= 3 together with i <= 2j. The last inequality (support of the wide member on or above the line through (0,0) and its bottom corner (2,1)) is carried in the paper by the polygon shape - \"the points of a gap column lie strictly above the bottom edge\" of N(Q), whose bottom edge runs from (0,0) to (k,1) - rather than by a displayed inequality; it is load-bearing for the gap kill and is made explicit here. (3) The saturation clause of (S2) (nonvanishing bottom-corner coefficients) and the uniqueness clause of (i) are not assumed: with the support inequalities they are consequences (the vertex equation a_{(1,0)}*b_{(2,1)} = 1 is derived from the bracket), so the formalized statement is harmlessly stronger. Strip lengths are un", "alignment": true, "original": false, "review": { "status": "self-assessed; statements audited against the source by the human author; the informal theorem previously passed campaign-internal adversarial review (independent ai re-derivations and exact-arithmetic replays); not yet externally peer-reviewed", "bucket": "self-assessed", "reviewers": [ "Dan Clemens Posch" ], "notes": "The key equations and the cascade formalized here were re-checked during formalization by an independent random-evaluation enumeration over a large prime field (all twelve keys and the identically-zero top stratum matched the bracket coefficients; the cascade's consistency and both obstruction events reproduced). The Lean proofs are kernel-checked; kernel-only (no native_decide), axioms exactly propext, Classical.choice, Quot.sound." }, "canonical": { "repo": "dcposch/jc2-lean", "directory": "vertex-gap" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-20-000001", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-20-000001", "commit": "3cee8100665bc59734511500b98728628ea190db", "trust": "high", "theorems": 4, "date": "2026-08-20" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "dcposch/kahn-kalai-lean/formalization.yaml", "repo": "dcposch/kahn-kalai-lean", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/dcposch/kahn-kalai-lean/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "kahn-kalai", "description": "Lean 4 / Mathlib formalization of the Park–Pham theorem (Kahn–Kalai expectation threshold) following Tran–Vu’s one-page inductive covering argument: for an ℓ-bounded family F of subsets of a finite ground set, p_c(F) = O(q(F) log ℓ). The compared statements are Tran–Vu’s strengthened covering theorem (EJC 31(3) #P3.2 (2024), Theorem 2.3, constant L = 1000) and the Park–Pham / Kahn–Kalai bound (their Theorem 1.1).", "authors": [ "Dan Clemens Posch" ], "maintainers": [ "Dan Clemens Posch" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "A Short Proof of Kahn-Kalai Conjecture", "id": "doi:10.37236/12266", "authors": [ "Phuc Tran", "Van Vu" ], "type": "article", "relationship": "independently-proves", "endorsement": "not-contacted" }, { "title": "A Proof of the Kahn-Kalai Conjecture", "id": "arXiv:2203.17207", "authors": [ "Jinyoung Park", "Huy Tuan Pham" ], "type": "article", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Thresholds versus expectation-thresholds", "id": "", "authors": [ "Jeff Kahn", "Gil Kalai" ], "type": "article", "relationship": "background", "endorsement": "n/a" } ], "related": [], "classification": { "arxiv": [ "math.CO", "math.PR" ], "msc2020": [ "05C80", "60C05" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "grok-4.6" ], "framework": "Grok Build TUI" } ], "strongest": "agent", "models": [ "grok-4.6" ], "spend": "not tracked", "notes": "Coordinator scaffolded the Palomar project; proof lanes fill covering_theorem then park_pham. Scratch files are verified with scripts/lean_verify.sh --file FILE --receipt-only, patterned on jc2-lean/max11-partial-y/scripts/box_lean_verify.sh." }, "scope": "Compared: Tran–Vu Theorem 2.3 (covering_theorem, L = 1000, log base 2) and Theorem 1.1 (park_pham, existential K). Not compared: the L ≈ 3.998 optimisation, Bell’s ε-covering theorem, and §§3–4. Ground sets are finite types; families are Finsets of Finsets.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "Logarithm is Real.logb 2 (Tran–Vu’s base-2 convention). Thresholds are sInf/sSup rather than “the unique p with equality 1/2”. covering_theorem fixes L = 1000 rather than “sufficiently large L”. park_pham uses Type (universe 0) rather than Type*.", "alignment": true, "original": true, "review": { "status": "self-assessed; in progress", "bucket": "self-assessed", "reviewers": [ "none" ], "notes": "Statement surface follows Tran–Vu Theorems 1.1 and 2.3." }, "canonical": { "repo": "dcposch/kahn-kalai-lean", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "dcposch/kakeya-lean/formalization.yaml", "repo": "dcposch/kakeya-lean", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/dcposch/kakeya-lean/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Finite-field Kakeya (Dvir)", "description": "A Palomar thin wrapper around Math Inc’s Lean 4 formalization of Dvir’s finite-field Kakeya theorem: for each dimension n there is a positive constant C, depending only on n, such that every Kakeya set K in F^n, with F a finite field of cardinality q, satisfies |K| ≥ C · q^n. A Kakeya set is a subset that contains an affine line in every direction. The compared declaration is Kakeya.kakeya_set_bound; the proof is discharged from KakeyaFiniteFields.kakeya_set_bound in math-inc/KakeyaFiniteFields.", "authors": [ "Dan Clemens Posch" ], "maintainers": [ "Dan Clemens Posch" ], "license": "Apache-2.0", "role": "thin-wrapper", "substantive": "math-inc/KakeyaFiniteFields", "sources": [ { "title": "On the size of Kakeya sets in finite fields", "id": "doi:10.1090/S0894-0347-08-00607-3", "authors": [ "Zeev Dvir" ], "type": "article", "relationship": "adapts", "endorsement": "not-contacted" }, { "title": "Kakeya sets in finite fields", "id": "https://github.com/math-inc/KakeyaFiniteFields", "authors": [], "type": "formalization", "relationship": "other", "endorsement": "not-contacted" } ], "related": [ { "id": "https://github.com/math-inc/KakeyaFiniteFields", "relationship": "adapts", "note": "This repository is a Palomar Challenge/Solution wrapper around that development at the pinned commit in repository.substantive_formalization." } ], "classification": { "arxiv": [ "math.CO", "math.CA" ], "msc2020": [ "52C17", "05B25" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "grok-4.6" ], "framework": "Grok Build TUI" } ], "strongest": "agent", "models": [ "grok-4.6" ], "spend": "not tracked", "notes": "This repository is a thin Comparator wrapper. The mathematical proof is the Math Inc / Gauss formalization in math-inc/KakeyaFiniteFields, produced from a LaTeX blueprint. The wrapper restates IsKakeyaSet and kakeya_set_bound in namespace Kakeya and discharges the theorem from KakeyaFiniteFields.kakeya_set_bound." }, "scope": "Compared: Kakeya.kakeya_set_bound, the existential form of Dvir’s bound |K| ≥ C_n · q^n for Kakeya sets in F^n over an arbitrary finite field F. Not compared: the explicit constant C_n = 1/n! and the binomial bound |K| ≥ binom(q+n−1, n) used internally; (δ,γ)-Kakeya sets; the Cartesian product argument for |K| ≥ C_{n,ε} q^{n−ε}; and later improvements by the method of multiplicities.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The compared statement is Dvir’s existential C_n form, not the explicit binomial lower bound proved in the source. Directions in IsKakeyaSet range over all vectors in F^n, including zero; the zero direction only requires K nonempty, which already follows from the existence of a line in a nonzero direction, so this matches the usual definition. Ambient spaces are Fin n → F with F any Field + Fintype, rather than a constructed F_q.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "none" ], "notes": "Statement surface copied from KakeyaFiniteFields.kakeya_set_bound / IsKakeyaSet and checked against Dvir, JAMS 22 (2009). No external referee review of the wrapper." }, "canonical": { "repo": "dcposch/kakeya-lean", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "dcposch/mason-flats-lean/formalization.yaml", "repo": "dcposch/mason-flats-lean", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/dcposch/mason-flats-lean/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "mason-flats", "description": "Lean 4 / Mathlib certificate that Mason’s 1972 conjecture on log-concavity of Whitney numbers of the second kind is false: Larson’s rank-76 dual of the rank-3 parallel extension of U_{3,4} with parallel classes of sizes 1, 26, 26, 26 satisfies W_74^2 < W_73 W_75. Flats are counted as cyclic sets in the dual; the closed forms are binomial identities. White’s conjecture from the same paper is not compared.", "authors": [ "Dan Clemens Posch" ], "maintainers": [ "Dan Clemens Posch" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Counterexamples to two conjectures about matroids", "id": "arXiv:2607.02208", "authors": [ "Matt Larson" ], "type": "article", "relationship": "independently-proves", "endorsement": "not-contacted" }, { "title": "Matroids: unimodal conjectures and Motzkin’s theorem", "id": "", "authors": [ "John H. Mason" ], "type": "article", "relationship": "background", "endorsement": "n/a" } ], "related": [], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05B35", "05C31" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "grok-4.6" ], "framework": "Grok Build TUI" } ], "strongest": "agent", "models": [ "grok-4.6" ], "spend": "not tracked", "notes": "Palomar certificate of Larson’s counterexample. Cyclic-set classification on the rank-3 dual plus binomial arithmetic; no enumeration of rank-76 flats." }, "scope": "Compared: MasonFlats.mason_log_concave_false, existence of a rank-76 matroid on 79 elements with W_74^2 < W_73 W_75. Not compared: White’s conjecture, unimodality of Whitney numbers, or ultra-log-concavity.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The compared matroid is constructed as the dual of an explicit rank-3 parallel extension of U_{3,4} on Fin 79, which is the dual of the graphic matroid of Larson’s generalized theta graph. Graphic-matroid API is not used; the dual identification of Whitney numbers with cyclic-set counts is proved.", "alignment": true, "original": true, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "none" ], "notes": "Statement surface follows Larson Theorem 2.3: W_74^2 < W_73 W_75." }, "canonical": { "repo": "dcposch/mason-flats-lean", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "dcposch/sharkovsky-lean/formalization.yaml", "repo": "dcposch/sharkovsky-lean", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/dcposch/sharkovsky-lean/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "sharkovsky", "description": "Lean 4 / Mathlib proof that a continuous self-map of an interval with a point of least period 3 has a point of least period n for every n ≥ 1. This is the periodic-point half of Li–Yorke (1975) and the period-3 case of Sharkovsky (1964). The compared statement is Sharkovsky.period_three_implies_all_periods.", "authors": [ "Dan Clemens Posch" ], "maintainers": [ "Dan Clemens Posch" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Period Three Implies Chaos", "id": "doi:10.2307/2318254", "authors": [ "Tien-Yien Li", "James A. Yorke" ], "type": "article", "relationship": "independently-proves", "endorsement": "not-contacted" }, { "title": "Co-existence of cycles of a continuous mapping of the line into itself", "id": "", "authors": [ "O. M. Sharkovsky" ], "type": "article", "relationship": "background", "endorsement": "n/a" } ], "related": [ { "id": "https://github.com/b-mehta/sharkovsky", "relationship": "independent", "note": "Mehta, CPP 2023, Lean 3 formalisation of the full Sharkovsky forcing and realization theorems. This project is a Lean 4 / Mathlib proof of the period-3 case only." } ], "classification": { "arxiv": [ "math.DS" ], "msc2020": [ "37E05", "26A18" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "grok-4.6" ], "framework": "Grok Build TUI" } ], "strongest": "agent", "models": [ "grok-4.6" ], "spend": "not tracked", "notes": "Coordinator specified wrap-if-Lean-4-pin-else-prove. Mehta’s CPP 2023 development is Lean 3; Mathlib Dynamics has periodic points but not Sharkovsky. This lane proved the period-3 theorem against Mathlib." }, "scope": "Compared: Sharkovsky.period_three_implies_all_periods, for a continuous self-map of an order-connected set I ⊆ ℝ with a point of least period 3. Not compared: the full Sharkovsky order, the realization theorem, or the Li–Yorke scrambled set.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "Stated for ContinuousOn a MapsTo-invariant OrdConnected set rather than a closed bounded interval; the classical Icc case is included. Least period is Function.minimalPeriod. The scrambled-set half of Li–Yorke and the rest of the Sharkovsky order are omitted.", "alignment": true, "original": true, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "none" ], "notes": "Statement surface follows Li–Yorke’s periodic-point theorem." }, "canonical": { "repo": "dcposch/sharkovsky-lean", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "DhyeyMavani2003/chip-firing-with-lean/formalization.yaml", "repo": "DhyeyMavani2003/chip-firing-with-lean", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/DhyeyMavani2003/chip-firing-with-lean/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Chip-Firing with Lean 4", "description": "A Lean 4 / Mathlib formalization of Baker and Norine's Riemann-Roch theorem for finite graphs. Chip configurations on a finite graph, modulo chip-firing moves, form an abelian group analogous to the group of line bundles on a compact Riemann surface. Each class can be assigned a rank, analogous to the dimension of a linear series. These ranks then satisfy a formula of the form r(D) - r(K-D) = deg D + 1 - g, in exact analogy to Riemann surfaces. This formula is formalized, along with the analog of Clifford's theorem for finite graphs.", "authors": [ "Dhyey Dharmendrakumar Mavani", "Nathan Pflueger" ], "maintainers": [ "Dhyey Dharmendrakumar Mavani", "Nathan Pflueger" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Riemann-Roch and Abel-Jacobi theory on a finite graph", "id": "https://doi.org/10.1016/j.aim.2007.04.012", "authors": [ "Matthew Baker", "Serguei Norine" ], "type": "article", "relationship": "formalizes", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05C57", "14T20" ] }, "automation": { "methods": [ { "method": "manual", "models": [], "framework": "" }, { "method": "copilot", "models": [ "not tracked" ], "framework": "n/a" }, { "method": "agent", "models": [ "not tracked" ], "framework": "n/a" } ], "strongest": "agent", "models": [ "not tracked" ], "spend": "not tracked", "notes": "Initial development was manual with autocomplete from GitHub Copilot. Agents were used in later phases for tasks including refactoring, simplifying proofs, debugging after version bumps, and writing documentation." }, "scope": "The Riemann-Roch formula, Theorem 1.12 in the original paper, is formalized, along with Corollary 3.5 (Clifford's theorem). These statements are presented using the predicate rank_eq rather than the rank function itself. The remaining results of the paper, most notably those concerning the Abel-Jacobi map and its implications, are not formalized here.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The statement module represents the source's rank function with the relational predicate rank_eq, and therefore states Riemann--Roch and Clifford with explicit rank witnesses. No other mathematical divergences are known.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Nathan Pflueger" ], "notes": "The definitions were compared against the definitions as stated in Baker--Norine (2007), and the theorem statements were compared against Theorem 1.12 and Corollary 3.5. The docstrings detail the precise comparison of Lean statements to wording in the paper. No independent external review was conducted." }, "canonical": { "repo": "dhyeymavani2003/chip-firing-with-lean", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-02-000013", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-02-000013", "commit": "5b422d855c02af60591121ba7c11b5e8242542b8", "trust": "high", "theorems": 2, "date": "2026-09-02" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "DiarHaidary/Improved-SparseStack-Constants/formalization.yaml", "repo": "DiarHaidary/Improved-SparseStack-Constants", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/DiarHaidary/Improved-SparseStack-Constants/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Improved SparseStack Constants", "description": "Complete Lean proof of the revised SparseStack main theorem through its retained-count route. For every fixed real isometry, the actual independent uniform signed-bucket sketch with the rounded C=80, B=169/20 prescription has Gram-norm failure probability at most delta. The same prescription satisfies s<26.35 q/eps and m<176.9(d+q)/eps^2. All physical channel estimates and the all-order scalar certificate are proved.", "authors": [ "Diar Heidary" ], "maintainers": [ "Diar Heidary" ], "license": "MIT", "role": "substantive-development", "substantive": "", "sources": [ { "title": "SparseStack Is an Optimal Oblivious Subspace Embedding, Version 3.0, retained-count revision", "id": "", "authors": [ "Diar Heidary" ], "type": "preprint", "relationship": "formalizes", "endorsement": "participated" }, { "title": "SparseStack hybrid constants template, updated retained-count manuscript", "id": "", "authors": [ "Diar Heidary" ], "type": "unpublished proof note", "relationship": "adapts", "endorsement": "participated" } ], "related": [ { "id": "https://github.com/DiarHaidary/Nelson-Nguyen-conjecture-upper-bound-palomar", "relationship": "builds-on", "note": "The V3 project vendors the author's V2 finite sample law, Parseval frame, Gram-error and matrix-tail modules under SparseStackV3/Base, retaining their original SparseFock namespaces and MIT provenance." } ], "classification": { "arxiv": [ "math.PR", "math.NA" ], "msc2020": [ "60B20", "68W20", "65F55" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "GPT-6" ], "framework": "OpenAI Codex" } ], "strongest": "agent", "models": [ "GPT-6" ], "spend": "not tracked", "notes": "The author supplied the mathematical source and existing formalization. This completion was AI assisted. Earlier development provenance remains in the source headers and README; its complete tool-use history is unavailable." }, "scope": "The revised real SparseStack main theorem and its retained-count route are complete. The final library theorem assumes only a fixed real isometry, d>=1, delta>0, and 00. The independent submission statement uses the strict event and the usual range 0 0. Some intermediate estimates use coarser constants and explicit finite counting; no sharp leading constant is claimed. Other claims in that report, including growing-moment, tail, almost-sure, and asymptotic limited-independence statements, are outside this formalization's compared scope. No additional unproved estimate hypotheses are added to the main theorem. Source reports retain historical remarks about the then-unfinished formalization; current verification records are separate.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "Internal agent reviews checked the graph model and Challenge statements; these are documented in docs/challenge-review.md and docs/challenge-signature-review.md. The local Lean build and endpoint audit passed. The all-theorem audit recorded 2,565 theorem declarations using only propext, Classical.choice, and Quot.sound. Saved diagnostic comparisons of Challenge and Solution are not the official Comparator check. The local Comparator/NanoDa runner was blocked before starting because its Linux host lacked the required Landlock ABI. GitHub Actions results must be inspected separately. No completed Palomar verification, independent-kernel replay, registration, or external human peer review is claimed by this metadata." }, "canonical": { "repo": "diarhaidary/sharp-bounds-for-graph-matrices", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-08-000005", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-08-000005", "commit": "01994362f8543743d556e7544d2538e93c62c02b", "trust": "high", "theorems": 3, "date": "2026-09-08" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "DiarHaidary/Spectral-norms-of-independent-entries-with-regular-moment-growth/formalization.yaml", "repo": "DiarHaidary/Spectral-norms-of-independent-entries-with-regular-moment-growth", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/DiarHaidary/Spectral-norms-of-independent-entries-with-regular-moment-growth/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "MI-32: Lean formalization of the parity and positive-polynomial upper bound", "description": "A Lean 4 formalization of the MI-32 upper bound, that is, of the Latała-Świątkowski conjecture that for every regularity parameter alpha >= 1 there is a constant C > 0, independent of the dimension and of the entry laws, with E||X|| <= C (M(X) + D(X)) for every square random matrix X with independent mean-zero real entries whose absolute moments satisfy ||X_ij||_{L_2r} <= alpha ||X_ij||_{L_r} at every real order r >= 1. Here ||.|| is the Euclidean operator norm, M(X) is the sum of the largest row and column standard deviations, and D(X) is the max-min weak moment at order log(k+1) taken over o", "authors": [ "Diar Heidary" ], "maintainers": [ "Diar Heidary" ], "license": "MIT", "role": "", "substantive": "", "sources": [ { "title": "MI-32 - Spectral norms of independent entries with regular moment growth, in Open Problems in Numerical Linear Algebra", "id": "https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/d7fcfe7bfe8c02cd98d9463fa5a8ba12850d5bdb/matrix-inequalities-and-norms/MI-32/README.md", "authors": [], "type": "web discussion", "relationship": "formalizes", "endorsement": "" }, { "title": "Norms of randomized circulant matrices", "id": "https://doi.org/10.1214/22-EJP799", "authors": [ "Rafał Latała", "Witold Świątkowski" ], "type": "paper", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "MI-32: a proof through parity counts and positive walk polynomials", "id": "", "authors": [ "Diar Heidary" ], "type": "other", "relationship": "formalizes", "endorsement": "" }, { "title": "Comparison of weak and strong moments for vectors with independent coordinates", "id": "https://arxiv.org/abs/1612.02407", "authors": [ "Rafał Latała", "Marta Strzelecka" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Dimension-free bounds and structured results in random matrix theory", "id": "https://web.math.princeton.edu/~rvan/dimfr180821.pdf", "authors": [ "Rafał Latała", "Ramon van Handel", "Pierre Youssef" ], "type": "paper", "relationship": "background", "endorsement": "" } ], "related": [ { "id": "https://github.com/DiarHaidary/Sharp-Bounds-for-Graph-Matrices/tree/01994362f8543743d556e7544d2538e93c62c02b", "relationship": "builds-on", "note": "Earlier Lean development by the same author, cited at the exact public commit DiarHaidary/Sharp-Bounds-for-Graph-Matrices@01994362f8543743d556e7544d2538e93c62c02b (MIT). It is a prior formalization reused as a Lean dependency, not a mathematical source: its own subject, sharp norm bounds for fixed-shape graph matrices, is unrelated to MI-32 and none of its graph-matrix results are reused or re-cla" } ], "classification": { "arxiv": [ "math.PR", "math.FA" ], "msc2020": [ "60B20", "60E15", "47A30" ] }, "automation": { "methods": [ { "method": "agentic", "models": [ "GPT-6 / Codex (earlier sessions)", "Claude Opus 5 (deletion chain)" ], "framework": "OpenAI Codex; Anthropic Claude Code" } ], "strongest": "other", "models": [ "Claude Opus 5 (deletion chain)", "GPT-6 / Codex (earlier sessions)" ], "spend": "Not tracked", "notes": "Human direction and responsibility are not a claim of manual authorship or independent human verification of every mathematical or Lean argument." }, "scope": "The selected theorem MI32.main_upper is proved with no proof gap and no added axiom. Solution imports no Challenge. Challenge.lean retains the deliberate Challenge hole by design.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The exact statement of the MI-32 problem is preserved verbatim in MI32/Statement.lean and Challenge.lean and is what is proved; no finite-law substitute, conditional comparison or assumed contraction is used, and no estimate is taken as an input hypothesis or a custom axiom. Two deliberate divergences from the informal draft's deletion argument: (1) the draft's far-remainder step invokes a variance-envelope consequence of Latała-van Handel-Youssef Theorem 4.4, whereas the Lean proof raises the per-coordinate screening threshold from b^2 to b^3 and uses an elementary centered-Gram second-moment estimate instead, so no general matrix fourth-moment inequality is needed; (2) the draft routes the deletion argument through the matrix-symmetric dilation H_Y, whose reflected entries repeat variables, whereas the Lean proof performs the whole decomposition on the original rectangular matrix with one shared index family on both axes, so every entry family used is genuinely independent and the paired dilation budget is not needed. Full details in docs/SOURCE_FIDELITY.md and docs/REMAINING_CHAIN_SPEC.md.", "alignment": true, "original": false, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [ "Codex statement-fidelity agent: independent read-only audit of the exact target, 14 September 2026" ], "notes": "The statement-fidelity audit found no material mismatch with the MI-32 problem statement. It is not independent review of the analytic proof. Palomar registered commit 762bd5ec5050a96f5e6ba3926b6cda4816fcd4b0 as entry PALOMAR-2026-09-14-000006 version 1 on 2026-09-14, after mechanical verification on its Linux infrastructure and an AI editorial review with outcome neutral, meaning no blocking problem was identified. An earlier round had requested changes to classification and provenance metadata and to the Challenge documentation, made before registration. The registered review carried one warning, that the informal account misidentified LocalLogMoment.mean_spectralNorm_le_literal as a step in the selected proof; it is a companion corollary, and the README, PROOF_OBLIGATIONS and this descr" }, "canonical": { "repo": "diarhaidary/spectral-norms-of-independent-entries-with-regular-moment-growth", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-14-000006", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-14-000006", "commit": "762bd5ec5050a96f5e6ba3926b6cda4816fcd4b0", "trust": "high", "theorems": 1, "date": "2026-09-14" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "diegode/proof-of-space-lean/drsample/formalization.yaml", "repo": "diegode/proof-of-space-lean", "path": "drsample/formalization.yaml", "directory": "drsample", "origins": [ "search" ], "url": "https://github.com/diegode/proof-of-space-lean/blob/HEAD/drsample/formalization.yaml", "version": "v0.4", "missing": [], "name": "Block and fractional depth robustness of DRSample and BucketSample", "description": "Exponent-dependent block-depth robustness for the exact independent DRSample and BucketSample distributions, the sampler-avoidance lemma with any fixed product fraction beta < 1, the two deterministic appendix lemmas, and block-to-fractional conversion with two-sided fractional bounds for both samplers, Valiant depth reduction, balanced optimality, and the general conditional-label multiscale theorem. The deletion budget is Theta(epsilon n/L(n)), surviving depth is Omega(n/(log_2 n)^epsilon), and block width is Theta(L(n)/epsilon), for every fixed 0 < epsilon < 1 and all sufficiently large n. ", "authors": [ "Diego de Estrada (@diegode)" ], "maintainers": [ "Diego de Estrada (@diegode)" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Data-Independent Memory Hard Functions: New Attacks and Stronger Constructions", "id": "https://eprint.iacr.org/2018/944", "authors": [ "Jeremiah Blocki", "Benjamin Harsha", "Siteng Kang", "Seunghoon Lee", "Lu Xing", "Samson Zhou" ], "type": "paper", "relationship": "adapts", "endorsement": "not-contacted" }, { "title": "Practical Graphs for Optimal Side-Channel Resistant Memory-Hard Functions", "id": "https://doi.org/10.1145/3133956.3134031", "authors": [ "Joel Alwen", "Jeremiah Blocki", "Ben Harsha" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "On the Depth-Robustness and Cumulative Pebbling Cost of Argon2i", "id": "https://doi.org/10.1007/978-3-319-70500-2_15", "authors": [ "Jeremiah Blocki", "Samson Zhou" ], "type": "paper", "relationship": "adapts", "endorsement": "not-contacted" }, { "title": "Randomized Search Trees", "id": "https://doi.org/10.1007/BF01940876", "authors": [ "Raimund Seidel", "Cecilia R. Aragon" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" } ], "related": [], "classification": { "arxiv": [ "cs.CR", "math.CO" ], "msc2020": [ "94A60", "05C80", "68Q25" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "OpenAI Codex" ], "framework": "Codex" } ], "strongest": "agent", "models": [ "OpenAI Codex" ], "spend": "", "notes": "Exact costs and wall times were not tracked; no prompt transcript is included." }, "scope": "Solution exposes eleven declarations covering all ten required labeled results from the current paper. Only drsample_conjecture2 is registered in Challenge and Comparator. Supporting results prove arbitrary-fraction graph multiscale bounds, block lifting, parameter selection with eb >= beta n, and asymptotic domination of the samplers' deletion budgets by their certified depths. The general multiscale theorem allows unbounded dependent integer labels. The cited BRG pebbling corollary is out of scope and is not used. No unproved paper result is assumed in the proofs.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "ProofOfSpaceStatement.drsample_conjecture2", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "ProofOfSpaceStatement.sampler_avoidance", "file": "ProofOfSpace/DRSample/PaperAvoidance.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "ProofOfSpaceStatement.bucketSample_block_robustness", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "ProofOfSpaceStatement.block_to_fractional", "file": "ProofOfSpace/DRSample/Fractional.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "ProofOfSpaceStatement.drsample_fractional_robustness", "file": "ProofOfSpace/DRSample/PaperFractional.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "ProofOfSpaceStatement.bucketSample_fractional_robustness", "file": "ProofOfSpace/DRSample/PaperFractional.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "ProofOfSpaceStatement.dr_subsequence", "file": "ProofOfSpace/DRSample/PaperSubsequence.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "ProofOfSpaceStatement.dr_labels", "file": "ProofOfSpace/DRSample/GeneralLabels.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "ProofOfSpaceStatement.valiant_depth_reduction", "file": "ProofOfSpace/DRSample/Valiant.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "ProofOfSpaceStatement.balanced_robustness_optimality", "file": "ProofOfSpace/DRSample/PaperOptimality.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "ProofOfSpaceStatement.conditional_multiscale", "file": "ProofOfSpace/DRSample/ConditionalLabels.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The exact independent sampler distributions, integer rounding, vertex-count path lengths, and adversarial deletion events are preserved. Public sampler statements now quantify over every exponent between zero and one, using positive constants to witness the paper's orders. Their thresholds may depend on the exponent. The general avoidance constants depend only on beta. Fractional constants depend only on f < 1/2. A residue-class cover replaces the greedy procedure in Blocki--Zhou Theorem 4, accounts for truncated blocks, and proves both directions by reversing the same sampled graph. The proof retains integer blocks of size 20a and the existing triangular port geometry, with a proportional to L(n)/epsilon. The density threshold in the label proof is (q-1)/q for a suitable q depending on alpha; these internal choices differ from the appendix but yield its stated bounds. Valiant's bound uses binary edge classes. Balanced reduction uses k = ceil(log_2 ceil(log_2 n)) instead of ceil(log_2 L(n)), yielding the same asymptotic order. The conditional label theorem uses a PMF on the joint array of all labels and states positivity on its support; its failure exponent is explicitly 2 - log(3)", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "Lean checks all ten required results and their supporting proofs. The full Lake build and statement-surface check pass with the main theorem as the only challenge. Comparator verifies drsample_conjecture2, and both NanoDa and the Lean default kernel accept its exported proof. All eleven declarations are exposed through Solution and their axiom audit reports only permitted axioms. No independent human mathematical review is claimed. Solution and supporting proof modules contain no theorem placeholders." }, "canonical": { "repo": "diegode/proof-of-space-lean", "directory": "drsample" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "diegode/proof-of-space-lean/pebbling_latency/formalization.yaml", "repo": "diegode/proof-of-space-lean", "path": "pebbling_latency/formalization.yaml", "directory": "pebbling_latency", "origins": [ "search" ], "url": "https://github.com/diegode/proof-of-space-lean/blob/HEAD/pebbling_latency/formalization.yaml", "version": "v0.4", "missing": [], "name": "Formalized latency bounds for stacked proofs of space", "description": "A Lean 4 reference-trajectory latency theorem for stacked depth-robust graphs, with concave expansion, explicit repair accounting, exact integer source counting, and a finite Chung port-model union bound. Under explicit graph and security assumptions, the Filecoin specialization gives at least 0.205n path vertices at 17 layers for n at least 10000.", "authors": [ "Diego de Estrada (@diegode)" ], "maintainers": [ "Diego de Estrada (@diegode)" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Proofs of space with maximal hardness", "id": "https://dblp.org/rec/conf/focs/Reyzin24", "authors": [ "Leonid Reyzin" ], "type": "paper", "relationship": "adapts", "endorsement": "not-contacted" }, { "title": "Tight proofs of space and replication", "id": "https://doi.org/10.1007/978-3-030-17656-3_12", "authors": [ "Ben Fisch" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" } ], "related": [], "classification": { "arxiv": [ "cs.CR" ], "msc2020": [ "94A60", "68Q25" ] }, "automation": { "methods": [ { "method": "manual", "models": [], "framework": "n/a" }, { "method": "agent", "models": [ "OpenAI Codex" ], "framework": "Codex" } ], "strongest": "agent", "models": [ "OpenAI Codex" ], "spend": "not tracked", "notes": "The mathematical development is human-directed. Lean and the pinned Palomar Comparator and NanoDa toolchain check the submitted result." }, "scope": "The single registered statement is pebbling_latency. It proves ceil(alpha_pi*n)+(z-1)q path vertices from concave interval expansion and ordinary depth robustness, where q=min(ceil(alpha_pi*n),ceil(alpha_pi*n)+m+1-(ceil(pi*n)+s)). Let p=m/n, sigma=s/n, F=beta-delta, g=F(p)-p, and kappa=F(sigma)-a, with positive band floor a. The scale tau interpolates the free trajectory from a through its first crossing of p. Set D=tau(p)-tau(sigma) and D0=tau(p)-tau(min(p,zeta-delta)). The sufficient layer bound is ell > 1+D0+(z-1)*(1+D)+rho/g+max(0,(rho-kappa)/g+2). Assumptions include monotonicity and conc", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "ProofOfSpaceStatement.pebbling_latency", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The theorem concerns path latency in a static pebbling snapshot, using ordinary depth robustness and a nondecreasing concave interval expansion bound. It adapts the local advisor draft using a floor-based interpolated delay and a global repair correction; it does not assert the draft's exact classification theorem or optimal adversarial strategy. ADVISOR_ALIGNMENT.md records these corrections. Its probability model samples one uniform permutation of all 8n ports and reuses it across layers; no equivalence with deployed Feistel wiring is asserted. Within-layer depth robustness and the exact finite security inequality remain assumptions. No time-indexed cryptographic reduction is formalized. Lean depth increases away from the challenge, whereas the explanation uses Reyzin's levels decreasing toward the top of the stack.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Diego de Estrada" ], "notes": "No independent mathematical review is recorded. The full project builds with Lean 4, and the pinned Palomar Comparator toolchain checks the submitted Challenge theorem against Solution.lean, rejects nonstandard axioms and sorryAx in its proof, and replays it through NanoDa." }, "canonical": { "repo": "diegode/proof-of-space-lean", "directory": "pebbling_latency" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "dsm054/kasami_cyclic_additive/formalization.yaml", "repo": "dsm054/kasami_cyclic_additive", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/dsm054/kasami_cyclic_additive/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Carlet's Kasami cyclic-additive conjecture", "description": "A Lean formalization of the Kasami cyclic-additive conjecture introduced by Claude Carlet. For the derivative image of the Kasami monomial x^(4^k-2^k+1) with gcd(k,n)=1, over a finite field of characteristic two with 2^n elements, the formalized result gives the exact three-variable coefficient count 2^(2n-3) for every pair of distinct nonzero coefficients.", "authors": [ "D.S. McNeil", "Gábor P. Nagy", "Attila Vajda" ], "maintainers": [ "D.S. McNeil" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Componentwise APNness, Walsh uniformity of APN functions, and cyclic-additive difference sets", "id": "https://eprint.iacr.org/2017/528", "authors": [ "Claude Carlet" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "On the Sixth International Olympiad in Cryptography NSUCRYPTO", "id": "https://doi.org/10.1134/S1990478920040031", "authors": [ "Anastasiya Gorodilova", "Natalia Tokareva", "Sergey Agievich", "Claude Carlet", "Evgeny Gorkunov", "Valeria Idrisova", "Nikolay Kolomeec", "Alexander Kutsenko", "Roman Lebedev", "Svetla Nikova", "Alexey Oblaukhov", "Irina Pankratova", "Marina Pudovkina", "Vincent Rijmen", "Aleksei Udovenko" ], "type": "paper", "relationship": "background", "endorsement": "n/a" }, { "title": "On a conjecture on the Kasami APN function", "id": "https://arxiv.org/abs/2608.18584", "authors": [ "Gábor P. Nagy", "Attila Vajda" ], "type": "paper", "relationship": "background", "endorsement": "participated" }, { "title": "The weight enumerators for several classes of subcodes of the second order binary Reed-Muller codes", "id": "", "authors": [ "Tadao Kasami" ], "type": "paper", "relationship": "background", "endorsement": "n/a" }, { "title": "Permutation polynomials and translation planes of even order", "id": "https://doi.org/10.1515/advgeom-2011-050", "authors": [ "Ulrich Dempwolff", "Peter Müller" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "New cyclic difference sets with Singer parameters", "id": "https://doi.org/10.1016/j.ffa.2003.09.003", "authors": [ "J. F. Dillon", "Hans Dobbertin" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Jacobi-like sums and difference sets with Singer parameters", "id": "", "authors": [ "J. F. Dillon", "Neeraj Kashyap" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Infinite families of 3-designs from APN functions", "id": "https://doi.org/10.1002/jcd.21685", "authors": [ "Chunming Tang" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Proof of Carlet's Kasami cyclic-additive conjecture", "id": "https://github.com/dsm054/kasami_cyclic_additive", "authors": [ "D.S. McNeil", "Gábor P. Nagy", "Attila Vajda" ], "type": "original-proof", "relationship": "other", "endorsement": "participated" } ], "related": [ { "id": "https://github.com/nagygp/kasami-apn-ab/tree/a92ff580e68a328d65515a16041763bbcb67d1c6", "relationship": "independent", "note": "Prior Lean formalization by Gábor P. Nagy and Attila Vajda of classical Kasami APN and AB theory. It includes Dempwolff-Müller permutation machinery, the Artin-Schreier/MCM treatment of the Kasami derivative, Walsh-moment arguments, and Frobenius transport between complementary exponents. It does not formalize Carlet's cyclic-additive triple-count conjecture or the Fermat-cubic/isogeny completion " }, { "id": "https://github.com/nagygp/conjecture-on-kasami/tree/aeb05ee1efe0e3446e5fb1dfaba5a43f01720713", "relationship": "independent", "note": "Companion Lean formalization by Gábor P. Nagy and Attila Vajda of the conjecture on the Kasami APN function, including reductions, structure theorems, special exponent classes, and computational verification." } ], "classification": { "arxiv": [ "math.CO", "math.NT" ], "msc2020": [ "11T06", "11T71", "94A60" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "GPT-5.6 Sol (OpenAI)" ], "framework": "ChatGPT (OpenAI)" }, { "method": "agent", "models": [ "GPT-5.6 Luna (OpenAI)" ], "framework": "Codex (OpenAI)" }, { "method": "agent", "models": [ "Claude Opus 5 (Anthropic)" ], "framework": "Claude Code (Anthropic)" }, { "method": "agent", "models": [ "Aristotle (Harmonic)" ], "framework": "Aristotle (Harmonic)" }, { "method": "other", "models": [], "framework": "Python scripts in scripts/" } ], "strongest": "agent", "models": [ "Aristotle (Harmonic)", "Claude Opus 5 (Anthropic)", "GPT-5.6 Luna (OpenAI)", "GPT-5.6 Sol (OpenAI)" ], "spend": "subscription-based across all methods", "notes": "The formal proof was developed collaboratively with AI systems under human direction; all final proof terms are checked by Lean. The Palomar port follows the official Challenge/Solution and Comparator conventions." }, "scope": "The complete proof is present in this repository. The theorem statement and Comparator-facing definitions are fixed in Palomar form. The only sorry in the project is the deliberate placeholder in Challenge.lean, which Solution.lean discharges; #print axioms on KasamiCyclicAdditive.carlet_kasami_cyclic_additive reports only propext, Classical.choice and Quot.sound.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "KasamiCyclicAdditive.carlet_kasami_cyclic_additive", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "No known mathematical divergence from the exact coefficient-form conjecture. The Challenge statement imposes neither the proof normalization k < n nor separate hypotheses 1 <= k or 2 <= n. ParameterReduction proves that |K|=2^n together with distinct nonzero coefficients forces n >= 2; then gcd(k,n)=1 makes k % n a positive representative below n. The derivative image and coefficient triple count are proved invariant under k -> k % n, and KasamiCyclicAdditive.carlet_kasami_cyclic_additive_literature reduces the exact statement to the normalized core theorem. The Lean statement works over an arbitrary finite field K equipped with CharP K 2 and a cardinality witness |K|=2^n, rather than choosing a concrete model of GF(2^n); this is mathematically equivalent.", "alignment": true, "original": true, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [ "Aristotle (Harmonic)" ], "notes": "The formal development has undergone repeated statement, dependency, and axiom audits and successful Lean builds. An independently structured, Mathlib-only literature specification and a formal semantic bridge to the implementation are included. An independent Aristotle review rechecked statement fidelity and Challenge/Solution correspondence and successfully replayed the real Landrun Comparator; Comparator, NanoDa, and the Lean kernel accepted the solution using only the three permitted axioms. The Comparator-facing Solution environment imports only KasamiCyclicAdditive.Main, excluding Validation computations from that environment." }, "canonical": { "repo": "dsm054/kasami_cyclic_additive", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-04-000005", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-04-000005", "commit": "c8ab49ebd8e9e53f5aa7585f1d43acf1ff68d10f", "trust": "high", "theorems": 1, "date": "2026-09-04" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "ehrlich-b/composition-algebras/formalization.yaml", "repo": "ehrlich-b/composition-algebras", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/ehrlich-b/composition-algebras/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Euclidean composition algebras over the reals: Hurwitz's theorem in dimension and classification form, with the octonions and Cayley-Dickson doubling", "description": "Hurwitz's theorem (1898), classification form: a nontrivial finite-dimensional Euclidean composition algebra over R - a unital, not necessarily associative or commutative bilinear product together with a symmetric positive-definite form whose quadratic form is multiplicative - is isomorphic to R, C, the quaternions or the octonions, by a linear isomorphism preserving the unit, the product and the norm form. All four targets are constructed, with Degen's eight-square identity discharging the octonion composition law, so every branch of the conclusion is attained. Machine checked, not peer revie", "authors": [ "Bryan Ehrlich" ], "maintainers": [ "Bryan Ehrlich" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Uber die Composition der quadratischen Formen von beliebig vielen Variablen", "id": "", "authors": [ "Adolf Hurwitz" ], "type": "paper", "relationship": "adapts", "endorsement": "n/a" }, { "title": "The Octonions", "id": "", "authors": [ "John C. 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This development registers two classification theorems for such products on the Hermitian matrix algebras. On H_N(R), the Hermitian N x N real matrices with N >= 1, every sequential product is the Lueders product sqrt(a) b sqrt(a): there is no free parameter. On H_N(C) with N >= 3, every sequential product has the twist normal form a^(1/2+it) b a^(", "authors": [ "Bryan Ehrlich" ], "maintainers": [ "Bryan Ehrlich" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Sequential Products on Euclidean Jordan Algebras: Classification in Rank at Least Three and the Complex Qubit", "id": "", "authors": [ "Bryan Ehrlich" ], "type": "paper", "relationship": "formalizes", "endorsement": "participated" }, { "title": "Sequential product spaces are Jordan algebras", "id": "", "authors": [ "John van de Wetering" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Power-associative rings", "id": "", "authors": [ "A. A. 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Every notion in the final sentence is internal to the first-order", "authors": [ "Elliot Glazer" ], "maintainers": [ "Elliot Glazer" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Ordinal-definable families in Cohen, random, and collapse extensions, version 21", "id": "docs/paper/ordinal_definable_families_rewritten_v21.pdf", "authors": [ "Elliot Glazer" ], "type": "paper", "relationship": "adapts", "endorsement": "participated" }, { "title": "Definability up to isomorphism versus definability of an isomorphic copy", "id": "https://mathoverflow.net/questions/382009/definability-up-to-isomorphism-versus-definability-of-an-isomorphic-copy", "authors": [ "Joel David Hamkins" ], "type": "web discussion", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Cohen's model yet again", "id": "https://mathoverflow.net/questions/244971/cohens-model-yet-again", "authors": [ "Asaf Karagila" ], "type": "web discussion", "relationship": "background", "endorsement": "not-contacted" } ], "related": [ { "id": "https://github.com/elliotglazer/erdos501/tree/218d1c1e46f77d4db80e566d1721782e85b94a17", "relationship": "builds-on", "note": "Supplies the pinned Lean 4 Flypitch framework, Boolean-valued models, forcing infrastructure, and the Mathlib first-order ZFC statement." }, { "id": "https://github.com/flypitch/flypitch", "relationship": "builds-on", "note": "Han and van Doorn's formal proof of the independence of CH, providing the original first-order, Boolean-valued-model, forcing, and completeness architecture later ported to Lean 4." }, { "id": "https://github.com/leanprover-community/mathlib4/tree/355bc1e0ed1d36e49525121e1a280ca13a058a92", "relationship": "builds-on", "note": "Provides Lean's mathematical library and the trusted Challenge's first-order model-theory vocabulary." } ], "classification": { "arxiv": [ "math.LO" ], "msc2020": [ "03C62", "03E35", "03E40" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "OpenAI GPT-5.6 (Sol), GPT-6 (Astra), and Codex", "Anthropic Claude Fable 5 and Claude Opus 4.8" ], "framework": "Iterated mathematical-research and Lean-agent sessions" }, { "method": "manual", "models": [], "framework": "" } ], "strongest": "agent", "models": [ "Anthropic Claude Fable 5 and Claude Opus 4.8", "OpenAI GPT-5.6 (Sol), GPT-6 (Astra), and Codex" ], "spend": "subscription-based usage", "notes": "Project authorship and maintenance are assigned only to the human author. Agent contributions are disclosed here. The Lean kernel checks all proof terms; the comparator checks exact agreement between the Mathlib-only Challenge and the proof-bearing Solution." }, "scope": "Complete formalization of the consistency of a parameter-free definable isomorphism type of countably saturated arithmetic extensions of size aleph one with no ordinal-definable presentation, together with its ZFC-only MathOverflow 382009 consequence. The one-Cohen-real forcing statement related to MathOverflow 244971 is deliberately excluded.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "DefinableSaturated.consistent_with_ch", "file": "DefinableSaturated/Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "DefinableSaturated.definable_up_to_isomorphism_without_od_copy_consistent", "file": "DefinableSaturated/Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "(1) The strongest target asserts first-order satisfiability of ZFC + CH plus the saturated-arithmetic sentence; it is a relative consistency result, not a theorem that ZFC proves its own consistency. (2) The first-order theory uses Flypitch's ZFC axiomatization: extensionality, empty set, ordered pairs, union, power set, infinity, regularity, Zorn's lemma, and strong collection. (3) The Lean construction begins with a closed collapse before the Cohen factor, which removes any hypothesis about CH in Lean's ambient universe. (4) Ordinal definability applies to the packed five-field presentation on every set-sized domain; uniqueness is uniqueness up to arithmetic isomorphism.", "alignment": true, "original": false, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [ "Elliot Glazer (trusted-statement and mathematical review)", "OpenAI and Anthropic audit sessions (proof, dependency, and axiom audits)" ], "notes": "No independent human peer review has yet been performed. Mechanical review comprises a full build, target axiom audit, Challenge import/size audit, and Challenge/source synchronization audit. 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The translated Metamath proof enters through eleven literal Hailperin sentences, which are used internally as a finite-basis bridge and are proved deductively equivalent to standard NF.", "authors": [ "Elliot Glazer" ], "maintainers": [ "Elliot Glazer" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Metamath New Foundations Explorer basis (nf.mm)", "id": "https://github.com/metamath/set.mm/blob/c1814c4fbaf88d329d7d73ec1c7358647570e435/nf.mm", "authors": [], "type": "", "relationship": "adapts", "endorsement": "" }, { "title": "Project Metamath WPP extension and accepted strict-scope endpoint source", "id": "provenance/metamath/accepted-translator-input.mm", "authors": [], "type": "", "relationship": "formalizes", "endorsement": "" }, { "title": "New Foundations Refutes the Weak Partition Principle", "id": "archive/paper/NF_Refutes_WPP_Unconditional_Proof.pdf", "authors": [], "type": "paper", "relationship": "formalizes", "endorsement": "" }, { "title": "The Axiom of Choice in Quine's New Foundations for Mathematical Logic", "id": "https://doi.org/10.1073/pnas.39.9.972", "authors": [ "Ernst P. Specker" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "The Axiom of Choice and the Partition Principle from Dialectica Categories", "id": "https://doi.org/10.1093/jigpal/jzaa023", "authors": [ "Samuel G. da Silva" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "MathOverflow discussion of the partition principle in NF", "id": "https://mathoverflow.net/a/449571/109573", "authors": [ "Thomas Forster" ], "type": "", "relationship": "background", "endorsement": "" }, { "title": "Literature search for prior proofs of NF ⊢ ¬WPP", "id": "LITERATURE_SEARCH.md", "authors": [], "type": "", "relationship": "background", "endorsement": "" } ], "related": [ { "id": "https://github.com/metamath/set.mm", "relationship": "adapts", "note": "The Lean development transports a theorem from the Metamath NF database into an explicit first-order syntactic deduction judgment." } ], "classification": { "arxiv": [ "math.LO" ], "msc2020": [ "03E70", "03E25", "03B35" ] }, "automation": { "methods": [ { "method": "other", "models": [], "framework": "Project Metamath-to-Lean translator" }, { "method": "agent", "models": [ "OpenAI Codex (GPT-5)" ], "framework": "OpenAI Codex" } ], "strongest": "agent", "models": [ "OpenAI Codex (GPT-5)" ], "spend": "Not tracked", "notes": "A primary Codex agent performed the translation and packaging with parallel agent passes over the proof bridge, Challenge/Solution boundary, standard-NF equivalence layer, provenance, and Palomar release surface. The human fixed the mathematical scope, confirmed authorship, maintenance responsibility, and the repository licence, authorized public upload to elliotglazer/nf-not-wpp, and retains authority over Palomar submission. No independent human peer review is claimed." }, "scope": "The advertised result is `NFNotWPP.NF_proves_not_WPP`: syntactic derivability of not-WPP from standard first-order NF, where `NF` consists of extensionality and every instance of the full stratified-comprehension schema. The eleven literal Hailperin sentences are an internal translated finite-basis bridge, not the assumptions of the public theorem; the repository proves their full deductive equivalence with standard NF. This is not a consistency theorem or an assertion that NF has a model. A documented search of the publicly accessible literature, completed on 2026-08-26, located no independen", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "NFNotWPP.NF_proves_not_WPP", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The accepted Metamath input follows the content-addressed 19-stage repair chain recorded in provenance/METAMATH_PROVENANCE.md. Most stages only change scoping, labels, compression, or proof terms; df-lnwmap has one recorded parenthesization correction, and 23 proved helper theorems were inserted as proofs rather than assumptions. In the self-contained Challenge surface and its solution-side mirror, membership is a primitive atom instead of Flypitch's two-node relation application, and intrinsically scoped closed formulas are erased to Nat-indexed de Bruijn syntax. Both are structural presentation isomorphisms. The exact source ordered pair and WPP expansion are retained. The exact eleven literal Hailperin formulas remain as the internal target of the translated C18 proof, but they are not the public theory: Challenge defines standard `NF` as extensionality plus all certified stratified-comprehension instances. The development proves that standard NF, the repository's `HailperinNF`, and the exact `LiteralHailperinNF` presentation have the same models and formal consequences. The final proof replaces every translated finite-basis axiom leaf by a derivation from standard NF.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "The accepted generated endpoint passed local Lean trust-zero, dependency, and exact-three-axiom audits. The standard-NF model equivalence and `NF ⊢ ¬WPP` endpoint have also passed local module and axiom audits, reporting exactly propext, Classical.choice, and Quot.sound. The earlier literal-basis Challenge/Solution and its nominal WPP AST comparison passed locally, as did an empty-origin, diagnosed/resumed build of the translated source closure. Public CI run #10 completed the standard-NF proof graph in about 2 h 42 min without an OOM and reached Comparator. Comparator exposed import-sensitive elaboration in the formerly duplicated statement surface. Challenge remains self-contained with only Mathlib imports; Solution now imports a byte-identical proof-free mirror compiled before its proof" }, "canonical": { "repo": "elliotglazer/nf-not-wpp", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-27-000005", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-27-000005", "commit": "fa58a7a895a73422042e167391c78e9149979913", "trust": "high", "theorems": 1, "date": "2026-08-27" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "enaslund/sarkozy-lower-bound-0.758/formalization.yaml", "repo": "enaslund/sarkozy-lower-bound-0.758", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/enaslund/sarkozy-lower-bound-0.758/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "An interval-moment criterion for square-difference-free integer sets", "description": "We prove a general interval-moment criterion for square-difference-free integer sets and apply it to obtain an improved unconditional lower bound in Sarkozy's theorem, with exponent 0.75806746. If D(N) is the largest size of a subset of {1, ..., N} containing no two distinct elements whose difference is a square, then for every epsilon > 0 and all sufficiently large N we prove D(N) >= N^(0.75806746 - epsilon). The exact exponent is 37903373/50000000. The reusable criterion starts from finite digit sets in pairwise coprime square bases, with ordered intervals and width-moment inequalities. A st", "authors": [ "Eric Naslund" ], "maintainers": [ "Eric Naslund" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Square-difference-free sets of exponent 0.75806746: Ordered intervals, composite digit codes, and a binary recursion", "id": "", "authors": [ "Eric Naslund" ], "type": "paper", "relationship": "other", "endorsement": "" }, { "title": "A simpler construction of square-difference-free sets beyond exponent 0.758", "id": "", "authors": [ "Eric Naslund" ], "type": "paper", "relationship": "other", "endorsement": "" }, { "title": "Ranked residues, intervals, and the integer construction", "id": "", "authors": [], "type": "other", "relationship": "adapts", "endorsement": "" }, { "title": "Square-Difference-Free Sets beyond the Three-Quarter Barrier", "id": "https://arxiv.org/abs/2608.01325v1", "authors": [ "Dmitry Krachun" ], "type": "paper", "relationship": "adapts", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.NT", "math.CO" ], "msc2020": [ "11B75", "05D05" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "GPT-6 (Codex)" ], "framework": "Codex" } ], "strongest": "agent", "models": [ "GPT-6 (Codex)" ], "spend": "", "notes": "Eric Naslund directed the research and chose the formalization scope. Codex agents wrote and checked the Lean development and prepared these files. AI assistance is credited here; human review of every proof step is not asserted. The accompanying manuscripts were written by GPT-6-Astra under Eric Naslund's prompting and supervision. Their opening AI-reading paragraph was supplied verbatim by Eric Naslund; their author and contact are Eric Naslund, naslund.math@gmail.com." }, "scope": "Proved: the general construction and all-large-N criterion; all six prime-chain certificates and moments; all odd free-digit/CRT lifts; the full 25-state/94-branch binary geometry, all numerical growth rows and fixed-depth inequality; the complete actual 4913-point 215 and 19683-point 437 witnesses, including every first-difference edge and all 1861 plus 18908 distinct-width power certificates. The exact exponent 37903373/50000000 has no remaining certificate assumptions. Comparator selects the criterion and the unconditional numerical result. Verification history: the September 11 NanoDa repl", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The interval-moment proof uses stopping words instead of entropy/Stirling estimates. Chain powers are rounded down to exact rationals with proved moments. Binary seed factor c=1/4 is absorbed by a proved depth margin of at least 27. The odd finite geometry uses conservative predecessor bit masks and endpoint cutoffs; odd width-power bounds use two direct 1024th-power endpoint comparisons and an eighth-order logarithm enclosure, cleared to integer polynomial comparisons. The endpoint checker has a general soundness proof for every positive integer degree. Scalar thresholds and binary estimates retain their integer square-root ladders. Histogram permutations connect numerical and geometric row orders. The full exponent, bases and component powers are preserved. Every certificate hypothesis is discharged. The smaller human-facing result is not formalized.", "alignment": true, "original": false, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [ "Codex (GPT-6)" ], "notes": "Agent review checked the construction, exact target parameters, compressed odd edge checker, polynomial log checker, histogram transport and unconditional submission statements. Reordered data retain the actual point/start/width tuples. Verification results and selected-statement coverage are recorded in verification/README.md. No human peer review or Palomar registration is claimed." }, "canonical": { "repo": "enaslund/sarkozy-lower-bound-0.758", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-19-000006", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-19-000006", "commit": "e5d693729e23762b063a55015ad79ccaf28a3217", "trust": "high", "theorems": 2, "date": "2026-09-19" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "Flaman55/RelMathApps/number_theory/structural_Bertrand/formalization.yaml", "repo": "Flaman55/RelMathApps", "path": "number_theory/structural_Bertrand/formalization.yaml", "directory": "number_theory/structural_Bertrand", "origins": [ "search" ], "url": "https://github.com/Flaman55/RelMathApps/blob/HEAD/number_theory/structural_Bertrand/formalization.yaml", "version": "v0.3", "missing": [], "name": "StructuralBertrand", "description": "A from-scratch structural proof of Bertrand's postulate (Chebyshev's strengthened form: for every N > 1 there is a prime p with N < p <= 2N) in Lean 4, avoiding Mathlib's existing Bertrand development entirely. The argument sieves every window (P_k, 2*P_k] against the primorial of primes below P_k, closed by an explicit Jacobsthal-gap bound and a finite small-window oracle, both via kernel-checked `decide` (no `native_decide`). Zero sorries; depends only on the three standard Lean axioms.", "authors": [ "Artur Flamandzki" ], "maintainers": [ "Artur Flamandzki" ], "license": "PolyForm-Noncommercial-1.0.0", "role": "", "substantive": "", "sources": [ { "title": "A Structural Sieve for Bertrand's Postulate", "id": "https://github.com/Flaman55/RelMathApps/blob/main/number_theory/structural_Bertrand/LaTex/A%20Structural%20Sieve%20for%20Bertrands%20Postulate.pdf", "authors": [ "Artur Flamandzki" ], "type": "preprint", "relationship": "independently-proves", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.NT", "cs.LO" ], "msc2020": [ "11A41", "11N05", "03B35" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "claude-sonnet-5" ], "framework": "Claude Code" }, { "method": "agent", "models": [ "claude-sonnet-5" ], "framework": "Claude Code" } ], "strongest": "agent", "models": [ "claude-sonnet-5" ], "spend": "not tracked (subscription-based usage)", "notes": "Development combined author-directed mathematical design (the structural sieve itself, prior to and independent of any AI assistance) with agent-drafted LaTeX prose and agent-assisted Lean formalization, debugging, and one specific proof-engineering technique (chunked `decide` + glue lemma) proposed by the agent to resolve native-decision-procedure resource limits." }, "scope": "A from-scratch structural proof of Bertrand's postulate (Chebyshev's strengthened form: for every N > 1 there is a prime p with N < p <= 2N), avoiding Mathlib.NumberTheory.Bertrand entirely. The argument covers every window (P_k, 2*P_k] for P_k prime via a coprimality sieve against the primorial of primes below P_k, closed by an explicit Jacobsthal-function gap bound on small moduli (chunked, kernel-checked `decide`, no `native_decide`) and a finite oracle over small windows (n < 512, also chunked `decide`).", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "StructuralBertrand.bertrand_chebyshev", "file": "StructuralBertrand/Main.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "No known divergences between the LaTeX paper and the Lean development; all \\lean{...} citations in the paper were cross-checked against declaration names in this repository. Note that the LaTeX prose itself was agent-drafted (see `automation` above); this check verifies citation accuracy, not independent authorship of the text.", "alignment": true, "original": true, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Artur Flamandzki" ], "notes": "Author directed the mathematical strategy throughout and verified every Lean change via successful kernel-checked builds (including confirming axiom-cleanliness of the main theorem via `#print axioms StructuralBertrand.bertrand_chebyshev`), rather than by independent line-by-line reading of agent-written Lean source. The LaTeX paper was reviewed by the author at the level of overall structure, scope, and format compliance with Ramanujan Journal conventions, not sentence-by-sentence; the paper-to-Lean citation mapping was checked mechanically (see `fidelity`)." }, "canonical": { "repo": "flaman55/relmathapps", "directory": "number_theory/structural_Bertrand" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "Flaman55/StructuralBertrand/formalization.yaml", "repo": "Flaman55/StructuralBertrand", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/Flaman55/StructuralBertrand/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "StructuralBertrand", "description": "Two structural, exact characterizations of primality via sieve-coverage -- proved for a prime base's own window (P_k, 2*P_k] (StructuralBertrand.prime_iff_uncovered_by_prev) and generalized to the base's full deterministic reach, the zone (P_k, P_k^2) (StructuralBertrand.void_iff_prime_in_deterministic_zone, with the reach shown exact by determinism_breaks_above) -- plus Bertrand's postulate (Chebyshev's strengthened form: for every N > 1 there is a prime p with N < p <= 2N) as a corollary of the first. The originating question is not Bertrand's (\"is there a prime here?\") but a different one: ", "authors": [ "Artur Flamandzki" ], "maintainers": [ "Artur Flamandzki" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "A Structural Sieve for Bertrand's Postulate", "id": "https://github.com/Flaman55/StructuralBertrand/blob/main/LaTex/A%20Structural%20Sieve%20for%20Bertrands%20Postulate.pdf", "authors": [ "Artur Flamandzki" ], "type": "preprint", "relationship": "formalizes", "endorsement": "n/a" }, { "title": "Bertrand's postulate (Chebyshev's strengthened form)", "id": "https://en.wikipedia.org/wiki/Bertrand%27s_postulate", "authors": [ "Joseph Bertrand", "Pafnuty Chebyshev" ], "type": "classical-result", "relationship": "independently-proves", "endorsement": "n/a" }, { "title": "real_main_inequality (Mathlib.NumberTheory.Bertrand)", "id": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/Bertrand.html", "authors": [ "The Mathlib Community" ], "type": "formalization", "relationship": "adapts", "endorsement": "n/a" } ], "related": [], "classification": { "arxiv": [ "math.NT", "cs.LO" ], "msc2020": [ "11A41", "11N05", "03B35" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "claude-sonnet-5" ], "framework": "Claude Code" }, { "method": "agent", "models": [ "claude-sonnet-5" ], "framework": "Claude Code" } ], "strongest": "agent", "models": [ "claude-sonnet-5" ], "spend": "not tracked (subscription-based usage)", "notes": "Development combined author-directed mathematical design (the structural sieve itself, prior to and independent of any AI assistance) with agent-drafted LaTeX prose and agent-assisted Lean formalization, debugging, and one specific proof-engineering technique (chunked `decide` + glue lemma) proposed by the agent to resolve native-decision-procedure resource limits." }, "scope": "Three declarations are proved, presented in the order that reflects the project's own history rather than fame — see Challenge.lean's header for the full account of why these are different questions with a shared answer. 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The originating question is not Bertrand's (\"is there a prime here?\") but a different one: \"up to", "authors": [ "Artur Flamandzki" ], "maintainers": [ "Artur Flamandzki" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "The Structural Sieve", "id": "https://github.com/Flaman55/StructuralSieve/blob/main/LaTex/The%20Structural%20Sieve.pdf", "authors": [ "Artur Flamandzki" ], "type": "preprint", "relationship": "formalizes", "endorsement": "n/a" }, { "title": "Bertrand's postulate (Chebyshev's strengthened form)", "id": "https://en.wikipedia.org/wiki/Bertrand%27s_postulate", "authors": [ "Joseph Bertrand", "Pafnuty Chebyshev" ], "type": "classical-result", "relationship": "independently-proves", "endorsement": "n/a" }, { "title": "real_main_inequality (Mathlib.NumberTheory.Bertrand)", "id": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/Bertrand.html", "authors": [ "The Mathlib Community" ], "type": "formalization", "relationship": "adapts", "endorsement": "n/a" } ], "related": [], "classification": { "arxiv": [ "math.NT", "cs.LO" ], "msc2020": [ "11A41", "11N05", "03B35" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "claude-sonnet-5" ], "framework": "Claude Code" }, { "method": "agent", "models": [ "claude-sonnet-5" ], "framework": "Claude Code" } ], "strongest": "agent", "models": [ "claude-sonnet-5" ], "spend": "not tracked (subscription-based usage)", "notes": "Development combined author-directed mathematical design (the structural sieve itself, prior to and independent of any AI assistance) with agent-drafted LaTeX prose and agent-assisted Lean formalization, debugging, and one specific proof-engineering technique (chunked `decide` + glue lemma) proposed by the agent to resolve native-decision-procedure resource limits." }, "scope": "Three declarations are proved, presented in the order that reflects the project's own history rather than fame — see Challenge.lean's header for the full account of why these are different questions with a shared answer. 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Note that the LaTeX prose itself was agent-drafted (see `automation` above); this check verifies citation accuracy, not independent authorship of the text.", "alignment": true, "original": true, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Artur Flamandzki" ], "notes": "Author directed the mathematical strategy throughout and verified every Lean change via successful kernel-checked builds (including confirming axiom-cleanliness of the main theorem via `#print axioms StructuralSieve.bertrand_chebyshev`), rather than by independent line-by-line reading of agent-written Lean source. 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The declarations registered from this repository are the Artzner–Delbaen–Eber–Heath representation theorem for coherent risk measures on a finite state space — a coherent risk measure is the least upper bound of expected loss over its representing set of probability vectors, obtained by separating a rejected position from the acceptance cone — together with the coherence of worst-case loss, which witnesses on a nonempty state space that the representation theorem's hypothesis", "authors": [ "Raphael Coelho" ], "maintainers": [ "Raphael Coelho" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Coherent Measures of Risk", "id": "artzner-delbaen-eber-heath-1999", "authors": [ "Philippe Artzner", "Freddy Delbaen", "Jean-Marc Eber", "David Heath" ], "type": "article", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "Stochastic Processes (lecture notes / textbook)", "id": "saporito-stochastic-processes", "authors": [ "Yuri F. Saporito" ], "type": "textbook", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "Standard mathematical-finance literature (Black-Scholes, CRR, Vasicek, Margrabe, FTAP, ...)", "id": "standard-mathematical-finance-literature", "authors": [], "type": "canonical results", "relationship": "background", "endorsement": "n/a" } ], "related": [], "classification": { "arxiv": [ "q-fin.RM", "q-fin.MF" ], "msc2020": [ "91G70", "46A22", "91G20", "60H05", "46N10" ] }, "automation": { "methods": [ { "method": "copilot", "models": [ "Claude Code (authoring assistant)" ], "framework": "Lean 4 + Mathlib + BrownianMotion; VS Code / lean-repl daemon" }, { "method": "agent", "models": [ "labs-leanstral-1-5", "magistral-medium" ], "framework": "formal-foundry: probe / vibe <-> lean-lsp-mcp" }, { "method": "manual", "models": [], "framework": "Lean 4 + Mathlib, this library's conventions; an Isabelle/HOL AFP entry consulted as a source for the classical result set" } ], "strongest": "agent", "models": [ "Claude Code (authoring assistant)", "labs-leanstral-1-5", "magistral-medium" ], "spend": "0 (Mistral Labs beta)", "notes": "scout, not author: the pipeline opens the PR; a human reviews and merges, so no machine proof enters the library unreviewed" }, "scope": "373 theorems across continuous-time stochastic processes and mathematical finance, built on Mathlib + BrownianMotion; 360 delivery-ready (full + library-wrapper).", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "MathFin.coherentRisk_isLUB", "file": "MathFin/RiskMeasures/AcceptanceSet.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [ { "statement": "Coherent-risk representation on a finite state space (Artzner-Delbaen-Eber-Heath 1999)", "source": "artzner-delbaen-eber-heath-1999" } ] }, { "declaration": "MathFin.worstCase_isCoherent", "file": "MathFin/RiskMeasures/WorstCaseRisk.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [ { "statement": "Worst-case loss is a coherent risk measure (witness that the axioms are satisfiable)", "source": "artzner-delbaen-eber-heath-1999" } ] } ], "comparator": false, "literature_dependencies": 2, "divergences": "13 reduced-core (a faithful but reduced formal core of the informal statement) and 18 library-wrappers (a thin restatement consuming a Mathlib/BrownianMotion lemma); 0 placeholders. 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Compared to the blueprint proof, we proved a general version of the Marcinkiewicz interpolation theorem, and there were minor divergences related to the Hardy--Littlewood's maximal function", "alignment": true, "original": false, "review": { "status": "self-assessed, author-reviewed, under peer-review", "bucket": "self-assessed", "reviewers": [], "notes": "" }, "canonical": { "repo": "fpvandoorn/carleson", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "g-pichler/bsc-averaging/formalization.yaml", "repo": "g-pichler/bsc-averaging", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/g-pichler/bsc-averaging/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "BSCAveraging", "description": "Three theorems about binary channels on a doubly symmetric binary source: a pair (𝑋, 𝑌) of Bernoulli(1/2) bits with P(𝑋 ≠ 𝑌) = 𝑝, together with Markov chains 𝑈 - 𝑋 - 𝑌 - 𝑉 in which 𝑈 and 𝑉 are the outputs of two independent binary channels applied to 𝑋 and to 𝑌. Let 𝒜 be the set of rate triples (𝑅₀, 𝑅₁, 𝑅₂) with 𝐼(𝑈;𝑋) ≤ 𝑅₁, 𝐼(𝑌;𝑉) ≤ 𝑅₂ and 𝑅₀ ≤ 𝐼(𝑈;𝑉) attainable by arbitrary binary channels on the two sides, and ℬ the same set with both channels restricted to be binary symmetric. (1) conv 𝒜 = conv ℬ for every 𝑝 in [0,1]. (2) At 𝑝 = 0, no pair of binary channels whose rates do not exceed those", "authors": [ "Georg Pichler" ], "maintainers": [ "Georg Pichler" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Three problems on binary channels for a doubly symmetric binary source: conv 𝒜 = conv ℬ, and the two p = 0 conjectures of Dikshtein-Ordentlich-Shamai", "id": "", "authors": [], "type": "original-proof", "relationship": "other", "endorsement": "" }, { "title": "The Double-Sided Information Bottleneck Function", "id": "", "authors": [ "Michael Dikshtein", "Or Ordentlich", "Shlomo Shamai (Shitz)" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Distributed Information-Theoretic Clustering", "id": "", "authors": [ "Georg Pichler", "Pablo Piantanida", "Gerald Matz" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Do averaged binary symmetric channels maximize mutual information?", "id": "", "authors": [ "Georg Pichler" ], "type": "other", "relationship": "background", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "cs.IT", "math.IT" ], "msc2020": [ "94A17", "94A15", "94A40", "94A34", "26D07" ] }, "automation": { "methods": [ { "method": "agent", "models": [], "framework": "" } ], "strongest": "agent", "models": [], "spend": "", "notes": "" }, "scope": "", "sorry_count": null, "sorry_in_definitions": null, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "", "alignment": false, "original": true, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "The author checked the three Challenge statements against the cited sources and reviewed the proof architecture; no separate reviewer was involved. Mechanical checks performed before submission: `lake build` of the whole development, and `#print axioms` for every theorem on the proof path (`BSCAveraging/Basic.lean`). Novelty has not been independently established. MathOverflow 285151 has no answer and no comments (StackExchange API, asked 2017-11-03). The Dikshtein-Ordentlich-Shamai paper has one recorded citing work, a 2024 Entropy editorial that mentions the setting but not the conjectures, and no resolution of either conjecture was found. Mathlib at the pinned revision contains no mutual information at all, and no prior Lean formalization of any of the three problems was found. That sea" }, "canonical": { "repo": "g-pichler/bsc-averaging", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-26-000006", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-26-000006", "commit": "288a1d8787c68010534969b53dcf01ba223e085e", "trust": "high", "theorems": 3, "date": "2026-09-15" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "gaearon/conway-refinement/formalization.yaml", "repo": "gaearon/conway-refinement", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/gaearon/conway-refinement/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "A proof of Conway's refinement conjecture", "description": "A Lean proof of Conway's 1976 refinement conjecture for omnific integers. For omnific integers a, b, c, d with ab = cd, it proves the existence of omnific integers e, f, g, h satisfying a = ef, b = gh, c = eg, and d = fh, including zero and degenerate cases. The proof proceeds through factorisation and refinement theory for generalised-power-series integer parts and connects the generalised-power-series and surreal-number formulations.", "authors": [ "Dan Abramov" ], "maintainers": [ "Dan Abramov" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "A proof of Conway's refinement conjecture for omnific integers", "id": "https://github.com/gaearon/conway-refinement", "authors": [ "Dan Abramov" ], "type": "original-proof", "relationship": "other", "endorsement": "participated" }, { "title": "On Numbers and Games", "id": "", "authors": [ "John H. Conway" ], "type": "book", "relationship": "background", "endorsement": "" }, { "title": "On a property of pre-Schreier domains", "id": "https://doi.org/10.1080/00927878708823512", "authors": [ "Muhammad Zafrullah" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Factorization in generalized power series", "id": "https://doi.org/10.1090/S0002-9947-99-02172-8", "authors": [ "Alessandro Berarducci" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Existence of prime elements in rings of generalized power series", "id": "https://doi.org/10.2307/2695102", "authors": [ "Daniel Pitteloud" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Algebraic properties of rings of generalized power series", "id": "https://doi.org/10.1016/S0168-0072(01)00099-9", "authors": [ "Daniel Pitteloud" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Unique factorization in generalized power series rings", "id": "https://doi.org/10.1090/S0002-9939-05-08162-1", "authors": [ "James Pommersheim", "Shahriar Shahriari" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Factorisation of germ-like series", "id": "https://doi.org/10.4115/jla.2017.9.3", "authors": [ "Sonia L'Innocente", "Vincenzo Mantova" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "A factorisation theory for generalised power series and omnific integers", "id": "https://doi.org/10.1016/j.aim.2024.109513", "authors": [ "Sonia L'Innocente", "Vincenzo Mantova" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Irreducibility in generalized power series", "id": "https://arxiv.org/abs/2405.13815", "authors": [ "Antongiulio Fornasiero", "Noa Lavi", "Sonia L'Innocente", "Vincenzo Mantova" ], "type": "paper", "relationship": "background", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.LO", "math.RA" ], "msc2020": [ "13F25", "13F15", "13A05", "03E10" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "ChatGPT 5.6 Sol", "Claude Fable 5", "Claude Opus 5" ], "framework": "Interactive ChatGPT, Claude, and OpenAI Codex sessions with Lean and Lake" }, { "method": "manual", "models": [], "framework": "Lean 4, Mathlib, CombinatorialGames, and repository-specific audits" } ], "strongest": "agent", "models": [ "ChatGPT 5.6 Sol", "Claude Fable 5", "Claude Opus 5" ], "spend": "Subscription-based; exact total not tracked", "notes": "The project is deliberately AI-assisted rather than an autonomous proof claim. Dan Abramov steered the process, supplied literature, and made editorial decisions; AI systems supplied all of the mathematical and Lean work; Lean's kernel checks the resulting proof terms." }, "scope": "The compared Challenge defines well-founded Conway games with option families in an arbitrary universe, Conway negation, addition, multiplication and order, surreal representatives as numeric games, and omnific integers by the cut equation x = {x - 1 | x + 1}. Its theorem proves the four-factor refinement statement for every a, b, c, d satisfying that equation, including zero and degenerate cases. Equality is Conway equivalence of numeric games, hence equality in their quotient. The statement is universe-polymorphic and uses set-sized option families at each instantiation; it does not construc", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "No algebraic divergence from Conway's displayed four-factor formula is known. The Challenge represents surreal numbers by numeric well-founded games and writes equality as Conway equivalence, rather than constructing one class-sized quotient. Omnific-integer membership is the cut equation x = {x - 1 | x + 1}; the full formalization also proves its equivalence with the generalised-power-series normal-form formulation. Lean's universes impose set-sized option families at each instantiation. No nonzero or nondegeneracy assumption is added.", "alignment": true, "original": true, "review": { "status": "agent-reviewed and self-assessed", "bucket": "agent-reviewed", "reviewers": [], "notes": "Lean checks the proof and repository audits enforce the admitted-axiom policy. AI-assisted review compared the proof guide's statements and proof prose with the Lean source. The standalone statement and its correspondence with Conway's displayed formula were separately re-audited before submission. No independent specialist or peer review is recorded. PALOMAR-PROVENANCE.md gives the source comparison and literature search." }, "canonical": { "repo": "gaearon/conway-refinement", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-03-000002", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-03-000002", "commit": "63797aae79c09df5be6bfa066f147fa39e63b6cd", "trust": "high", "theorems": 1, "date": "2026-09-03" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "gdahia/DensityHalesJewett/formalization.yaml", "repo": "gdahia/DensityHalesJewett", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/gdahia/DensityHalesJewett/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "The density Hales–Jewett theorem and Szemerédi's theorem", "description": "A Lean 4 formalization of the density Hales–Jewett theorem: for every finite alphabet α and every density δ > 0, all sufficiently large n have the property that any set of at least δ·|α|^n words of length n over α contains a combinatorial line. The basis for the formalization is the combinatorial proof of Dodos, Kanellopoulos, and Tyros. As a consequence, we also formalize Szemerédi's theorem on the integers: for every k ≥ 3 and δ > 0, the following holds for all sufficiently large n. Any subset of {0, …, n−1} of size at least δ·n contains a nonconstant arithmetic progression of length k.", "authors": [ "Gabriel Dahia" ], "maintainers": [ "Gabriel Dahia" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "A Simple Proof of the Density Hales--Jewett Theorem", "id": "https://doi.org/10.1093/imrn/rnt041", "authors": [ "Pandelis Dodos", "Vassilis Kanellopoulos", "Konstantinos Tyros" ], "type": "paper", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "Ramsey's theorem for n-parameter sets", "id": "https://doi.org/10.2307/1995354", "authors": [ "Ronald L. Graham", "Bruce L. Rothschild" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" } ], "related": [ { "id": "https://github.com/leanprover-community/mathlib4", "relationship": "builds-on", "note": "Mathlib supplies `Combinatorics.Line` and the colouring Hales--Jewett theorem, which this development uses and on which the compared statements are phrased. Mathlib does not contain the density version; that is what this repository adds." } ], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05D10", "05A05", "11B75", "68R15" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "GPT-5.6 Sol", "GPT-5.6 Terra", "GPT-5.6 Lua", "Opus 5" ], "framework": "n/a" } ], "strongest": "agent", "models": [ "GPT-5.6 Lua", "GPT-5.6 Sol", "GPT-5.6 Terra", "Opus 5" ], "spend": "subscription-based", "notes": "" }, "scope": "The density Hales--Jewett theorem and Szemeredi's theorem on the integers are proved in full, with no `sorry` outside the deliberate holes in `Challenge.lean`. Both are proved asymptotically in `Challenge.lean`. The formalization also includes the finite-unions theorem, the Graham--Rothschild theorem for combinatorial lines, block canonization, and Varnavides' averaging argument, none of which are among the compared declarations. Szemeredi's theorem is formalized for subsets of an initial segment of the naturals only; no version for general abelian groups or for the primes is included.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "Combinatorics.Line.exists_of_density", "file": "DensityHalesJewett/Main.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "Combinatorics.ArithmeticProgression.exists_of_density_nat", "file": "DensityHalesJewett/Szemeredi.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] } ], "comparator": false, "literature_dependencies": 0, "divergences": "- The paper sets `eta = delta * theta / 48` and `gamma = delta * eta^2 / k`. The formalization replaces these by smaller minima that also enforce `eta <= theta / 4`, `eta <= delta / 6`, `gamma <= eta^2 / 2`, and `gamma <= 3 * eta`. Consequently its numerical thresholds and density increment are weaker, while the qualitative theorem is unchanged. - Several displayed paper bounds (`M_0`, `F`, and `N`) are replaced by sufficient bounds selected with `Nat.find` from eventual-existence lemmas. This makes the formalized thresholds non-explicit, so the compared statements are asymptotic rather than quantitative.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Gabriel Dahia" ], "notes": "Reviewed by the author for mathematical correctness, including a statement audit of the two compared declarations against the cited paper. No independent reviewer has checked the development, and there is no novelty whatsoever, as the theorems are known results. The only contribution is the formalization." }, "canonical": { "repo": "gdahia/densityhalesjewett", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-06-000005", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-06-000005", "commit": "1285e649d0256a848d4693377dd1244065ac49ff", "trust": "high", "theorems": 2, "date": "2026-09-06" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "gdahia/komlos/formalization.yaml", "repo": "gdahia/komlos", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/gdahia/komlos/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Komlós conjecture", "description": "Lean4 formalization of the Komlós bound with constant 36 following the Lovett—Karingula exposition. The Komlós statement gives signs for finitely many vectors of Euclidean norm at most 1 so that the signed sum has supremum norm at most 36. The Beck–Fiala statement gives a vertex colouring with edge discrepancy at most 36√t when every vertex belongs to at most t edges.", "authors": [ "Gabriel Dahia" ], "maintainers": [ "Gabriel Dahia" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "An elementary proof of the Komlós conjecture", "id": "ECCC TR26-188", "authors": [ "Sankeerth Rao Karingula", "Shachar Lovett" ], "type": "article", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "Vector balancing via directional total variation", "id": "arXiv:2609.11189", "authors": [ "S. Guo", "E. X. Fang", "J. Lu" ], "type": "article", "relationship": "background", "endorsement": "not-contacted" } ], "related": [], "classification": { "arxiv": [ "math.CO", "cs.DM" ], "msc2020": [ "11K38", "05C65", "05D40", "68R05" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Claude Opus 5" ], "framework": "Claude Code" }, { "method": "agent", "models": [ "GPT-6" ], "framework": "Codex" } ], "strongest": "agent", "models": [ "Claude Opus 5", "GPT-6" ], "spend": "not tracked", "notes": "Claude Code produced the initial implementation. Codex revised proofs and documentation. Gabriel Dahia set the goals and directed both stages. The library and Solution.lean build without sorry; the two Challenge.lean declarations are placeholders. The solution theorems use propext, Classical.choice, and Quot.sound, with no additional axioms." }, "scope": "The project proves the Komlós bound with constant 36 for arbitrary finite vector and coordinate index types, and the Beck–Fiala bound with constant 36 for Mathlib hypergraphs with finitely many vertices. The library includes Lemmas 1.4 and 1.5 and the matrix form of Theorem 1.2. It does not formalize prefix discrepancy bounds or the paper's algorithmic remarks. The sorry counts below refer to the library and Solution.lean; Challenge.lean contains two statement placeholders.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "For Lemma 1.5, the paper bounds the directional total variation of a continuous product density and then rounds to a grid. This proof starts with discrete squared tent weights. Telescoping and Cauchy–Schwarz give the one-dimensional bound. Factorisation of correlations and Weierstrass' product inequality give the product bound. The resulting shift-distance bound is ‖v‖₂/√12. For Theorem 1.2 over real vectors, the paper takes a subsequence of rational approximations. This proof instead establishes discrepancy at most 36 + η for every η > 0 and applies le_of_forall_pos_le_add. The Komlós statement uses arbitrary finite index types rather than only Fin n and Fin d. The Beck–Fiala bound is proved as a separate theorem.", "alignment": true, "original": false, "review": { "status": "reviewed", "bucket": "other", "reviewers": [ "Gabriel Dahia" ], "notes": "Reviewed the main statements and the style of the proofs." }, "canonical": { "repo": "gdahia/komlos", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-18-000003", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-18-000003", "commit": "d802857449234318d557361b1dbb0a27e0258528", "trust": "high", "theorems": 2, "date": "2026-09-18" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "gexahedron/sabidussi-lean/formalization.yaml", "repo": "gexahedron/sabidussi-lean", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/gexahedron/sabidussi-lean/blob/HEAD/formalization.yaml", "version": "v0.3", "missing": [], "name": "Sabidussi compatibility in Lean", "description": "", "authors": [ "Nikolay Ulyanov" ], "maintainers": [ "Nikolay Ulyanov" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Graph Puzzles III.1: A Proof of Sabidussi's Compatibility Conjecture", "id": "https://arxiv.org/abs/2607.13225", "authors": [ "Nikolay Ulyanov" ], "type": "paper", "relationship": "formalizes", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05C45", "05C70" ] }, "automation": { "methods": [ { "method": "other", "models": [], "framework": "not recorded" } ], "strongest": "other", "models": [], "spend": "not recorded", "notes": "Production details for the Lean code are not recorded in repository evidence. Separately, the bundled manuscript states that its mathematical proof is entirely due to GPT 5.6 Pro and that its writeup was prepared with Codex using GPT 5.6 Sol. That statement does not by itself establish that the same tools produced the Lean code." }, "scope": "Lean 4 and Mathlib formalization of Theorem 1.1 (Sabidussi compatibility) and the proof-driving results in Sections 2–7 of the source manuscript. The model permits labelled parallel edges and loops, counts both ends of a loop in vertex degree, and concludes with ordinary nonempty connected 2-regular circuits. The expository circuit-double-cover discussion in Section 8 and the dimer/spin interpretation in Section 9 are outside the formalization's scope. The sorry counts exclude the statement-only Comparator challenge fixture.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "Sabidussi.LoopMultigraph.loop_sabidussi_compatibility_ordinary", "file": "Sabidussi/OrdinaryCircuit.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The public endpoint matches Theorem 1.1. Finite vertices and labelled edges are represented by Fintype instances and an endpoint map E → Fin 2 → V. Rather than carrying separate connectedness and even-degree hypotheses, the theorem receives an explicit nonempty cyclic Euler tour; together with minimum degree at least four, this represents the source's finite Eulerian-graph assumptions. A circuit decomposition is represented by a list of finite edge sets in which every labelled edge occurs exactly once, and compatibility forbids any member from containing both the previous and current tour edges. Some proof engineering intentionally differs from the manuscript. In particular, the formalization first decomposes binary-even edge sets into inclusion-minimal pieces and then proves by a finite-dimensional rank-nullity argument that such pieces are connected and 2-regular. The manuscript instead invokes Euler's theorem and repeatedly splits closed trails. Several internal Lean lemmas are strengthened or specialized to Fin-indexed representations, while the final ordinary-circuit statement is unchanged. Sections 8 and 9 are not formalized. In particular, the interpretive Propositions 9.1–9", "alignment": true, "original": false, "review": { "status": "self-assessed; agent-audited", "bucket": "self-assessed", "reviewers": [ "Nikolay Ulyanov", "OpenAI Codex (statement and fidelity audit, 2026-07-14)" ], "notes": "VERIFICATION.md records a successful full build on 2026-07-14, an empty production-source scan for sorry, admit, native_decide, axiom, opaque, and unsafe declarations, and exact final-theorem dependencies on propext, Classical.choice, and Quot.sound. The 2026-07-14 agent audit compared the bundled manuscript's principal statements with their Lean declarations. These checks are not independent external mathematical peer review." }, "canonical": { "repo": "gexahedron/sabidussi-lean", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-17-000003", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-17-000003", "commit": "58307d74da99ec6e4fe28e012c30c83e369c54c3", "trust": "high", "theorems": 1, "date": "2026-08-17" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "gnostrich/certified-positivity/formalization.yaml", "repo": "gnostrich/certified-positivity", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/gnostrich/certified-positivity/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Certified Positivity and Certified Frontiers: machine-checked positivity windows for truncated Weil-type quadratic forms and the Suzuki screw kernel", "description": "This Lean 4 and Mathlib development produces exact, machine-checked positivity certificates for truncated Weil-type quadratic forms and for the finite Gram matrices of Suzuki's zeta screw function. Weil's explicit formula recasts the Riemann Hypothesis as positivity of a quadratic form assembled from prime powers, and a recent line of work truncates that form and studies the resulting finite matrices. The objects formalized here are the screw function Psi, built from an archimedean profile minus a von Mangoldt weighted prime sum; its Gram kernel G(t,u) = Psi(t) + Psi(u) - Psi(t-u); positive de", "authors": [ "Rohan Badade" ], "maintainers": [ "Rohan Badade" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Explicit machine-checked positivity windows for the genuine Suzuki zeta screw kernel, the exact prime-side identity for its prime sum, and the certified-frontier object with its three-site full-band coverage theorem and two machine-refuted pre-registered hypotheses", "id": "", "authors": [ "Rohan Badade" ], "type": "original-proof", "relationship": "other", "endorsement": "participated" }, { "title": "Certified Positivity: Formally Verified Horizon Thresholds for Truncated Weil Quadratic Forms in Lean", "id": "", "authors": [ "Rohan Badade" ], "type": "paper", "relationship": "background", "endorsement": "participated" }, { "title": "Certified Frontiers: Self-Expanding Positivity Certificates with a Verified Coverage Theorem", "id": "", "authors": [ "Rohan Badade" ], "type": "paper", "relationship": "background", "endorsement": "participated" }, { "title": "Screw functions and the Weil quadratic form", "id": "arXiv:2606.09096", "authors": [ "Masatoshi Suzuki" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Quadratic forms, real zeros and echoes of the spectral action", "id": "arXiv:2511.23257", "authors": [ "Alain Connes", "Walter D. van Suijlekom" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Zeta spectral triples", "id": "arXiv:2511.22755", "authors": [ "Alain Connes", "Caterina Consani", "Henri Moscovici" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Sur les \"formules explicites\" de la theorie des nombres premiers", "id": "", "authors": [ "Andre Weil" ], "type": "paper", "relationship": "background", "endorsement": "n/a" }, { "title": "mathlib4: the Lean mathematical library", "id": "https://github.com/leanprover-community/mathlib4", "authors": [ "The Mathlib Community" ], "type": "other", "relationship": "background", "endorsement": "n/a" } ], "related": [], "classification": { "arxiv": [ "math.NT", "math.LO" ], "msc2020": [ "11M26", "15B48", "68V20", "65G20", "15A63", "11M06" ] }, "automation": { "methods": [ { "method": "autonomous", "models": [ "Aristotle" ], "framework": "Aristotle (Harmonic)" }, { "method": "copilot", "models": [ "Claude (Anthropic)" ], "framework": "Claude" }, { "method": "manual", "models": [], "framework": "n/a" } ], "strongest": "autonomous", "models": [ "Aristotle", "Claude (Anthropic)" ], "spend": "not tracked", "notes": "The division of labour is: human problem-setting, specification design, pre-registration and decisions; AI-assisted specification drafting and statement-level audit; automated proof production by Aristotle (Harmonic). No human has reviewed the proof terms line by line. The trust model is Lean's kernel plus the axiom audit: the consolidated all-files lake build was re-run outside the prover's environment on 2026-07-26 on the pinned toolchain (Lean 4 v4.28.0, Mathlib v4.28.0), building all 75 modules then present with zero errors, exactly the three disclosed sorry warnings, and #print axioms out" }, "scope": "Formalized: exactly the eighteen statements listed in comparator.json and stated in Challenge.lean, with sorry-free proofs in RequestProject/, all inside the wider 76-module development in that directory. These are positivity and non-positivity certificates about explicit finite objects at explicit rationals. Sorry sites. The development is sorry-free. It formerly contained exactly three documented sorry sites, in RequestProject/D4.lean, RequestProject/F3.lean and RequestProject/F3R.lean, with one shared cause: unitary diagonalizability of a Hermitian matrix is not available in Mathlib in the ", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "V5_5.true_kernel_grid_margin", "file": "RequestProject/V5_5.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "V5_6.true_kernel_first_prime_posdef", "file": "RequestProject/V5_6.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "V5_7.weil_triangle_prime_side", "file": "RequestProject/V5_7.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "TierR.GramState.expand_pd", "file": "RequestProject/R_A3_A4.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "TierR.GramState.halt_not_pd", "file": "RequestProject/R_A3_A4.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "R5.gershgorin_margin", "file": "RequestProject/R5Prime.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "R5.coverage_band_final", "file": "RequestProject/R5Final.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "G5.G5_c_prime", "file": "RequestProject/G5.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "There is no cited source whose statements this development reproduces, so the fidelity question here is between the sorry-free repository modules and their restatement in Challenge.lean, which is what Palomar compares. One systematic transformation: Challenge.lean imports only Mathlib and therefore contains inlined copies of every custom definition the eighteen statements need, copied from the named repository files. Because those copies all live in the single namespace Challenge, the namespace prefixes of the originals (Horizon., V5_1., V5_5., V5_6., TierR., R5. and so on) are necessarily dropped. This applies to every inlined definition and statement. Two individual statement deviations, both marked at the theorem in Challenge.lean. (1) expand_pd: the repository states IsPDq (G.expand b d hs).M, where expand is a GramState-valued constructor whose hsymm and hpd fields carry proofs and so cannot be inlined without them; Challenge.lean states IsPDq (border G.M b d) instead. Since (G.expand b d hs).M is definitionally border G.M b d, this is the same claim with the constructor unfolded. (2) frontier_covers_band_final: notation only. The repository writes the kernel as V5_1.G, but in", "alignment": true, "original": true, "review": { "status": "self-assessed; automated mechanical checks (consolidated lake build in ci with sorry census; leanprover/comparator statement match and both lean-kernel and nanoda replay against challenge.lean for all 18 compared theorems); statement-level audit of prover output was ai-assisted; no line-by-line human review of proof terms; no external mathematical peer review yet", "bucket": "self-assessed", "reviewers": [ "Rohan Badade (self)" ], "notes": "What was checked mechanically: the consolidated all-files lake build on the pinned toolchain, re-run independently of the prover's environment and re-run in CI, with no sorry warnings and no errors (as of the 2026-09-14 closure of the three previously disclosed sorry sites; see status.scope above); #print axioms on the headline theorems; and leanprover/comparator (pinned) against Challenge.lean for the eighteen theorems in comparator.json, covering statement match, byte-identical definition graphs, the axiom allowlist (propext, Quot.sound, Classical.choice only) and both Lean-kernel and NanoDa replay. What was checked by a human: the choice and wording of every statement, the scope and non-claim lists in STATEMENTS.md and TIER_R_FINAL.md, the axiom and sorry disclosures, and the contents o" }, "canonical": { "repo": "gnostrich/certified-positivity", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "GregoryMorse/unrestricted-boolean-mul/formalization.yaml", "repo": "GregoryMorse/unrestricted-boolean-mul", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/GregoryMorse/unrestricted-boolean-mul/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Unrestricted Boolean multiplicative complexity of four-term polynomial multiplication", "description": "Formalizes the exact minimum AND-gate counts 0, 1, 3, 6 and 9 for binary polynomial multiplication with zero through four coefficients per operand. XOR and constants are free, and AND inputs may reuse arbitrary earlier nonlinear wires. The principal result excludes eight-gate four-term circuits and matches an explicit nine-gate construction. Separate research workflows check selected n5 quadratic results and n6 upper bounds and reductions; their open unrestricted/bilinear lower bounds are not claims of this submission.", "authors": [ "Gregory Morse" ], "maintainers": [ "Gregory Morse" ], "license": "MIT", "role": "", "substantive": "", "sources": [ { "title": "Unrestricted Boolean Multiplicative Complexity of Four-Term Binary Polynomial Multiplication: Rational Places, Hasse Jets, and the Failure of Nonlinear Feedback", "id": "arXiv:2608.30238v1", "authors": [ "Gregory Morse" ], "type": "paper", "relationship": "formalizes", "endorsement": "participated" } ], "related": [ { "id": "https://github.com/GregoryMorse/unrestricted-boolean-mul/tree/2ebc0cf42039e4b5de84cb7f9d97cad4791c669d", "relationship": "builds-on", "note": "The same author's n4-arxiv-v2 development on Lean 4.32.1. This update migrates to 4.33.1 and isolates the existing theorems for Comparator and dedicated CI; it is not an independent formalization." } ], "classification": { "arxiv": [ "cs.CC", "cs.DS" ], "msc2020": [ "68Q06", "68Q17", "68W30", "15A75", "94D10" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "OpenAI GPT-5.6 Sol (original development, as disclosed in the paper)", "Anthropic Opus 5 (critical referee, as disclosed in the paper)", "Codex (release preparation; exact current model not recorded)" ], "framework": "Codex and Lean 4" } ], "strongest": "agent", "models": [ "Anthropic Opus 5 (critical referee, as disclosed in the paper)", "Codex (release preparation; exact current model not recorded)", "OpenAI GPT-5.6 Sol (original development, as disclosed in the paper)" ], "spend": "", "notes": "Gregory Morse directed and reviewed the research and assumes responsibility. The source paper reports the original model roles. Release automation separates CI, checks the statement interface, and records verification evidence; it does not supply independent human peer review." }, "scope": "Comparator selects six original declarations: the five exact complexities for n=0 through 4 and the n4 eight-gate impossibility. The Boolean ANF circuit model includes constants, arbitrary linear recombination and nonlinear reuse. Natural subtraction gives no outputs when n=0. The minimum's fallback zero is unused because these targets have circuits. The six deliberate Challenge theorem holes are not imported by Solution. Selected n5/n6 work elsewhere is excluded from the default proof closure and this Comparator selection. No unrestricted n5 or exact bilinear n6 lower bound is asserted.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "No statement divergence is intended for the six selected results after source-level comparison with the cited sections. The n3 proof method differs as noted above. The paper pins n4-arxiv-v2 on Lean 4.32.1; this release uses Lean 4.33.1 without claiming to update that published snapshot. The remaining source-alignment and official verification gates are reported in n4/PALOMAR_AUDIT.md, not inferred from compilation alone.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "Author responsibility confirmed during preparation. The paper discloses an AI referee. VibeMathed reports an independent build, axiom audit and leanchecker replay of n4-arxiv-v2 on 2026-09-03, while explicitly not claiming independent statement anchoring. That historical replay is not evidence for this new commit. No Palomar editorial review or registration is claimed." }, "canonical": { "repo": "gregorymorse/unrestricted-boolean-mul", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-21-000005", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-21-000005", "commit": "56837b8ed4ffe6390a7a84d9a77f1489bfd9d872", "trust": "high", "theorems": 6, "date": "2026-09-21" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "GrgAakash/factorial-hypergraph-auxiliary-erdos/formalization.yaml", "repo": "GrgAakash/factorial-hypergraph-auxiliary-erdos", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/GrgAakash/factorial-hypergraph-auxiliary-erdos/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Factorial-residue hypergraphs and an auxiliary problem of Erdos", "description": "A Lean 4 formalization of a quantitative solution to the auxiliary problem suggested by Erdos alongside Problem 1059. Conditional on an explicitly stated standard dimension-one upper-bound sieve theorem, the formalization proves that for every sufficiently large L there are at least ((L+1)!)^eta/(100 log L) integers n in (2L!, (L+1)!] whose prime factors all exceed L and for which n-k! is composite for every 1 <= k <= L. The formalization does not prove that n is prime and therefore does not solve the original Erdos prime problem.", "authors": [ "Aakash Gurung" ], "maintainers": [ "Aakash Gurung" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Factorial-residue hypergraphs and an auxiliary problem of Erdos", "id": "", "authors": [ "Aakash Gurung" ], "type": "paper", "relationship": "formalizes", "endorsement": "participated" }, { "title": "Unsolved Problems in Number Theory, third edition, Problem A2", "id": "https://doi.org/10.1007/978-0-387-26677-0", "authors": [ "Richard K. Guy" ], "type": "book", "relationship": "background", "endorsement": "n/a" }, { "title": "The Distribution of Prime Numbers", "id": "https://doi.org/10.1090/gsm/203", "authors": [ "Dimitris Koukoulopoulos" ], "type": "book", "relationship": "background", "endorsement": "n/a" }, { "title": "Erdos Problem 1059", "id": "https://www.erdosproblems.com/1059", "authors": [ "Thomas F. Bloom" ], "type": "web discussion", "relationship": "background", "endorsement": "n/a" } ], "related": [], "classification": { "arxiv": [ "math.NT", "math.CO" ], "msc2020": [ "11N36", "11B65", "11A41", "05C65" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Aristotle (Harmonic)" ], "framework": "Aristotle" }, { "method": "agent", "models": [ "OpenAI ChatGPT 5.6 Sol Pro", "OpenAI Codex" ], "framework": "AI-assisted open-problem and verification workflow" } ], "strongest": "agent", "models": [ "Aristotle (Harmonic)", "OpenAI ChatGPT 5.6 Sol Pro", "OpenAI Codex" ], "spend": "not tracked", "notes": "The mathematical manuscript was developed through a human-directed pipeline/one-shot workflow. Aristotle produced and extended the Lean development. The author directed the target and is responsible for the submitted result. Codex performed an independent clean build and packaging audit of the supplied archive." }, "scope": "The compared declaration formalizes the manuscript's quantitative auxiliary theorem. It is conditional on the explicit proposition UpperBoundSieveDimOne, a source-matched finite weighted form of the dimension-one upper-bound sieve; the fundamental lemma itself is not proved in Lean. All problem-specific arguments downstream of that theorem are formalized. The large-prime-factor intermediate estimate used by the final proof has threshold constant 1/40 instead of the manuscript's 1/8; this changes only a sufficiently-large-L threshold and not the compared conclusion. The original prime form of E", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "FactorialHypergraph.Palomar.quantitative_auxiliary_erdos", "file": "", "description": "", "axioms": [], "sorry_count": null, "comparator": false, "literature": [] } ], "comparator": false, "literature_dependencies": 0, "divergences": "The compared Lean theorem exposes UpperBoundSieveDimOne as an explicit hypothesis, whereas the manuscript cites the established fundamental lemma as an external theorem. That fundamental lemma is not formalized in Lean here. The downstream large-prime-factor argument uses the weaker threshold (i log i)/40 rather than the manuscript's (i log i)/8; the final quantitative statement is unchanged. The Lean transference theorem obtains X^(eta/2) where the manuscript obtains X^eta, because Lean uses the elementary estimate W <= 4^L in place of Mertens's product estimate. Since the compared theorem asserts only the existence of some absolute positive exponent, this rescaling leaves its statement unchanged. An unused singleton-baseline lemma has 21 in place of the manuscript's 20 after bounding an integer ceiling; it is not in the final dependency chain. The manuscript sets kappa(L) to positive infinity when no rainbow cover exists, while Lean defines real-valued kappa by sInf, which is zero for an empty cover family. Every result used here constructs a cover explicitly, so the empty-family convention is never used by the compared theorem. The theorem concerns rough composite candidates n, ", "alignment": true, "original": false, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [ "Aristotle proof and dependency audit", "OpenAI Codex independent archive and build audit", "Three independent OpenAI Codex presubmission referee audits" ], "notes": "Clean builds accepted all proof modules under Lean 4.28.0 during the original independent archive audit and under Lean 4.33.0 with Mathlib revision db584cd6d46c92f209a44c0f1c829460d327499d after the Palomar compatibility migration. The migration changed three proof bodies but no declaration statement. Source scans found no sorry, admit, unsafe declaration, opaque placeholder, or custom axiom outside the deliberate Challenge statement hole. Exported theorems report only propext, Classical.choice, and Quot.sound. The sieve interface was compared line by line with Koukoulopoulos, Theorem 18.11(a), as documented in docs/SIEVE_SOURCE_AUDIT.md. Three independent presubmission agents separately reviewed mathematical fidelity, Lean integrity, and Palomar compliance; their reconciled report is in d" }, "canonical": { "repo": "grgaakash/factorial-hypergraph-auxiliary-erdos", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-27-000003", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-27-000003", "commit": "67b4088fc84422c84f7e5e205552757c3c0c9a71", "trust": "high", "theorems": 1, "date": "2026-08-27" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "GrgAakash/moh-p3-set-theoretic-complete-intersection/formalization.yaml", "repo": "GrgAakash/moh-p3-set-theoretic-complete-intersection", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/GrgAakash/moh-p3-set-theoretic-complete-intersection/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Moh's P3 curve is a set-theoretic complete intersection", "description": "A Lean 4 formalization proving that the Moh curve parametrized by x=t^6+t^31, y=t^8, z=t^10 is cut out set-theoretically by two explicit equations over every characteristic-zero field. The development computes the polynomial and formal-power-series parametrization kernels, proves the radical equality for the two-generated witness ideal, and proves that the curve ideal has height two and arithmetic rank two, both globally and after completion.", "authors": [ "Aakash Gurung" ], "maintainers": [ "Aakash Gurung" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Moh's P3 Curve Is a Global Set-Theoretic Complete Intersection", "id": "", "authors": [ "Aakash Gurung" ], "type": "paper", "relationship": "formalizes", "endorsement": "participated" }, { "title": "Prime ideals of Moh and the characteristic of the field", "id": "arXiv:2407.21692", "authors": [ "Luis González", "Francesc Planas-Vilanova" ], "type": "article", "relationship": "background", "endorsement": "n/a" }, { "title": "Symbolic powers, set-theoretic complete intersection and certain invariants", "id": "arXiv:2003.09101", "authors": [ "Clare D'Cruz" ], "type": "article", "relationship": "background", "endorsement": "n/a" }, { "title": "On the unboundedness of generators of prime ideals in power series rings of three variables", "id": "", "authors": [ "T. T. Moh" ], "type": "article", "relationship": "background", "endorsement": "n/a" } ], "related": [], "classification": { "arxiv": [ "math.AC", "math.AG" ], "msc2020": [ "13C40", "14M10", "13H05", "13P10" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Aristotle (Harmonic)" ], "framework": "Aristotle" }, { "method": "agent", "models": [ "OpenAI ChatGPT", "OpenAI Codex" ], "framework": "AI-assisted mathematical and verification workflow" } ], "strongest": "agent", "models": [ "Aristotle (Harmonic)", "OpenAI ChatGPT", "OpenAI Codex" ], "spend": "not tracked", "notes": "The author selected and directed the mathematical target. Aristotle produced and extended the Lean formalization. Codex performed a fresh archive extraction, full build, assumption scan, statement audit, and Palomar packaging pass. No independent human expert review is claimed." }, "scope": "The two compared declarations state the global polynomial theorem and the formal-power-series theorem over an arbitrary characteristic-zero field. Each includes the exact parametrization kernel, the radical equality for the two explicit equations, height two, and arithmetic rank two. All are unconditional under Field and CharZero. The proof development also checks the explicit square-containment certificate. It does not formalize the manuscript's Hilbert-Burch matrix symmetrization as matrix algebra, and the accompanying Macaulay2 computations are not part of the Lean trust chain.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "MohP3.Palomar.global_setTheoretic_completeIntersection", "file": "", "description": "", "axioms": [], "sorry_count": null, "comparator": false, "literature": [] }, { "declaration": "MohP3.Palomar.formalLocal_setTheoretic_completeIntersection", "file": "", "description": "", "axioms": [], "sorry_count": null, "comparator": false, "literature": [] } ], "comparator": false, "literature_dependencies": 0, "divergences": "The advertised theorem statements agree with the manuscript. The Lean proof of the formal-power-series kernel uses a convergent formal division algorithm for the weighted filtration wt(x)=6, wt(y)=8, wt(z)=10 rather than the manuscript's normalization and adic-completion argument. Lean formalizes the resulting H1,H2 certificate identities but not the Hilbert-Burch row operation and symmetric-adjugate construction that motivate those equations in the manuscript. No theorem-level weakening or additional hypothesis is present.", "alignment": true, "original": false, "review": { "status": "agent-reviewed and author-supervised", "bucket": "agent-reviewed", "reviewers": [ "Aristotle proof and dependency audit", "OpenAI Codex independent archive, build, and Palomar packaging audit" ], "notes": "The original supplied project was independently built under Lean 4.28.0 and Mathlib revision 8f9d9cff6bd728b17a24e163c9402775d9e6a365. The submission package was subsequently migrated to Lean 4.33.0 and Mathlib revision db584cd6d46c92f209a44c0f1c829460d327499d; the complete 8,729-job build succeeds. The migration changes no theorem statement or assumption and makes only compatibility edits for renamed or restricted Mathlib APIs. Searches found no proof placeholders or custom axioms in the substantive development. The exported results report only propext, Classical.choice, and Quot.sound. The Palomar package adds only the two deliberate Challenge statement holes. No independent human expert Lean or mathematical peer review has yet been conducted." }, "canonical": { "repo": "grgaakash/moh-p3-set-theoretic-complete-intersection", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-07-000012", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-07-000012", "commit": "e1dd554d07425ae9822389427795bd83485a5b2c", "trust": "high", "theorems": 2, "date": "2026-09-07" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "GrgAakash/tunnell-two-to-one-map/formalization.yaml", "repo": "GrgAakash/tunnell-two-to-one-map", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/GrgAakash/tunnell-two-to-one-map/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "A two-to-one executable map between Tunnell representation sets under the Tunnell balance", "description": "Let B(n) and A(n) be the integer solution sets of 2x^2 + y^2 + 8z^2 = n and 2u^2 + v^2 + 32w^2 = n, respectively. For every odd positive squarefree n satisfying |B(n)| = 2|A(n)|, we construct a deterministic computable map Phi_n : B(n) -> A(n) that sends exactly two source points to each target point. The construction combines an elementary even-coordinate map, three explicit quarter-turn correspondences, and a unique stable matching of the remaining antipodal orbits, with preferences determined by primitive midpoint directions. The map, its two-to-one property, the branch formulas, and the in", "authors": [ "Aakash Gurung" ], "maintainers": [ "Aakash Gurung" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Stable Matching and a Two-to-One Map for Tunnell's Ternary Forms", "id": "", "authors": [ "Aakash Gurung", "Kyungyong Lee" ], "type": "paper", "relationship": "formalizes", "endorsement": "participated" }, { "title": "A classical Diophantine problem and modular forms of weight 3/2", "id": "", "authors": [ "J. B. Tunnell" ], "type": "paper", "relationship": "background", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.NT", "math.CO" ], "msc2020": [ "05C70", "11E25", "11D85", "68W40", "03B35" ] }, "automation": { "methods": [ { "method": "agent", "models": [], "framework": "OpenAI ChatGPT, OpenAI Codex, and Aristotle (Harmonic)" } ], "strongest": "agent", "models": [], "spend": "not tracked", "notes": "Human-directed, AI-assisted. Humans set the specification (the source manuscript and the required shape of each statement) and directed the development; automated assistants wrote substantial portions of the Lean definitions and proofs and produced the initial traceability audit. Human review of the mathematical arguments, cited sources, exposition, and manuscript-to-Lean correspondence is ongoing. The submitted Solution modules and the complete project have been rebuilt locally with the Lean kernel. Palomar's protected Comparator and NanoDa run is intentionally left to the registry's mechanic" }, "scope": "FORMALIZED: the definitions of B(n) and A(n); the even branch; the three quarter-turn branches and their disjointness; the antipodal orbit construction and the primitive-midpoint edge key; existence and uniqueness of the stable matching for globally ranked pairs; the deterministic orthogonal frame and its Lagrange-Gauss reduction; the stateful one-h-block arithmetic generator with soundness, completeness, exactly-once emission, strict key ordering, one-block storage bound, and equality with the extensional incident stream; record-driven deferred acceptance with termination and equality with th", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "(1) Two wording corrections were applied to the source manuscript, both established before this run and summarized in FORMALIZATION_NOTES.md: the n = 41 control-layer count is \"six generator advances and six proposal-processing steps, of which exactly two involve a key comparison against a held record\", and the complexity proof asserts \"at most one key comparison\" per proposal rather than exactly one. (2) Cardinality statements use noncomputable Fintype instances, which never occur inside the executable map. (3) Literal n = 41 evaluations are performed by the Lean kernel on fuelled kernel-reducible mirrors that are proved equal to the real definitions. (4) The Challenge module restates the representation sets over ℤ × ℤ × ℤ rather than the project's `Triple` structure, with explicit coordinate bridges in the Solution. (5) The registered residual theorem specifies the stable matching through its primitive-midpoint projective ranking and preferred-sign lift, and proves uniqueness among stable perfect matchings (equivalences). It does not duplicate the generator's internal queue, cursor, or held-table state. The underlying project separately proves both the manuscript's reduction from", "alignment": true, "original": false, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [], "notes": "An AI-assisted internal statement audit matched the manuscript's definitions, theorems, algorithms, complexity claims, and worked examples to Lean declarations. The whole source was scanned for sorry, admit, sorryAx, axiom declarations, unsafe, opaque, extern, implemented_by, Lean.ofReduceBool, Lean.trustCompiler and native_decide; and #print axioms was run on the registered statements and the listed principal supporting declarations. Human review of the mathematical arguments, cited sources, exposition, and manuscript-to-Lean correspondence is ongoing. No external peer review has taken place." }, "canonical": { "repo": "grgaakash/tunnell-two-to-one-map", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-07-000010", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-07-000010", "commit": "d888a7c73c0ef1695dab8c48ccde629fbebb2d46", "trust": "high", "theorems": 10, "date": "2026-09-07" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "guanyangwang/ktv-swap-lean-comparator/formalization.yaml", "repo": "guanyangwang/ktv-swap-lean-comparator", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/guanyangwang/ktv-swap-lean-comparator/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "KTV Swap Chain Comparator", "description": "Lean 4 formalization and Comparator verification of the spectral-gap theorem for the lazy swap chain on binary contingency tables with fixed margins.", "authors": [ "Guanyang Wang" ], "maintainers": [ "Guanyang Wang" ], "license": "MIT", "role": "thin-wrapper", "substantive": "https://github.com/guanyangwang/ktv-swap-lean", "sources": [ { "title": "Spectral Gap for the Binary Fixed-Margin Swap Chain", "id": "arXiv:2606.22636", "authors": [ "Weibo Fu", "Qian Qin", "Guanyang Wang" ], "type": "preprint", "relationship": "formalizes", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.PR", "cs.DS", "math.CO", "stat.CO" ], "msc2020": [ "60J10" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "GPT-5 / Codex" ], "framework": "OpenAI Codex / ChatGPT coding agent" } ], "strongest": "agent", "models": [ "GPT-5 / Codex" ], "spend": "not recorded", "notes": "" }, "scope": "Comparator wrapper for the Mathlib-only statement of the fixed-margin lazy swap-chain spectral-gap theorem. Challenge.lean is a self-contained Mathlib-only challenge containing the definitions needed to state the theorem. Solution.lean imports ChallengeDefs.lean and the ktv-swap-lean proof library to prove the same statement.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "KTV.Johnson.MathlibOnly.binaryFixedMarginLazySwapChain_spectralGap_card", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The comparator theorem states the result as an OperatorSpectralGapAtLeast Poincare/Dirichlet-form inequality over rational functions on a finite state space, rather than as an eigenvalue statement. The statement uses ordered distinct row and column pairs; for this swap operator this is equivalent to the paper's unordered-pair proposal because each unordered rectangle has four ordered representatives producing the same swap result. Row and column index sets are arbitrary finite types.", "alignment": true, "original": false, "review": { "status": "comparator-passed", "bucket": "other", "reviewers": [], "notes": "The GitHub Actions workflow runs ./verify.sh on Ubuntu using landrun, lean4export, and the permitted axiom list in comparator.json; see the workflow badge in README.md for the current comparator status." }, "canonical": { "repo": "guanyangwang/ktv-swap-lean-comparator", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-20-000004", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-20-000004", "commit": "fd4dc985900b1fba59791067dcf3bf9341a94bda", "trust": "high", "theorems": 1, "date": "2026-08-20" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "Inspectordaniel/metric-surgery-lean/formalization.yaml", "repo": "Inspectordaniel/metric-surgery-lean", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/Inspectordaniel/metric-surgery-lean/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "An atomic decomposition of one-dimensional metric currents without boundary", "description": "This project formalizes all five main results of arXiv:2502.09871, \"An atomic decomposition of one-dimensional metric currents without boundary\" (Chen, Goodman, Hernandez, Spector), in Lean 4 against Mathlib.", "authors": [ "Daniel Spector (NTNU, Taipei)" ], "maintainers": [ "Daniel Spector (NTNU, Taipei)" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "An atomic decomposition of one-dimensional metric currents without boundary", "id": "arXiv:2502.09871", "authors": [ "You-Wei Benson Chen", "Jesse Goodman", "Felipe Hernandez", "Daniel Spector" ], "type": "preprint", "relationship": "formalizes", "endorsement": "participated" }, { "title": "Currents in metric spaces", "id": "doi:10.1007/BF02392711", "authors": [ "Luigi Ambrosio", "Bernd Kirchheim" ], "type": "article", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Structure of metric cycles and normal one-dimensional currents", "id": "doi:10.1016/j.jfa.2012.12.007", "authors": [ "Emanuele Paolini", "Eugene Stepanov" ], "type": "article", "relationship": "background", "endorsement": "not-contacted" } ], "related": [], "classification": { "arxiv": [ "math.FA" ], "msc2020": [ "49Q15", "30L99", "28A75" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Claude Sonnet", "Claude Opus" ], "framework": "Trellis (https://github.com/wpegden/trellis)" } ], "strongest": "agent", "models": [ "Claude Opus", "Claude Sonnet" ], "spend": "not tracked", "notes": "The Lean development was produced with Trellis driving Claude, as described under automation.methods, and every proof is machine-checked. The human contribution is mathematical rather than tactic-level: the author selected and stated the five compared theorems and the two citation hypotheses, and reviewed those statements against the paper. The proofs themselves were not written by hand." }, "scope": "All five main results of arXiv:2502.09871 are formalized in Lean 4 against Mathlib, and all five are proved: ApproximationMetric (Theorem 1.1), BBAssertion (Theorem 4.4), SurgeryEta (Corollary 3.2), C1SmallBall (Corollary 2.6) and EtaRepeats (Corollary A.6, psi display). No statement is weakened and no hypothesis is added beyond the instance-level nonemptiness noted in fidelity.divergences. Two results the paper cites rather than proves — Ambrosio-Kirchheim Proposition 2.7 (mass supremum formula, hard direction) and Paolini-Stepanov Theorem 3.1 with Proposition 4.2 (existence of a mass-realizi", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "Four deliberate, documented divergences, none of them weakenings. (1) Universal constants are pinned to numerals rather than left existential: C = 4*522 = 2088 in Theorem 1.1, C' = 522 in Corollary 3.2, and the paper's prescription epsilon = c*eta, n = ceil(epsilon^-2) is pinned at c = 1/15, sharp at eta = 1 for this argument. The paper quantifies these constants universally and outside the data, so a pinned numeral is at least as strong as the paper's existential form. (2) Corollary 2.6 is stated for an arbitrary metric space rather than for a complete, separable, geodesic one, since its proof uses none of those hypotheses; this is a strengthening. (3) Theorems 1.1 and 4.4 carry a [Nonempty E] instance hypothesis. This is an artifact of the Lean formulation, not of the mathematics: the conclusions assert total functions into Curve E, and Curve E is uninhabited exactly when E is. (4) Ambrosio-Kirchheim Proposition 2.7 and Paolini-Stepanov Theorem 3.1 with Proposition 4.2, which the paper cites rather than proves, are explicit hypotheses of Theorems 1.1 and 4.4 rather than axioms, so they are visible in the statements instead of being assumed silently; Corollaries 2.6, 3.2 and A.6 a", "alignment": true, "original": false, "review": { "status": "preprint under review; formalization not independently peer reviewed", "bucket": "other", "reviewers": [ "Daniel Spector (NTNU, Taipei)" ], "notes": "The five theorem statements in Challenge.lean and the two citation hypotheses MassSupFormulaStatement and PaoliniStepanovExistsStatement were audited against the paper by the author, who is also an author of the paper. All Lean proofs are machine-checked by the Lean 4 kernel, and each of the five compared declarations depends only on propext, Classical.choice and Quot.sound. The underlying preprint is under review; the formalization itself has had no independent peer review, and no reviewer other than the author has audited the statement surface." }, "canonical": { "repo": "inspectordaniel/metric-surgery-lean", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-21-000001", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-21-000001", "commit": "91c2fc65884c9b284fdc6ba4071aa63c12ded471", "trust": "high", "theorems": 5, "date": "2026-09-21" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "itpplasma/global-stafford-formal/formalization.yaml", "repo": "itpplasma/global-stafford-formal", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/itpplasma/global-stafford-formal/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Global Stafford", "description": "Lean formalization of the same-divisor identity 1 = d*R + F*d*S in the ring of k-linear finite-order differential operators of every smooth integral finitely generated algebra over a characteristic-zero field, extending Stafford's Conjecture 3.8 from Weyl algebras to smooth affine varieties.", "authors": [], "maintainers": [], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "Global Stafford theorem: project proof record", "id": "", "authors": [], "type": "original-proof", "relationship": "other", "endorsement": "participated" }, { "title": "Module Structure of Weyl Algebras", "id": "https://doi.org/10.1112/jlms/s2-18.3.429", "authors": [ "J. T. Stafford" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "The Bernstein class of modules on algebraic manifolds", "id": "", "authors": [ "J.-E. Björk" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Module Structure of Rings of Differential Operators", "id": "https://doi.org/10.1112/plms/s3-57.3.417", "authors": [ "S. C. Coutinho", "M. P. Holland" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Module structure of Weyl algebras", "id": "https://doi.org/10.1112/jlms.70373", "authors": [ "Gwyn Bellamy" ], "type": "paper", "relationship": "background", "endorsement": "" } ], "related": [ { "id": "https://doi.org/10.5281/zenodo.22667041", "relationship": "depends-on", "note": "The reviewed Global Stafford build consumes stafford38-formal commit e77e176c381ca2d6b20c030f9227f1b031415d2e. The older 784b5992 revision is retained in research history only as a historical inspection pin." } ], "classification": { "arxiv": [ "math.RA", "math.AG" ], "msc2020": [ "16S32", "14F10" ] }, "automation": { "methods": [ { "method": "manual", "models": [], "framework": "Lean 4 and Mathlib" }, { "method": "agent", "models": [ "claude-fable-5-1", "claude-opus-5", "claude-sonnet-5" ], "framework": "Claude Code" } ], "strongest": "agent", "models": [ "claude-fable-5-1", "claude-opus-5", "claude-sonnet-5" ], "spend": "not tracked", "notes": "Model use records process assistance and does not assign authorship of a mathematical idea." }, "scope": "The universal same-divisor theorem over every smooth integral affine algebra in characteristic zero is proved and transported to the Mathlib-only Challenge. The current dependency pins passed isolated Phase I/full verification, endpoint and consumer audits, and Comparator with NanoDa and Lean acceptance. Palomar's canonical-Challenge comparison also passed locally; its outer systemd wrapper was unavailable in that local replay. The registry record is PALOMAR-2026-09-05-000007 version 2. The unique submission Challenge and retained legacy Challenge are byte-identical alternative templates, each", "sorry_count": 2, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "GlobalStafford.universalStatement", "file": "GlobalStafford/Assembly/Closure.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "GlobalStaffordChallenge.universalStatement", "file": "GlobalStaffordSolution.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The formal proof follows the right-moved conjugation variant (paper Remark 4.2) and a polynomial form of the chart hypothesis; the produced certificate is the paper's. Smoothness is Mathlib's Algebra.Smooth.", "alignment": true, "original": true, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [], "notes": "Mechanical verification succeeded. The internal informal argument passed two independent-context model reviews and the formal development has kernel checking, scoped agent audits, and statement correspondence in docs/paper-lean-specification.md. The 2026-09-09 provenance/literature audit addresses the stale dependency-revision and novelty-evidence comments by distinguishing the consumed e77e176 Stafford38 pin from the historical 784b5992 inspection pin and documenting the prior-art search. Human expert review of the paper and formal statement remains separate." }, "canonical": { "repo": "itpplasma/global-stafford-formal", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-09-000001", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-09-000001", "commit": "e676caed1253e5d09fc6699d5dc683febbabf7bb", "trust": "high", "theorems": 1, "date": "2026-09-09" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "itpplasma/stafford38-formal/formalization.yaml", "repo": "itpplasma/stafford38-formal", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/itpplasma/stafford38-formal/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Stafford 3.8", "description": "Lean formalization of Stafford's Conjecture 3.8 for every characteristic-zero field and every Weyl-algebra rank, proving that each nonzero d admits F, R, and S with 1 = d*R + F*d*S. The development also proves an exact Bernstein-degree fixed-source strengthening and derived localization, adjoint, and evolutionary results.", "authors": [], "maintainers": [], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "Original proof of Stafford's Conjecture 3.8 in this formalization", "id": "", "authors": [], "type": "original-proof", "relationship": "other", "endorsement": "participated" }, { "title": "Module structure of Weyl algebras", "id": "", "authors": [ "J. T. Stafford" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Module structure of Weyl algebras", "id": "10.1112/jlms.70373", "authors": [ "Gwyn Bellamy" ], "type": "paper", "relationship": "background", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.RA" ], "msc2020": [ "16S32" ] }, "automation": { "methods": [ { "method": "manual", "models": [], "framework": "Lean 4 and Mathlib" }, { "method": "agent", "models": [ "gpt-5.5", "gpt-5.6-luna", "gpt-5.6-sol", "gpt-6-astra", "claude-fable-5", "claude-haiku-4-5-20251001", "claude-opus-5[1m]", "claude-opus-5" ], "framework": "Codex and Claude Code" } ], "strongest": "agent", "models": [ "claude-fable-5", "claude-haiku-4-5-20251001", "claude-opus-5", "claude-opus-5[1m]", "gpt-5.5", "gpt-5.6-luna", "gpt-5.6-sol", "gpt-6-astra" ], "spend": "not tracked", "notes": "Model use records process assistance and does not assign authorship of a mathematical idea. The private evidence register preserves the exact project-scoped metadata and its coverage limits." }, "scope": "Universal Stafford identity in every rank and exact Bernstein-degree source in positive rank. The controller-host build, 20 endpoint reports, 14 consumer reports, import audits, and both Comparator configurations passed, including NanoDa and Lean kernel acceptance. A separate isolated-clone replay is pending. Each of the two Challenge templates has one deliberate compared placeholder; these are excluded from the proved-declaration count below.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "Stafford38.universalStatement", "file": "Stafford38/FoundationClosure.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Stafford38.universalFixedSourceStatement", "file": "Stafford38/FoundationClosure.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The formal theorem keeps the written noncommutative order 1 = d*R + F*d*S and works over every characteristic-zero field. It includes n = 0 and replaces the manuscript's nonzero-nonunit restriction by the slightly stronger d != 0 restriction, since units are immediate. The fixed-source theorem is stated separately and requires positive rank for its linear Weyl coordinate.", "alignment": true, "original": true, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [], "notes": "The proof has extensive automated replay, kernel checking, scoped agent audits, and source-correspondence checks. Independent human expert review remains open. The author authorized publication on 2026-09-08; the release requires the recorded automated checks." }, "canonical": { "repo": "itpplasma/stafford38-formal", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-05-000007", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-05-000007", "commit": "914d6ad0c8ac9851e782fa61302ba340996d9472", "trust": "high", "theorems": 1, "date": "2026-09-08" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "JaTaSn/lattice-line-covers/lean/formalization.yaml", "repo": "JaTaSn/lattice-line-covers", "path": "lean/formalization.yaml", "directory": "lean", "origins": [ "palomar" ], "url": "https://github.com/JaTaSn/lattice-line-covers/blob/HEAD/lean/formalization.yaml", "version": "v0.4", "missing": [], "name": "lattice-line-covers", "description": "A Lean 4 formalization of Theorem 1 of arXiv:2608.22550: there is a family of lattice lines covering the integer plane whose set of realized directions is dense. Concretely, a family F of subsets of Z^2 is exhibited such that (i) every member is a genuine lattice line, the nonempty solution set of Px + Qy = c with P, Q coprime; (ii) the members cover Z^2; (iii) two members of different direction are disjoint, so no lattice point is covered twice in different directions; and (iv) the directions that occur are dense in the space RP^1 of line directions, that is, every angle in [0, pi) is approxi", "authors": [ "Jan Snellman" ], "maintainers": [ "Jan Snellman " ], "license": "MIT", "role": "substantive-development", "substantive": "", "sources": [ { "title": "On the directions occurring in lattice-line coverings of the integer plane", "id": "arXiv:2608.22550", "authors": [ "Jan Snellman" ], "type": "preprint", "relationship": "formalizes", "endorsement": "participated" } ], "related": [], "classification": { "arxiv": [ "math.CO", "math.NT" ], "msc2020": [ "05B40", "11H31", "52C15", "11B25", "11B57" ] }, "automation": { "methods": [ { "method": "agent", "models": [], "framework": "Claude Code" } ], "strongest": "agent", "models": [], "spend": "", "notes": "The mathematics was developed in a Typst working draft (../article/lattice_line_covers_pedantic.typ, kept in this repository unmodified) that doubled as the formalization's own guide, with inline \"formalization status\" annotations tracking progress lemma by lemma. See STRATEGY.md." }, "scope": "A family of straight lines, each through infinitely many points of Z^2, covering every lattice point, with no two lines of different direction sharing a lattice point, whose realized directions are dense in the space of line directions (RP^1). The full paper's other remarks and discussion are not formalized -- only Theorem 1 itself.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "Palomar.main_theorem", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "STATEMENT: none. The Lean statement is a verbatim restatement of Theorem 1 of the paper, in the Palomar namespace; see the alignment note below. No hypothesis was strengthened, no conclusion weakened, and nothing in Theorem 1 is left informal. PROOF: two divergences, both deliberate. (1) Disjointness of lines of different direction is argued differently. The article derives it from the Rigidity and Coset lemmas of section 2 -- a criterion for when two individual lines meet. The formalization instead partitions each region into sub-cosets and carries a nested-region invariant: splitting_partition -> RsubCoset_disjoint_of_ne -> claims_disjoint, together with det_eq_zero_of_lline_eq, feeding family_valid. Rigidity and Coset are nonetheless formalized, as LatticeLineCovers.rigidity and LatticeLineCovers.coset in Basic.lean, for fidelity with section 2; they have no call site and main_theorem does not depend on them. (2) The concrete target sequence differs. The article's figures and verification code use an equidistributed golden-angle sequence; the formalization uses thetaRat (Density.lean), an enumeration of the rationals clamped to [0, pi). Theorem 1 asserts only that the realized d", "alignment": true, "original": false, "review": { "status": "author-verified", "bucket": "author-verified", "reviewers": [ "Jan Snellman (author; verified the final manuscript and the build)", "Claude (Anthropic) -- AUTOMATED review by fresh subagent instances, not a human reviewer; run from Claude Code session f5f0edbb-abe5-4e27-97b6-0084ee0b3b7a on 2026-08-02; specific model version not recorded, consistent with automation.methods[].models above" ], "notes": "NO INDEPENDENT HUMAN HAS REVIEWED THIS. The status author-verified means what it says: the author checked it. Two separate things are easily conflated here, so both are set out. (1) The automated review, and its basis. Before and alongside formalization, the informal proof (the Typst draft article/lattice_line_covers_pedantic.typ) was put through adversarial review rounds performed by fresh Claude subagent instances -- separate contexts launched in parallel with an identical brief and no shared history, so that agreement between them was evidence rather than an echo. Scope: the proof's own reasoning -- recursive bookkeeping, coverage, disjointness, density -- with the Splitting and Steering lemmas handed over as machine-verified black boxes to be trusted rather than rechecked, and attentio" }, "canonical": { "repo": "jatasn/lattice-line-covers", "directory": "lean" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-26-000003", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-26-000003", "commit": "68f90ff9d90cb71af2c76eb1291a6ddf564cf4fb", "trust": "high", "theorems": 1, "date": "2026-09-01" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "jaumededios/cantor-frames-palomar/formalization.yaml", "repo": "jaumededios/cantor-frames-palomar", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/jaumededios/cantor-frames-palomar/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Cantor measures with odd base do not admit Fourier frames", "description": "", "authors": [ "Jaume de Dios Pont", "Lukas Liehr", "Mitchell A. Taylor" ], "maintainers": [ "Jaume de Dios Pont" ], "license": "Apache-2.0", "role": "", "substantive": "https://github.com/jaumededios/Cantor_Measure_Frames", "sources": [ { "title": "Cantor measures with odd base do not admit Fourier frames", "id": "arXiv:2607.08656v1", "authors": [ "Jaume de Dios Pont", "Lukas Liehr", "Mitchell A. Taylor" ], "type": "paper", "relationship": "formalizes", "endorsement": "participated" } ], "related": [], "classification": { "arxiv": [ "math.FA" ], "msc2020": [ "28A80", "42B05", "46B15" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "GPT-5.5" ], "framework": "ChatGPT web interface and Codex" }, { "method": "agent", "models": [ "GPT-5" ], "framework": "Codex" } ], "strongest": "agent", "models": [ "GPT-5", "GPT-5.5" ], "spend": "", "notes": "" }, "scope": "The compared result is Theorem 1.1 of arXiv:2607.08656v1: for odd integers b > 1, the fair {0, 2} base-b Cantor measure admits no Fourier frame. The project does not compare every intermediate result in the paper or the even-base half of the broader classification. The sorry count refers to the proof-backed Solution and its pinned dependency; the single sorry in Challenge.lean is Comparator's intentional statement placeholder.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "CantorFramesPalomar.NoFourierFrameExists", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "No known mathematical divergence. Challenge.lean uses the appendix's `Fin 2` digit space and compares the headline odd-base nonexistence theorem. It does not compare every intermediate result or the prior even-base theorem. This scope choice is detailed below.", "alignment": true, "original": false, "review": { "status": "human-authored-statement-and-definitions", "bucket": "other", "reviewers": [ "Jaume de Dios Pont", "Lukas Liehr", "Mitchell A. Taylor" ], "notes": "The final paper, public Lean statement, and statement-facing definitions are human-written. The generated internal Lean proof is not claimed to have been written or reviewed line by line by humans. Details follow." }, "canonical": { "repo": "jaumededios/cantor-frames-palomar", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-17-000004", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-17-000004", "commit": "e1ce015224e702a4e93ae59e8ca73616daaa8d5a", "trust": "high", "theorems": 1, "date": "2026-08-17" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "JD-Jones-ASES/dr-lean/formalization.yaml", "repo": "JD-Jones-ASES/dr-lean", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/JD-Jones-ASES/dr-lean/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "dr-lean: Dittert and rectangular semimatching inequalities", "description": "A Lean 4 alternative proof of the square Dittert inequality in every positive dimension, and sharp rectangular semimatching inequalities. Rectangular results include every admissible rectangle at K=2 and K=3, every four-row rectangle at K=4, the 5-by-5 and 20-by-20 K=4 cases, all-order large-board thresholds, the stated endpoint strips, and their small-side corollaries, and K=n-1 on every square of order n>=21. The probability event is distinct rows or distinct columns in independent draws with replacement. Every theorem includes nonnegative boundary matrices and unique equality at the uniform", "authors": [ "JD Jones" ], "maintainers": [ "JD Jones" ], "license": "MIT", "role": "", "substantive": "", "sources": [ { "title": "Rybin P2: independent-cell rectangular semimatching problem", "id": "", "authors": [ "Dmitry Rybin" ], "type": "web discussion", "relationship": "background", "endorsement": "not-contacted" }, { "title": "An interpretation of the Dittert conjecture in terms of semi-matchings", "id": "doi:10.1016/j.disc.2007.01.008", "authors": [ "G.-S. Cheon", "Ian M. Wanless" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Public Dittert conjecture proof project", "id": "", "authors": [ "Pedro Paulo Marques do Nascimento" ], "type": "other", "relationship": "independently-proves", "endorsement": "not-contacted" }, { "title": "Van der Waerden/Schrijver-Valiant like Conjectures and Stable (aka Hyperbolic) Homogeneous Polynomials: One Theorem for all", "id": "arXiv:0711.3496v2", "authors": [ "Leonid Gurvits" ], "type": "paper", "relationship": "adapts", "endorsement": "not-contacted" }, { "title": "On Leonid Gurvits's proof for permanents", "id": "", "authors": [ "Monique Laurent", "Alexander Schrijver" ], "type": "paper", "relationship": "adapts", "endorsement": "not-contacted" }, { "title": "Minimum permanents on two faces of the polytope of doubly stochastic matrices", "id": "", "authors": [ "Kyle Pula", "Seok-Zun Song", "Ian M. Wanless" ], "type": "paper", "relationship": "adapts", "endorsement": "not-contacted" }, { "title": "Some results towards the Dittert conjecture on permanents", "id": "doi:10.1016/j.laa.2010.08.041", "authors": [ "Gi-Sang Cheon", "Ian M. Wanless" ], "type": "paper", "relationship": "adapts", "endorsement": "not-contacted" }, { "title": "Proof of Dittert's conjecture for dimensions n >= 17", "id": "arXiv:2606.01531v1", "authors": [ "Zhekai Pang" ], "type": "paper", "relationship": "adapts", "endorsement": "not-contacted" }, { "title": "New Constructive Aspects of the Lovasz Local Lemma", "id": "arXiv:1001.1231v5", "authors": [ "Bernhard Haeupler", "Barna Saha", "Aravind Srinivasan" ], "type": "paper", "relationship": "adapts", "endorsement": "not-contacted" } ], "related": [], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05A20", "15A15" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "OpenAI Codex (GPT-6)" ], "framework": "Codex with parallel mathematical and Lean agents" } ], "strongest": "agent", "models": [ "OpenAI Codex (GPT-6)" ], "spend": "not tracked", "notes": "See DISCLOSURE.md for AI assistance and human responsibility; detailed monetary accounting is unavailable." }, "scope": "The twenty principal declarations have the precise ranges recorded in docs/THEOREMS.md and release-targets.json. Rectangular statements impose only nonnegativity and total mass one; Dittert uses total mass n. Endpoint families include transpose and all statements give sharp inequality and iff uniform equality. K=1 is separated from the general uniqueness claim. The deliberate Challenge statement placeholders are excluded from proof counts. Solution does not import Challenge. No full arbitrary-rectangle P2 or unrestricted new Dittert priority claim is made.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The square proof is an alternative spectral and stationary-cut argument. The rectangular proof uses actual collision laws, matrix averaging, retained kernels, finite certificates and permanent boundary bounds. Algebraic identities retain signed inputs where proved, but their probabilistic conclusions require nonnegative matrices. Original-row and deleted-row avoidance probabilities remain distinct. All-integer ranges and exact closed thresholds replace no theorem by a finite sample.", "alignment": false, "original": true, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "AI agents developed and independently cross-reviewed the proof chains. No independent human expert refereeing or source-author endorsement is claimed. docs/VERIFICATION.md and exact-commit receipts distinguish source checks, Lean kernel compilation, Comparator and NanoDa replay, independent CI. Mechanical acceptance alone establishes neither novelty nor mathematical significance." }, "canonical": { "repo": "jd-jones-ases/dr-lean", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "JD-Jones-ASES/fs-formal/formalization.yaml", "repo": "JD-Jones-ASES/fs-formal", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/JD-Jones-ASES/fs-formal/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "fs-formal: an improved Furstenberg-Sárközy lower bound", "description": "A Lean 4 formalization of an explicit construction for the Furstenberg-Sárközy problem. If D(N) is the largest size of a subset of {1,...,N} with no nonzero square difference, the development proves that an explicit eleven-block constant alphaInf is at most liminf log D(N)/log N, that every exponent below alphaInf holds eventually pointwise, and that 0.7537 < alphaInf. The proof includes the finite certificates, the composite even-digit lift, CRT gluing, asymptotic passage, and kernel-checked numerical inequalities. This strictly improves Krachun's preceding explicit exponent 0.752796455874514", "authors": [ "JD Jones" ], "maintainers": [ "JD Jones" ], "license": "MIT", "role": "", "substantive": "", "sources": [ { "title": "An improved lower bound for the Furstenberg-Sárközy problem", "id": "https://github.com/JD-Jones-ASES/fs-lower-bound/tree/c2c06875ee4544e069e8da16564ee429d870a13e", "authors": [ "JD Jones" ], "type": "other", "relationship": "formalizes", "endorsement": "participated" }, { "title": "Square-difference-free sets beyond the three-quarter barrier", "id": "arXiv:2608.01325", "authors": [ "D. Krachun" ], "type": "paper", "relationship": "background", "endorsement": "" } ], "related": [ { "id": "https://github.com/JD-Jones-ASES/fs-lower-bound-lean/tree/2e6c33c82e57415e5238467620a44a8e1d9b5904", "relationship": "adapts", "note": "Canonical completed Lean development from which this Palomar-ready repository was copied. This repository adds the independent Mathlib-only Challenge, proved Solution bridge, Comparator configuration, registry metadata, and immutable CI pins, and ports the project from unsupported patch release v4.33.1 to v4.33.0." } ], "classification": { "arxiv": [ "math.CO", "math.NT" ], "msc2020": [ "11B30", "11B75" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Claude Code (Fable 5)", "Codex (GPT 5.6 Sol)", "Grok Build (Grok 4.6)", "OpenAI Codex (Palomar adaptation; exact model not recorded)" ], "framework": "mixed AI-agent workflow" } ], "strongest": "agent", "models": [ "Claude Code (Fable 5)", "Codex (GPT 5.6 Sol)", "Grok Build (Grok 4.6)", "OpenAI Codex (Palomar adaptation; exact model not recorded)" ], "spend": "not tracked", "notes": "AI systems generated and audited substantial mathematics and code; JD Jones directed the work and is the responsible author and maintainer. Git history and the source DISCLOSURE.md preserve the recorded division of work." }, "scope": "The three compared theorems prove alphaInf <= liminf log D(N)/log N, the eventual pointwise bound for every rho < alphaInf, and 0.7537 < alphaInf. The formalization does not state all digits of the informal approximation 0.753741541837329405..., an endpoint estimate with a fixed multiplicative constant, optimality of the pool or certificates, or the induced point-line matching corollary in the source note. Deliberate sorries occur only in Challenge.lean and are excluded from the proof counts below; Solution and the proof development are sorry-free.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "FsFormal.sdf_liminf_ge", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "FsFormal.sdf_pointwise", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "FsFormal.alphaInf_gt", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The compared Lean statements match the three targets in the canonical formalization. The rational inequality 0.7537 < alphaInf is formalized, but the longer decimal shown informally is not a Lean theorem. The informal induced-matching corollary is omitted. The Palomar port changes Lean and Mathlib from v4.33.1 to v4.33.0 for exact exporter compatibility; it does not intentionally change any mathematical declaration.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "JD Jones" ], "notes": "No independent human expert or peer review is recorded. The completed upstream project passed Lean CI. This port's transitive axiom audit covers 129 declarations, and its source guards also pass. Palomar's pinned Comparator accepted the three declarations and Lean-kernel replay in a local macOS development run; that run used Comparator's unsandboxed development shim and disabled NanoDa. The real Landrun and NanoDa replay remain the Linux CI and Palomar-verifier checks. Those checks establish proof replay and statement equality, not novelty, significance, or correctness of the informal interpretation." }, "canonical": { "repo": "jd-jones-ases/fs-formal", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-26-000004", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-26-000004", "commit": "4e796565301b1425c53e807a6edf408535b2e922", "trust": "high", "theorems": 3, "date": "2026-08-26" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "JD-Jones-ASES/gn-lean/formalization.yaml", "repo": "JD-Jones-ASES/gn-lean", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/JD-Jones-ASES/gn-lean/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "gn-lean: the Gurvich–Naumova partition conjecture", "description": "A Lean 4 proof that for every nonnegative integer n, the distinct integers 1,...,n can be partitioned into nonempty sets of at most three elements, each having sum equal to a nonnegative integer power of three. This proves Conjecture 1 of Gurvich and Naumova, arXiv:2508.00946v3. The development constructs permutations with all prescribed nonzero displacements, derives signed zero-sum triple partitions, proves a contraction and complement-pair extension, and completes strong induction on n. All required finite certificates and unbounded sequence constructions are proved in Lean.", "authors": [ "JD Jones" ], "maintainers": [ "JD Jones" ], "license": "MIT", "role": "", "substantive": "", "sources": [ { "title": "Partitioning set [n]={1,...,n} into subsets of size at most m such that all sums are powers of m", "id": "arXiv:2508.00946v3", "authors": [ "Vladimir Gurvich", "Mariya Naumova" ], "type": "paper", "relationship": "adapts", "endorsement": "not-contacted" }, { "title": "Hooked extended Langford sequences of small and large defects", "id": "doi:10.2478/s12175-014-0242-6", "authors": [ "S. Mor", "V. Linek" ], "type": "paper", "relationship": "adapts", "endorsement": "not-contacted" } ], "related": [], "classification": { "arxiv": [ "math.CO", "math.NT" ], "msc2020": [ "05A17", "11B75" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "OpenAI Codex (GPT-6)" ], "framework": "Codex with parallel mathematical and Lean agents" } ], "strongest": "agent", "models": [ "OpenAI Codex (GPT-6)" ], "spend": "not tracked", "notes": "See DISCLOSURE.md. Detailed prompt, token, and monetary accounting was not retained." }, "scope": "The single compared theorem proves the m=3 partition conjecture for every n, including the auxiliary empty interval. Blocks are finite sets, are nonempty and pairwise disjoint, cover exactly {1,...,n}, have cardinality at most three, and have sum 3^k for some natural k. The formalization makes no claim about partitions for general m, optimality, uniqueness, or counts. The intentional placeholder in Challenge.lean is excluded from the proof counts. Solution does not import Challenge.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "GurvichNaumova.partition_into_powers_of_three", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The statement uses finite sets of integers constrained by their union to equal the positive interval {1,...,n}; hence no negative or repeated labels are admitted. The auxiliary n=0 case is included. The unbounded permutation lemma is proved by six explicit Langford families and finite small cases, rather than assuming a published sequence-existence theorem. Only the conjecture for m=3 is addressed.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "No independent human expert review, source-author endorsement, or completed novelty survey is recorded. AI agents developed and cross-checked the proof. See VERIFICATION.md for the precise mechanical checks completed at this snapshot. 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With N_u = {1, 2, 3, 4} union {3^t - 4, 3^t - 2, 3^t - 1, 3^t, 3^t + 1, 3^t + 2, 3^t + 3, 3^t + 5 : t >= 2} and E_2 = {13} union {3^t - 3 : t >= 2} union {3^t - 6 : t >= 3}, the four compared theorems say that for n >= 1 the count is 1 exactly on N_u, is 2 exactly on E_2, and is at least 3 elsewhere, and that t", "authors": [ "JD Jones" ], "maintainers": [ "JD Jones" ], "license": "MIT", "role": "", "substantive": "", "sources": [ { "title": "Partitioning set [n]={1,...,n} into subsets of size at most m such that all sums are powers of m", "id": "arXiv:2508.00946v3", "authors": [ "Vladimir Gurvich", "Mariya Naumova" ], "type": "paper", "relationship": "adapts", "endorsement": "not-contacted" }, { "title": "Hooked extended Langford sequences of small and large defects", "id": "doi:10.2478/s12175-014-0242-6", "authors": [ "S. Mor", "V. Linek" ], "type": "paper", "relationship": "adapts", "endorsement": "not-contacted" }, { "title": "On the number of additive permutations and Skolem-type sequences", "id": "doi:10.26493/1855-3974.1098.ca0", "authors": [ "Diane M. Donovan", "Michael J. Grannell" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" } ], "related": [ { "id": "PALOMAR-2026-09-07-000013", "relationship": "builds-on", "note": "gn-lean, by the same author: every {1, ..., n} has a good partition. The directory GN/ of this repository is an unchanged copy of that development at commit 73bb87b16da8c053fd07e0d4dc43732b74335a3b, MIT, and the counting layer in GNM/ uses its objects and theorems." } ], "classification": { "arxiv": [ "math.CO", "math.NT" ], "msc2020": [ "05A17", "11B75" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Claude (Anthropic)", "GPT-6 (OpenAI)", "Grok (xAI)" ], "framework": "Claude Code with parallel subagents" } ], "strongest": "agent", "models": [ "Claude (Anthropic)", "GPT-6 (OpenAI)", "Grok (xAI)" ], "spend": "not tracked", "notes": "See DISCLOSURE.md." }, "scope": "Four compared declarations over four compared definitions. They classify the number of good partitions of {1, ..., n} for n >= 1: exactly one on N_u, exactly two on E_2, at least three elsewhere, and exactly two at every 3^t - 6 with t >= 3. No asymptotics and no closed form for count n are proved, no value outside the classification is claimed, and nothing is claimed at any base other than three. The value n = 0 lies outside the classification and is excluded by hypothesis in two of the theorems. The intentional placeholders of Challenge.lean are excluded from proof counts; Solution.lean does", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "GNM.count_eq_one_iff", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "GNM.count_eq_two_iff", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "GNM.three_le_count_iff", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "GNM.count_eq_two_of_add_six", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "Intervals and centred intervals are stated as bounds on the elements rather than as interval constants, so that no statement type depends on an instance that can diverge. The two exceptional sets are written with sums (n + 4 = 3^t rather than n = 3^t - 4), which also keeps them inside the natural numbers. count n is the cardinality of a subtype of finite sets of finite sets of integers, as Nat.card; the good partitions are shown to form a finite set, and count n is its Finset cardinality. The value n = 0 has the empty partition and nothing else, so count 0 = 1; it lies outside the classification, which is why two of the four theorems assume 1 <= n. Blocks are finite sets of integers whose union is constrained to be the positive interval, so no repeated or negative labels are admitted. The source's own proofs are not formalized; its Theorem 3 is re-proved here independently.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "No independent human expert review and no source-author endorsement is recorded; AI agents developed and cross-reviewed the proof, the statements were checked against the source text, and VERIFICATION.md states what is checked and what is not. 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Theorem A characterises, by an if-and-only-if, when a Goethals-Seidel array of four sign-valued sequences over a finite abelian group extends to a Hadamard matrix through a border of width 4s whose strips are constant on the cosets of a subgroup. Theorem B is its house form, whose hypothesis is exactly a two-tier periodic-autocorrelation profile, and carries two classical degenerations at an arbitrary refl", "authors": [ "JD Jones" ], "maintainers": [ "JD Jones" ], "license": "MIT", "role": "", "substantive": "", "sources": [ { "title": "Hadamard-B: bordered Goethals-Seidel arrays - characterisation, classification, and three equivalence classes at order 668", "id": "https://github.com/JD-Jones-ASES/Hadamard-B/tree/01a7f0614208ca68d69e64a7cd5b560aadafbb8a", "authors": [ "JD Jones" ], "type": "other", "relationship": "formalizes", "endorsement": "participated" }, { "title": "A skew Hadamard matrix of order 36", "id": "J. Austral. Math. Soc. 11 (1970) 343-344", "authors": [ "J. M. Goethals", "J. J. Seidel" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Some classes of Hadamard matrices with constant diagonal", "id": "Bull. Austral. Math. Soc. 7 (1972) 233-249", "authors": [ "J. Wallis", "A. L. Whiteman" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Hadamard matrices from relative difference sets", "id": "J. Combin. Theory Ser. A 19 (1975) 287-300", "authors": [ "E. Spence" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "The Dokovic-Kotsireas compression device (two papers)", "id": "arXiv:1302.0571 and arXiv:1801.07627", "authors": [ "D. Z. Dokovic", "I. S. Kotsireas" ], "type": "paper", "relationship": "background", "endorsement": "" } ], "related": [ { "id": "https://github.com/leanprover-community/mathlib4/blob/db584cd6d46c92f209a44c0f1c829460d327499d/Mathlib/LinearAlgebra/Matrix/HadamardMatrix.lean", "relationship": "builds-on", "note": "At the exact Mathlib commit pinned by lake-manifest.json, which is the commit the tag v4.33.0 named when this package was built. Mathlib's Hadamard-matrix module supplies the Matrix.IsHadamard predicate that every compared conclusion is stated in, together with Matrix.IsHadamard.of_mul_conjTranspose and the transpose-side conjTranspose_mul field that the Theorem-D argument uses. It exhibits no Had" }, { "id": "https://github.com/leanprover-community/mathlib4/blob/db584cd6d46c92f209a44c0f1c829460d327499d/Mathlib/LinearAlgebra/Matrix/Circulant.lean", "relationship": "builds-on", "note": "At the same pinned Mathlib commit. Mathlib's circulant module is the nearest existing object to the development's type-1 development dev x, which is the transpose of Matrix.circulant x. The module carries circulant multiplication and commutation over an additive commutative group but no development over a general abelian group indexed by a quotient, and nothing on the block array; the proofs here " }, { "id": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-31-000001&version=1", "relationship": "builds-on", "note": "The sibling Lean repository Hadamard-formal from the same laboratory, https://github.com/JD-Jones-ASES/Hadamard-formal, registered 2026-08-31 as PALOMAR-2026-08-31-000001. Its developed-matrix algebra and its sixteen-block Goethals-Seidel Gram lemma are the ancestors of the corresponding lemmas here, ported with two changes: the six transposed blocks are negated, because that repository uses the S" }, { "id": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-17-000002&version=1", "relationship": "other", "note": "Paul-Lez/hadamard-668-comparator on GitHub, created 2026-08-14 and registered 2026-08-17 as PALOMAR-2026-08-17-000002. A Lean formalization that verifies a single Hadamard matrix of order 668 built from the publicly posted data of 2026-08-12; its Challenge.lean independently exhibits the bordered structure at that order - circulants on Fin 166, a width-4 border, per-block-constant strips, and the " }, { "id": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-29-000009&version=1", "relationship": "other", "note": "Arthur742Ramos/hadamard-668-lean on GitHub, created 2026-08-28 and registered 2026-08-29 as PALOMAR-2026-08-29-000009. A Lean formalization (Ramos-Hulak-de Queiroz) that verifies one supplied Hadamard matrix of order 668 from the same public data. It states no general construction theorem. As the source note puts it, neither of the two public order-668 formalizations states the general constructio" } ], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05B20" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Claude Code (Fable 5)", "Claude Opus (subagent lanes)" ], "framework": "multi-agent workflow (plan lane, staged implementation lanes, adversarial audit)" } ], "strongest": "agent", "models": [ "Claude Code (Fable 5)", "Claude Opus (subagent lanes)" ], "spend": "not tracked", "notes": "AI systems generated and audited the mathematics and the Lean code; JD Jones directed the work and is the responsible author and maintainer. AI assistance is disclosed at the project level, in DISCLOSURE.md, and in the source repository." }, "scope": "Formalized and compared: Theorem A in both directions and its sufficiency half separately; Theorem B (the house profile); the classical Goethals-Seidel theorem over an abelian group and the Wallis-Whiteman/Spence bordered construction, both at an arbitrary reflection shift; the row-sum rows D5 and D6; Lemma T, the s = 1 profile bijection, the psi(rho) = 1 conjugation and its Hadamard-iff corollary; Theorem D clauses (D-a)/(D-b) (the forced Gram table and pair-negated column table), (D-c) (the forced row table and Hadamard corner), (D-d) (the collapse of (H4) to a single 4x4 equation), the inte", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "HadamardBFormal.theoremA", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", 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"description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "HadamardBFormal.borderedGS_index_one", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "HadamardBFormal.lemmaT", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "HadamardBFormal.twist_profile_iff", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "HadamardBFormal.twist_isConjugation", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { 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Every statement is for the note's standard orientation only; the transpose-negated companion instance T2 of the order-20 record is out of scope by construction and the exporter refuses to emit it. The quotient G/K is modelled as a surjective additive hom kappa : G -> Gbar rather than as a quotient type, with K = ker kappa, i = card Gbar and w = card K, and the constancy of the fiber size is taken as a hypothesis rather than derived from surjectivity; the border tables are indexed by Fin 4 x Gbar rather than by Fin (4i). Theorem C is represented only by its two row-sum rows D5 and D6: the forced-parameter clauses D1-D4 and the classification corollary are not formalized. Clause (D-a') is stated with the trivial quotient carried as any additive hom into a subsingleton type, matching borderedGS_subsingleton, and the doubling of an i = 1 border is encoded by the two pair-equality hypotheses rather than by a separate doubling operator. The section-3 separation at order 668 is not formalized here; its exact 4-profile computations are outside this registration's scope and impractical under direct kernel reduction. Where the note writes Fin 4 for a corner at s = 1, several", "alignment": true, "original": false, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [ "Claude (Anthropic) - statement-fidelity audit lane, 2026-09-01", "GPT 5.6 (OpenAI, Codex desk) - independent package review, 2026-09-01" ], "notes": "BOTH REVIEWS WERE PERFORMED BY AI SYSTEMS, AND NO INDEPENDENT HUMAN EXPERT REVIEW HAS OCCURRED. The Claude lane audited every compared statement against the source note's current text and produced the divergence list under fidelity.divergences. The Codex review was independent of that lane: it rebuilt the package from an isolated checkout of the tracked files with the pinned Lean 4.33.0 toolchain, replayed the transitive axiom audit over every compared declaration, re-ran the exporter in --check mode, and reviewed the publication surface; its findings were adjudicated and applied before this commit. JD Jones is the responsible owner, publisher and maintainer and appears in authors and responsible_maintainers; he is NOT a reviewer and made no mathematical contribution. What has actually bee" }, "canonical": { "repo": "jd-jones-ases/hadamard-b-formal", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-01-000006", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-01-000006", "commit": "d99c0180872e4213e54272f9699d28f28032a2a3", "trust": "high", "theorems": 26, "date": "2026-09-01" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "JD-Jones-ASES/md-lean/formalization.yaml", "repo": "JD-Jones-ASES/md-lean", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/JD-Jones-ASES/md-lean/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "md-lean: exact height span, dart mass, and adjugate scaling for Mazur's greedy construction", "description": "Exact height-span of Mazur's scheme grammar, a local dart-mass bound, and an adjugate identity for the Dirichlet matrix of that construction; not a proof of the strong Papadimitriou–Ratajczak conjecture and not a theorem that every 3-connected plane graph attains the scheme maximum. Mazur builds injective integer heights by inductive recurrences on rooted pieces (edge, cycle, binary series, two-range join) and then forms a weighted Dirichlet matrix for the horizontal coordinate; the paper does not bound the height span or the bit length of the coordinates. For a well-formed scheme on n >= 2 ve", "authors": [ "JD Jones" ], "maintainers": [ "JD Jones" ], "license": "MIT", "role": "", "substantive": "", "sources": [ { "title": "A Proof of the Strong Papadimitriou–Ratajczak Conjecture", "id": "https://proofatlas.ai/formalizations/strong-papadimitriou-ratajczak-conjecture/", "authors": [ "Lech Mazur" ], "type": "paper", "relationship": "adapts", "endorsement": "not-contacted" }, { "title": "The strong Papadimitriou–Ratajczak conjecture, open since 2004, is now a theorem", "id": "https://x.com/LechMazur/status/2098915169799733339", "authors": [ "Lech Mazur" ], "type": "web discussion", "relationship": "background", "endorsement": "not-contacted" }, { "title": "On a conjecture related to geometric routing", "id": "doi:10.1016/j.tcs.2005.06.022", "authors": [ "Christos H. Papadimitriou", "David Ratajczak" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Some results on greedy embeddings in metric spaces", "id": "doi:10.1007/s00454-009-9227-6", "authors": [ "Tom Leighton", "Ankur Moitra" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "On succinct convex greedy drawing of 3-connected plane graphs", "id": "doi:10.1137/1.9781611973082.115", "authors": [ "Xin He", "Huaming Zhang" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "On the area requirements of planar greedy drawings of triconnected planar graphs", "id": "doi:10.1007/978-3-030-58150-3_35", "authors": [ "Giordano Da Lozzo", "Anthony D'Angelo", "Fabrizio Frati" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" } ], "related": [ { "id": "https://proofatlas.ai/formalizations/strong-papadimitriou-ratajczak-conjecture/", "relationship": "other", "note": "Mazur's Lean existence development for SPRC (ProofAtlas page of 2026-09-09; announced on X 2026-09-12). Not imported. This repository does not re-prove the existence theorem and does not share any Lean source with that development." } ], "classification": { "arxiv": [ "math.CO", "math.MG" ], "msc2020": [ "05C10", "05C62" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Grok 4.6 (xAI)" ], "framework": "Grok Build" } ], "strongest": "agent", "models": [ "Grok 4.6 (xAI)" ], "spend": "not tracked", "notes": "See DISCLOSURE.md. Prompt, token and monetary accounting was not retained." }, "scope": "Fifteen compared theorems over the compared definitions, plus the inductive type Scheme. span_succ_le_join_budget and exists_span_eq_join_budget are the two-parameter budget T+1 <= 2^j (n-2j), attained at every admissible pair (n,j)=(2,0) or 2j+3 <= n. span_le_exact and exists_span_eq_exact are the exact one-parameter scheme maximum F(n). span_le_two_pow is the coarser height-span bound. natAbs_det_le_prod_rowSum and natAbs_det_le_of_rowSum_le are the elementary Leibniz ell_1 determinant bounds. dirichlet_rowSum_le and dirichlet_det_le are the dart-weight row-sum and determinant bounds. dartMa", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "MazurSpan.span_succ_le_join_budget", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "MazurSpan.exists_span_eq_join_budget", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "MazurSpan.span_le_exact", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "MazurSpan.exists_span_eq_exact", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "MazurSpan.span_le_two_pow", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "MazurSpan.dirichlet_det_le", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "MazurSpan.dartMass_le", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "MazurSpan.dartWeight_interior_sum_le", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "MazurSpan.coord_denom", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "MazurSpan.dirichlet_adjugate_clears", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The compared theorems are consequences of Mazur's recurrences and weight recipe, not a formalization of his existence theorem. Join span uses the worst attachment height c = 1, so the scheme span upper-bounds the constructed span. The (n-1) factor in the row-sum bound is in-house: Mazur writes weights <= n T; dropping the zero self-term and bounding degree by n-1 is this development. The ell_1 determinant bound is Leibniz/Laplace, weaker than Hadamard's Euclidean form and of the same 2^{O(n^2)} order. Invertibility of M is Mazur's maximum principle and is not a compared theorem. The adjugate identity produces integer z with Mz = Delta u regardless; uniqueness requires Delta != 0, which is not proved. The products T^2 |det M| and T^2 |det M| T are motivated by Mazur's boundary values b_s = h_s(T-h_s)/T^2; the Lean sizes those products against T <= 2^{n-2} and separately exhibits the adjugate vector.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "No independent human expert review and no source-author endorsement is recorded. The statements were checked against Mazur §§2–3 for dropped hypotheses, and the Python censuses recheck the arithmetic and a finite weight census independently of Lean. Mechanical correctness does not establish priority or significance." }, "canonical": { "repo": "jd-jones-ases/md-lean", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-14-000001", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-14-000001", "commit": "b7f9f9deb5f5e520c99fd0d48893808d6ad082d6", "trust": "high", "theorems": 15, "date": "2026-09-14" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "JD-Jones-ASES/nk-lean/formalization.yaml", "repo": "JD-Jones-ASES/nk-lean", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/JD-Jones-ASES/nk-lean/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Lower bounds for sets without perfect-power differences", "description": "Let D_k(N) be the largest size of a subset of {1, ..., N} with no positive k-th-power difference. We prove D_k(N) >= c_k N^alpha_k for every N >= 1, with positive constants c_k and exponents 0.75806770413, 0.9142, and 0.95295 for squares, fourth powers, and sixth powers. The proofs use a general interval-moment criterion for every positive integer k. We also prove arithmetic lifting results, exact product and prime-depth laws for interval capacity, and existence of the binary square capacity limit. The square construction refines Naslund's published example while retaining its binary transitio", "authors": [ "JD Jones" ], "maintainers": [ "JD Jones" ], "license": "MIT", "role": "", "substantive": "", "sources": [ { "title": "Square-difference-free sets of exponent 0.75806746", "id": "", "authors": [ "Eric Naslund" ], "type": "paper", "relationship": "adapts", "endorsement": "not-contacted" }, { "title": "Square-Difference-Free Sets beyond the Three-Quarter Barrier", "id": "arXiv:2608.01325", "authors": [ "Dmitry Krachun" ], "type": "paper", "relationship": "adapts", "endorsement": "not-contacted" }, { "title": "Difference sets without squares", "id": "doi:10.1007/BF02454169", "authors": [ "Imre Z. Ruzsa" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Lower bounds in the polynomial Szemeredi theorem", "id": "arXiv:1908.06058", "authors": [ "Khalid Younis" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Prime powers units and finite subgroups of GL_n(Q)", "id": "https://kconrad.math.uconn.edu/blurbs/gradnumthy/primepowerunitsandGLnQ.pdf", "authors": [ "Keith Conrad" ], "type": "other", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Waring numbers over finite commutative local rings", "id": "arXiv:2212.12396", "authors": [ "Ricardo A. Podestá", "Denis E. Videla" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "On certain properties of the p-unitary Cayley graph over a finite ring", "id": "arXiv:2403.05635", "authors": [ "Tung T. Nguyen", "Nguyen Duy Tân" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "On k-th unitary Cayley graphs over finite commutative rings: structure and decompositions", "id": "arXiv:2606.06774", "authors": [ "Ricardo A. Podestá", "Denis E. Videla" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" } ], "related": [ { "id": "https://github.com/JD-Jones-ASES/rk-lean/tree/4bf2e3a56b6a522764a4d03e5dc606c12376185e", "relationship": "builds-on", "note": "Pinned MIT dependency for ranked modular blocks, word constructions, and CRT. The interval transfer and the exact numerical certificates are proved in this repository." }, { "id": "https://github.com/JD-Jones-ASES/fs-lower-bound/tree/d1294519dfdcc5f7e139e77336a7d042f390b9e7", "relationship": "builds-on", "note": "Square predecessor and ancestry of rk-lean; not a direct dependency." }, { "id": "https://github.com/JD-Jones-ASES/ns-lean/tree/035e9b0c147630e35631e4401433660f695d1fba", "relationship": "other", "note": "Related function-field result cited by Naslund. No theorem from this project is used as a dependency." } ], "classification": { "arxiv": [ "math.NT", "math.CO" ], "msc2020": [ "11B30", "11B75", "05C20" ] }, "automation": { "methods": [ { "method": "agent", "models": [], "framework": "OpenAI Codex" }, { "method": "copilot", "models": [ "GPT 6 Pro", "Claude", "Grok" ], "framework": "" } ], "strongest": "agent", "models": [ "Claude", "GPT 6 Pro", "Grok" ], "spend": "not tracked", "notes": "JD Jones is responsible for the statements and submission. AI assistance and the limits of review are disclosed in DISCLOSURE.md." }, "scope": "All 19 selected theorem statements are proved, including the three unconditional numerical bounds. The proof-bearing NKSolution has no admissions or definition holes. NKChallenge has 19 intentional theorem placeholders. The complete official local Comparator replay accepted proof commit 409bcef149ae96f40f69d38a17cf8bd7330dee98. See docs/VERIFICATION.md for later checks and the macOS host scope. The first Palomar submission failed because its module names collided with a dependency; the renamed modules require a new intake run. No global optimum, computed binary capacity, Bellman identity, or e", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The interval criterion is generalized from squares to every positive integer k. Canonical closure permits arbitrary seed moduli. The square application changes odd supports, widths, and moment exponents while retaining Naslund's binary transition rules. Fourth- and sixth-power applications give the stated exact bounds. Unit conductors are sufficient bounds, without a minimality claim. Prime-depth capacity ranges over all interval alphabets and excludes binary squares. Binary capacity is defined from actual alphabets; no optimizer or Bellman identity is assumed.", "alignment": false, "original": false, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [ "Codex theorem-scope reviewer", "Codex independent definition and signature reviewer" ], "notes": "Separate agents reviewed the theorem statements, definitions, proof scope, and certificate assembly. This is agent review, not human peer review or source-author endorsement. Mechanical evidence is in docs/VERIFICATION.md." }, "canonical": { "repo": "jd-jones-ases/nk-lean", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-21-000004", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-21-000004", "commit": "55acbf0def926afb15eb243a3c3f2a3b42439bce", "trust": "high", "theorems": 19, "date": "2026-09-21" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "JD-Jones-ASES/ns-lean/formalization.yaml", "repo": "JD-Jones-ASES/ns-lean", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/JD-Jones-ASES/ns-lean/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "ns-lean: Naslund's Conjecture 13 is false at q = 3, k = 2, for every n congruent to 0 mod 4 with n >= 8", "description": "A Lean 4 proof that square-difference-free subsets of the polynomials of degree below n over F_3 can be larger than 3^(3n/4). Call A square-difference-free when no two of its elements differ by a nonzero square, and let D_3(n) be the largest size of such a set inside P_{3,n}. A ten-word code S in F_3^4, in which no two words differ by a vector with every coordinate in {0, 1}, drives a lift that carries a square-difference-free subset of P_{3,m} to one of P_{3,m+8} with 810 times as many elements. Iterating it from the bases {0} and B_4 = {a T^3 + b T + c (1 - T^2)} gives square-difference-free", "authors": [ "JD Jones" ], "maintainers": [ "JD Jones" ], "license": "MIT", "role": "", "substantive": "", "sources": [ { "title": "Paley graphs and Sárközy's theorem in function fields", "id": "doi:10.1093/qmath/haac035", "authors": [ "Eric Naslund" ], "type": "article", "relationship": "adapts", "endorsement": "not-contacted" }, { "title": "The Sperner capacity of linear and nonlinear codes for the cyclic triangle", "id": "doi:10.1023/A:1022424630332", "authors": [ "A. R. Calderbank", "P. Frankl", "R. L. Graham", "W.-C. W. Li", "L. A. Shepp" ], "type": "article", "relationship": "background", "endorsement": "not-contacted" }, { "title": "On the Sperner capacity of the cyclic triangle", "id": "doi:10.1023/A:1022455407000", "authors": [ "Aart Blokhuis" ], "type": "article", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Difference sets without squares", "id": "doi:10.1007/BF02454169", "authors": [ "Imre Z. Ruzsa" ], "type": "article", "relationship": "background", "endorsement": "not-contacted" } ], "related": [ { "id": "PALOMAR-2026-09-12-000004", "relationship": "other", "note": "rk-lean: the repository layout pattern; no mathematics shared." }, { "id": "PALOMAR-2026-09-11-000003", "relationship": "other", "note": "ry-lean: the repository layout pattern; no mathematics shared." } ], "classification": { "arxiv": [ "math.NT", "math.CO" ], "msc2020": [ "11T55", "11B30", "05B40" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Claude (Anthropic)", "GPT-6 (OpenAI)", "Grok" ], "framework": "Claude Code with parallel subagents" } ], "strongest": "agent", "models": [ "Claude (Anthropic)", "GPT-6 (OpenAI)", "Grok" ], "spend": "not tracked", "notes": "See DISCLOSURE.md." }, "scope": "Six compared declarations over four compared definitions. Four of them exhibit square-difference-free sets of polynomials over F_3 (810 elements in degree below 8, 810^e in degree below 8e, 27 * 810^e in degree below 8e + 4) and two of them draw the consequences: the conjectured bound 3^(3n/4) fails at every n divisible by 4 with n >= 8, and 16/21 <= liminf log D_3(n) / (n log 3). No upper bound on D_3(n) is proved, no value of D_3(n) is determined, the ten-word code is not claimed to be optimal, and nothing is claimed at any q other than 3. The intentional placeholders of Challenge.lean are e", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "NS.D3_8_ge_810", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "NS.conjecture13_fails_3_2_8", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "NS.D3_8e_ge", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "NS.D3_8e4_ge", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "NS.conjecture13_fails_3_2_all", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "NS.liminf_ge", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The compared statements are the negation of the source's Conjecture 13 at q = 3, k = 2, not a formalization of it, and the source's Theorem 1 is not re-derived here. Degree bounds are stated as natDegree f < n, a comparison of natural numbers, so that no statement type depends on an instance that can diverge; since the zero polynomial has natural degree 0, that form cannot express P_{3,0} = {0}, and the recursion therefore works internally with degree in WithBot, converting at n >= 1, where the two agree. The conjectured exponent is written 3 * n / 4 with natural division, which is exact on the hypothesis 4 divides n.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "No independent human expert review and no source-author endorsement is recorded; AI agents developed and cross-reviewed the proof, the statements were checked against the source text, and VERIFICATION.md states what is checked and what is not." }, "canonical": { "repo": "jd-jones-ases/ns-lean", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-17-000003", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-17-000003", "commit": "035e9b0c147630e35631e4401433660f695d1fba", "trust": "high", "theorems": 6, "date": "2026-09-17" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "JD-Jones-ASES/rk-lean/formalization.yaml", "repo": "JD-Jones-ASES/rk-lean", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/JD-Jones-ASES/rk-lean/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "rk-lean: the directed construction for k-th-power-difference-free sets at every k, with the exponents 0.912145 (k = 4) and 0.950826 (k = 6)", "description": "A Lean 4 proof of Krachun's directed (acyclic-block) lower-bound construction for sets of integers with no k-th-power difference, stated and proved for every exponent k >= 2. Let D_k(N) be the largest size of a subset of {1,...,N} no two of whose elements differ by a nonzero perfect k-th power. A pool for k is a nonempty list of blocks (m, sup, H) with pairwise coprime square-free moduli m >= 2, heights H >= 2, and sup a nonempty list of (vertex, rank) pairs with distinct vertices below m, ranks below H, and a strict rank drop along every ordered pair of vertices whose difference is a nonzero ", "authors": [ "JD Jones" ], "maintainers": [ "JD Jones" ], "license": "MIT", "role": "", "substantive": "", "sources": [ { "title": "Square-Difference-Free Sets beyond the Three-Quarter Barrier", "id": "arXiv:2608.01325", "authors": [ "Dmitry Krachun" ], "type": "article", "relationship": "adapts", "endorsement": "not-contacted" }, { "title": "Difference sets without squares", "id": "doi:10.1007/BF02454169", "authors": [ "Imre Z. Ruzsa" ], "type": "article", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Lower bounds in the polynomial Szemerédi theorem", "id": "arXiv:1908.06058", "authors": [ "Khalid Younis" ], "type": "article", "relationship": "background", "endorsement": "not-contacted" }, { "title": "An improved lower bound related to the Furstenberg–Sárközy theorem", "id": "doi:10.37236/4656", "authors": [ "Mark Lewko" ], "type": "article", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Square-Difference-Free Sets of Size Ω(n^0.7334...)", "id": "arXiv:0804.4892", "authors": [ "Richard Beigel", "William Gasarch" ], "type": "article", "relationship": "background", "endorsement": "not-contacted" }, { "title": "On the Shannon capacity of a graph", "id": "doi:10.1109/TIT.1979.1055985", "authors": [ "László Lovász" ], "type": "article", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Difference sets without κ-th powers", "id": "doi:10.1007/BF01874311", "authors": [ "Antal Balog", "József Pelikán", "János Pintz", "Endre Szemerédi" ], "type": "article", "relationship": "background", "endorsement": "not-contacted" }, { "title": "New bounds for the Furstenberg–Sárközy theorem", "id": "arXiv:2411.17448", "authors": [ "Ben Green", "Mehtaab Sawhney" ], "type": "article", "relationship": "background", "endorsement": "not-contacted" } ], "related": [ { "id": "PALOMAR-2026-08-26-000004", "relationship": "builds-on", "note": "fs-formal (JD Jones, registered 2026-08-26): the k = 2 case of the same construction, an eleven-block pool and the exponent 0.7537415..., proved to exceed Krachun's 0.752796455875.... This repository is the general-k development on that architecture (the digit lift, the word construction, the CRT glue, the allocation and the numeric layer, each with the digit stride 2 replaced by k); the eleven-bl" }, { "id": "PALOMAR-2026-09-11-000003", "relationship": "other", "note": "ry-lean (JD Jones, registered 2026-09-11): a different configuration (nonlinear Roth {x, x+y, x+y^2}, not k-th-power differences); cited as prior art in this problem area and as the pattern for the one-repository registry layout, the axiom audit and the source guard used here. No mathematics is shared." } ], "classification": { "arxiv": [ "math.NT", "math.CO" ], "msc2020": [ "11B30", "11B75", "05C20" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Claude Fable 5.1 (Anthropic)", "Claude Opus (Anthropic)" ], "framework": "Claude Code with parallel subagents" } ], "strongest": "agent", "models": [ "Claude Fable 5.1 (Anthropic)", "Claude Opus (Anthropic)" ], "spend": "not tracked", "notes": "See DISCLOSURE.md. Prompt, token and monetary accounting was not retained." }, "scope": "Ten compared declarations over nine compared definitions. directed_liminf and directed_pointwise are the general theorem: for every k >= 2 and every pool for k (a nonempty list of blocks on pairwise coprime square-free moduli, each with a nonempty support carrying a ranking strictly decreasing along every k-th-power-residue arc), alpha k P <= liminf log D_k(N)/log N, and N^rho <= D_k(N) eventually for every rho < alpha k P. pool4_valid and pool6_valid are the finite verifications of the two concrete pools. alpha4_gt_transfer, alpha4_gt, alpha6_gt_transfer and alpha6_gt are inequalities between", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "KthPower.directed_liminf", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "KthPower.directed_pointwise", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "KthPower.pool4_valid", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "KthPower.pool6_valid", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "KthPower.alpha4_gt_transfer", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "KthPower.alpha4_gt", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "KthPower.alpha6_gt_transfer", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "KthPower.alpha6_gt", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "KthPower.fourth_power_liminf", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "KthPower.sixth_power_liminf", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The compared theorem is stated for every k >= 2 with no parity condition, where the source is k = 2 throughout. The additive formulation of k-th-power-difference-freeness (a + z^k not in A for z > 0) covers both signs of a difference, so Lemma B needs no odd-k branch; at odd k, however, -1 = (-1)^k is a k-th-power residue, the arc relation is symmetric, every valid block is an independent set, and the theorem is then weaker than Ruzsa's transfer. The general theorem is therefore of interest at even k, and the two instances are at k = 4 and k = 6. Acyclicity of a block is not stated as a hypothesis: the ranking, which is an explicit part of the data, is equivalent to it, and a strictly decreasing ranking is what the proofs consume. The set of nonzero k-th-power residues is the full image of z -> z^k with 0 removed, non-units included; restricting to units would make the lift false. Sets of integers are Finset ℕ and residues are natural numbers below the modulus, so that every finite claim stays decidable; D k N is a supremum of Finset.card over the k-th-power-difference-free subsets of Finset.Icc 1 N, and no range is stated as an interval in ℤ (instance diamonds are avoided by stati", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "No independent human expert review and no source-author endorsement is recorded. AI agents developed and cross-reviewed the proof; the statements were reviewed against the source text for dropped hypotheses and junk values, and against the prior-art report for every numerical comparison. VERIFICATION.md records the mechanical checks completed at this snapshot and states explicitly what is not checked (maximality of supports, minimality of heights, optimality of the pools, the two residue sets behind the transfer values, the theta ceiling, and the prior-art searches). Mechanical correctness and statement agreement do not establish priority, significance, or the correctness of any informal interpretation." }, "canonical": { "repo": "jd-jones-ases/rk-lean", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-12-000004", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-12-000004", "commit": "4bf2e3a56b6a522764a4d03e5dc606c12376185e", "trust": "high", "theorems": 10, "date": "2026-09-12" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "JD-Jones-ASES/ry-lean/formalization.yaml", "repo": "JD-Jones-ASES/ry-lean", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/JD-Jones-ASES/ry-lean/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "ry-lean: sets avoiding {x, x+y, x+y²} — Younis's two-set construction and the exponent 0.77028", "description": "A Lean 4 proof of Younis's lower-bound construction for subsets of {1,...,N} with no configuration {x, x+y, x+y^2} (y nonzero), in its two-set form: for every square-free modulus m and residue sets R1, R2 modulo m with R1 square-difference-free and no nonzero difference of R2 equal to the square of a difference of R1, the interval {1,...,N} contains a configuration-free set of size at least C(rho) N^rho for every rho below 1/2 + log|R1|/(3 log m) + log|R2|/(6 log m). The theorem is instantiated at m = 145 with |R1| = 10 and |R2| = 32, giving the exponent 0.770287..., which is proved to exceed ", "authors": [ "JD Jones" ], "maintainers": [ "JD Jones" ], "license": "MIT", "role": "", "substantive": "", "sources": [ { "title": "Lower bounds in the polynomial Szemerédi theorem", "id": "arXiv:1908.06058", "authors": [ "Khalid Younis" ], "type": "article", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "Difference sets without squares", "id": "doi:10.1007/BF02454169", "authors": [ "Imre Z. Ruzsa" ], "type": "article", "relationship": "background", "endorsement": "not-contacted" }, { "title": "An improved lower bound related to the Furstenberg–Sárközy theorem", "id": "", "authors": [ "Mark Lewko" ], "type": "article", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Square-difference-free sets of size Ω(n^0.7334...)", "id": "arXiv:0804.4892", "authors": [ "Richard Beigel", "William Gasarch" ], "type": "article", "relationship": "background", "endorsement": "not-contacted" }, { "title": "A polylogarithmic bound in the nonlinear Roth theorem", "id": "doi:10.1093/imrn/rnaa261", "authors": [ "Sarah Peluse", "Sean Prendiville" ], "type": "article", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Polynomial extensions of van der Waerden's and Szemerédi's theorems", "id": "doi:10.1090/S0894-0347-96-00194-4", "authors": [ "Vitaly Bergelson", "Alexander Leibman" ], "type": "article", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Square-Difference-Free Sets beyond the Three-Quarter Barrier", "id": "arXiv:2608.01325", "authors": [ "Dmitry Krachun" ], "type": "article", "relationship": "background", "endorsement": "not-contacted" } ], "related": [], "classification": { "arxiv": [ "math.NT", "math.CO" ], "msc2020": [ "11B30", "11B75", "05D10" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Claude Fable 5.1 (Anthropic)", "Claude Opus (Anthropic)" ], "framework": "Claude Code with parallel subagents" } ], "strongest": "agent", "models": [ "Claude Fable 5.1 (Anthropic)", "Claude Opus (Anthropic)" ], "spend": "not tracked", "notes": "See DISCLOSURE.md. Prompt, token, and monetary accounting was not retained." }, "scope": "Five compared declarations. younis_period_two is the two-set form of Younis's Theorem 1.5 for every square-free modulus m >= 2 and every nonempty admissible pair (R1, R2), in the pointwise form \"for every rho below the exponent there is C > 0 with a configuration-free subset of {1,...,N} of size at least C N^rho for all N >= 1\". record_pointwise and record_liminf are its instance at m = 145 (|R1| = 10, |R2| = 32), the latter as a liminf bound for the counting function D. younis_lt_record and record_exponent_bounds are exact inequalities between real numbers defined by logarithms. Configuration", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "NonlinearRoth.younis_period_two", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "NonlinearRoth.record_pointwise", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "NonlinearRoth.record_liminf", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "NonlinearRoth.younis_lt_record", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "NonlinearRoth.record_exponent_bounds", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The compared theorem is the k = 2, period-two, R_0 = Z/m specialisation of Younis's Theorem 1.5, stated for arbitrary square-free m >= 2 and arbitrary nonempty admissible (R_1, R_2); Younis's general chains (arbitrary k, arbitrary sequences R_n) are not formalized. The hypotheses are the two edge conditions R_0 -> R_1 and R_1 -> R_2 of the chain condition; the remaining edges of the periodic chain follow from the first and are derived, not assumed. The conclusion is stated with an arbitrary exponent rho below the displayed exponent and a constant depending on rho, which is Younis's |A| >>_{m, eps} N^{gamma - eps}. Sets live in Finset ℤ inside Finset.Icc 1 N. The construction is analysed through a periodic truncation of the digit rule (each depth gives a slightly smaller exponent, converging to the displayed one); this is a proof device and does not change the statements.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "No independent human expert review or source-author endorsement is recorded. AI agents developed and cross-reviewed the proof; the statements were reviewed against the source text for dropped hypotheses and junk values. VERIFICATION.md records the mechanical checks completed at this snapshot. Mechanical correctness and statement agreement do not establish priority, significance, or the correctness of any informal interpretation." }, "canonical": { "repo": "jd-jones-ases/ry-lean", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-11-000003", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-11-000003", "commit": "a4edb0ea61422813cfd23b319cb28c62f09129bf", "trust": "high", "theorems": 5, "date": "2026-09-11" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "JD-Jones-ASES/x-ray-lean/formalization.yaml", "repo": "JD-Jones-ASES/x-ray-lean", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/JD-Jones-ASES/x-ray-lean/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Constructive families and an extension theorem for binary permutation X-rays", "description": "We prove permutation realization for every admissible binary X-ray profile whose rank deviations from the odd staircase lie in [-2, 2], at every order and without a bound on density slack. We also prove the one-sided families with all deviations at least -1 or all deviations at most 1, with at least 2^u realizations, where u counts positive or negative deviations respectively. For n >= 1, if a label multiset L is realizable at order n, then Q(L) = {3, 2n+1} multiset-union (L+2) is realizable at order n+2. If every source label lies in [2, 2n-2], then h(L) <= h(Q(L)), where h counts permutation", "authors": [ "JD Jones" ], "maintainers": [ "JD Jones" ], "license": "MIT", "role": "", "substantive": "", "sources": [ { "title": "Constructive families and an extension theorem for binary permutation X-rays", "id": "", "authors": [ "JD Jones" ], "type": "original-proof", "relationship": "other", "endorsement": "" }, { "title": "On the X-rays of permutations", "id": "", "authors": [ "C. Bebeacua", "T. Mansour", "A. Postnikov", "S. Severini" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Perfect Skolem sets", "id": "", "authors": [ "G. Nordh" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "A note on X-rays of permutations and a problem of Brualdi and Fritscher", "id": "arXiv:1707.03928", "authors": [ "G. Nordh" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Hankel and Toeplitz X-rays of permutations", "id": "doi:10.1016/j.laa.2014.02.037", "authors": [ "R. A. Brualdi", "E. Fritscher" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Binary X-rays of doubly stochastic matrices", "id": "arXiv:2608.01442", "authors": [ "Bangzheng Li", "Yuewei Liu", "Yuan Yao" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Relating tournaments and permutations with xrays", "id": "arXiv:2606.21532", "authors": [ "Matthew Davis", "Michael W. Schroeder" ], "type": "paper", "relationship": "background", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05A05", "05C70" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "OpenAI GPT-6 Astra" ], "framework": "OpenAI Codex" } ], "strongest": "agent", "models": [ "OpenAI GPT-6 Astra" ], "spend": "", "notes": "See DISCLOSURE.md. No GitHub Actions workflows were used." }, "scope": "All eight principal statements in Challenge.lean are proved in Solution.lean. The lower one-sided proof additionally retains its minimum edge. Ranked-cell realizability is proved equivalent to literal permutation-matrix labels. The example deviation words in PROOF.md are informal consequences; no finite census is a premise. The unrestricted conjecture is not proved.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "XRay.two_sided", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "XRay.one_sided_lower", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "XRay.one_sided_upper", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "XRay.one_sided_lower_count", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "XRay.one_sided_upper_count", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "XRay.ordinary_extension", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "XRay.extension_count", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "XRay.least_counterexample", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "These are restricted realization theorems and an extension principle, not a proof of the unrestricted binary X-ray conjecture. Definitions use zero-based Fin indices with one-based antidiagonal labels. The extension is compared as a multiset, so its slot order is immaterial. The selected extension statements assert existence and a conditional cardinality inequality, without specifying a pointwise extension of a prescribed permutation. All definitions are explicit in Challenge.lean; the comparison has no definition holes.", "alignment": false, "original": true, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "Local verification is recorded in VERIFICATION.md. No independent human expert review or source-author endorsement is claimed. Mechanical checks do not establish novelty or settle the remaining prior-art comparison." }, "canonical": { "repo": "jd-jones-ases/x-ray-lean", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-15-000005", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-15-000005", "commit": "c0a7d9f6119ed1126a292081689fdf8927ac12ea", "trust": "high", "theorems": 8, "date": "2026-09-15" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "jenkins-byu/nonnegativity-cutoffs-lean/formalization.yaml", "repo": "jenkins-byu/nonnegativity-cutoffs-lean", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/jenkins-byu/nonnegativity-cutoffs-lean/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Nonnegativity cutoffs for level-one modular forms", "description": "A Lean formalization of the qualitative well-definedness of the nonnegativity Sturm bound A(k) for real level-one modular forms. For every weight k at least 12 and divisible by four, the formalization proves that a finite initial segment detects coefficientwise nonnegativity and that its optimal endpoint is attained as the greatest possible first negative Fourier-coefficient index.", "authors": [ "Paul Jenkins" ], "maintainers": [ "Paul Jenkins" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Modular Forms with Only Nonnegative Coefficients", "id": "doi:10.1007/s40993-026-00744-z", "authors": [ "Paul Jenkins", "Jeremy Rouse" ], "type": "paper", "relationship": "formalizes", "endorsement": "other" } ], "related": [], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11F11", "11F30" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "OpenAI Codex (underlying model identifier not recorded)" ], "framework": "OpenAI Codex desktop app" }, { "method": "manual", "models": [], "framework": "n/a" } ], "strongest": "agent", "models": [ "OpenAI Codex (underlying model identifier not recorded)" ], "spend": "subscription-based; not tracked", "notes": "OpenAI Codex generated and revised substantial portions of the Lean code and documentation. It is disclosed as automation and is not listed as an author." }, "scope": "The full development formalizes the qualitative level-one argument that the nonnegative normalized forms form a bounded finite intersection of coefficient half-spaces, and derives a finite coefficient test and the well-definedness of A(k). The single Palomar declaration records the consequence for arbitrary level-one modular forms with real Fourier coefficients when k is at least 12 and divisible by four: an inclusive finite nonnegativity cutoff exists, is attained as a first negative index, and bounds every such first negative index. The advertised declaration does not include the low weights", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "NonnegativeModularForms.exists_optimal_nonnegativity_cutoff", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The Palomar headline is a streamlined consequence of Section 4 rather than the full convex-polytope formulation of Theorem 5. It is restricted to k >= 12 with k divisible by four, makes the real-coefficient condition explicit, includes the constant coefficient in the arbitrary-form cutoff test, and combines finite detection with the maximum characterization of A(k). The low-weight conventions A(4) = A(8) = 1 and all quantitative claims are omitted from the headline.", "alignment": true, "original": false, "review": { "status": "author-verified", "bucket": "author-verified", "reviewers": [], "notes": "The underlying Lean development has passed repeated clean builds and axiom audits. The author has completed a focused semantic audit of the Palomar Challenge statement and confirmed that it expresses the intended theorem. The formalization has not received external mathematical review." }, "canonical": { "repo": "jenkins-byu/nonnegativity-cutoffs-lean", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-25-000011", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-25-000011", "commit": "b3217f0e8367a677426daa461f8f79953532d80c", "trust": "high", "theorems": 1, "date": "2026-09-25" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "jeremypparker/mahler-hansen-cograph/formalization.yaml", "repo": "jeremypparker/mahler-hansen-cograph", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/jeremypparker/mahler-hansen-cograph/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Three Sharp Bounds for Hansen Polytopes of Cographs", "description": "A Lean 4/Mathlib formalisation of three sharp lower bounds for Hansen polytopes of finite nonempty cographs: normalized Mahler volume at least one, at least 3^d nonempty faces, and at least 2^d d! full flags in dimension d. In each result equality holds exactly for threshold graphs.", "authors": [ "Jeremy P Parker" ], "maintainers": [ "Jeremy P Parker" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "Sharp Mahler inequality for Hansen polytopes of cographs, with equality exactly for threshold graphs", "id": "", "authors": [ "Jeremy Parker" ], "type": "original-proof", "relationship": "other", "endorsement": "" }, { "title": "Sharp 3^d face bound for Hansen polytopes of cographs, with equality exactly for threshold graphs", "id": "", "authors": [ "Jeremy Parker" ], "type": "original-proof", "relationship": "other", "endorsement": "" }, { "title": "Sharp full-flag bound for Hansen polytopes of cographs, with equality exactly for threshold graphs", "id": "", "authors": [ "Jeremy Parker" ], "type": "original-proof", "relationship": "other", "endorsement": "" }, { "title": "Face numbers of centrally symmetric polytopes produced from split graphs", "id": "doi:10.37236/3315", "authors": [ "Ragnar Freij", "Matthias Henze", "Moritz W. Schmitt", "Günter M. Ziegler" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Kalai's 3^d conjecture for unconditional and locally anti-blocking polytopes", "id": "doi:10.1090/proc/16909", "authors": [ "Raman Sanyal", "Martin Winter" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Volume growth of Funk geometry and the flags of polytopes", "id": "doi:10.2140/gt.2025.29.3773", "authors": [ "Dmitry Faifman", "Constantin Vernicos", "Cormac Walsh" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Kalai's flag conjecture for locally anti-blocking polytopes", "id": "arxiv:2507.22284", "authors": [ "Arnon Chor" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Geometric Inequalities for Anti-Blocking Bodies", "id": "doi:10.1142/S0219199721501133", "authors": [ "Shiri Artstein-Avidan", "Shay Sadovsky", "Raman Sanyal" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Ein Übertragungsprinzip für konvexe Körper", "id": "doi:10.21136/CPMF.1939.109441", "authors": [ "Kurt Mahler" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Minimal volume product near Hanner polytopes", "id": "doi:10.1016/j.jfa.2013.08.008", "authors": [ "Jaegil Kim" ], "type": "paper", "relationship": "background", "endorsement": "" } ], "related": [ { "id": "https://github.com/Vilin97/lean-pool/tree/9c296f447f48f3242df5e65e0b6120ddffcd79a7/LeanPool/Isoperimetric", "relationship": "builds-on", "note": "This project reuses three Lean files authored by Jonathan Ho and vendored from this exact Lean Pool revision: Basic.lean, PrekopaLeindler.lean, and BrunnMinkowski.lean, stored locally as PrekopaLeindlerBasic.lean, PrekopaLeindler.lean, and BrunnMinkowski.lean. Ho's contribution is the formalization of the Prekopa-Leindler and Brunn-Minkowski analytic infrastructure used by this project. In particu" } ], "classification": { "arxiv": [ "math.MG", "math.CO" ], "msc2020": [ "52A40", "52B05", "05C75" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "GPT-5.6 Sol", "GPT-5.6 Terra" ], "framework": "ChatGPT" }, { "method": "manual", "models": [], "framework": "" } ], "strongest": "agent", "models": [ "GPT-5.6 Sol", "GPT-5.6 Terra" ], "spend": "", "notes": "" }, "scope": "The project defines Hansen polytopes, their normalized Mahler ratio, and a finite Hansen face lattice from the signed stable-set/clique incidence matrix. It proves the Mahler, 3^d nonempty-face, and 2^d d! full-flag lower bounds for graphs represented by cotrees, with equality in each case exactly for the cotree-based threshold graphs. The cotree datatype starts from a one-vertex leaf, so the null graph is outside the formalized cograph and threshold conventions. The results do not prove the corresponding conjectures for arbitrary centrally symmetric polytopes. The six Challenge declarations c", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "", "alignment": true, "original": true, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [ "ChatGPT (GPT-5.6 Sol)" ], "notes": "" }, "canonical": { "repo": "jeremypparker/mahler-hansen-cograph", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-24-000004", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-24-000004", "commit": "c7639c5060f40d69b613bcbd637b1750f27b7e3f", "trust": "high", "theorems": 6, "date": "2026-09-10" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "jfairfaxball-348/Fischer-Zero-Forcing-Counterexample/formalization.yaml", "repo": "jfairfaxball-348/Fischer-Zero-Forcing-Counterexample", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/jfairfaxball-348/Fischer-Zero-Forcing-Counterexample/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Fischer's 24-vertex zero-forcing counterexample", "description": "A Lean 4 formalization of Mikko Fischer's explicit connected 24-vertex subcubic graph with independence number 9 and zero-forcing number 11. The formalization verifies the exact graph and both numerical invariants, yielding an existential counterexample to the proposed bound Z(G) <= alpha(G) + 1 for connected graphs of maximum degree at most 3.", "authors": [], "maintainers": [ "John Fairfax-Ball" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "A counterexample to the zero forcing versus independence conjecture for cubic and subcubic graphs", "id": "arXiv:2607.23664v1", "authors": [ "Mikko Fischer" ], "type": "paper", "relationship": "formalizes", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05C50", "05C69" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "OpenAI GPT-5.6 Sol" ], "framework": "ChatGPT" } ], "strongest": "agent", "models": [ "OpenAI GPT-5.6 Sol" ], "spend": "subscription-based; exact API-equivalent spend not tracked", "notes": "The human maintainer selected the target and scope, supplied acceptance criteria, and directed iterative audits. AI assistance was used extensively for Lean development, debugging, repository operations, and documentation. The final finite graph claims are also reconstructed by a standard-library Python checker independent of Lean." }, "scope": "Proves that there exists a connected simple graph on 24 vertices with maximum degree at most 3, independence number 9, and zero-forcing number 11, using Fischer's explicit 24-vertex graph. It does not formalize the 36-vertex cubic example, any minimality result, or a general zero-forcing theory beyond what is needed for this certificate.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "No known mathematical divergence for the formalized 24-vertex result. Scope is deliberately narrower than the source paper: the 36-vertex cubic example and any minimality question are not formalized.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "none" ], "notes": "The completed theorem has undergone a hostile theorem-level Lean axiom audit and an independent Python reconstruction of the source graph, independence number, zero-forcing upper certificate, fort reduction, and all 17,712 lower-bound candidates. No independent human referee review is claimed." }, "canonical": { "repo": "jfairfaxball-348/fischer-zero-forcing-counterexample", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-20-000009", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-20-000009", "commit": "4ee012d6941e221d294f26520aff92043d715140", "trust": "high", "theorems": 1, "date": "2026-09-20" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "jfairfaxball-348/pascal-minus-one/formalization.yaml", "repo": "jfairfaxball-348/pascal-minus-one", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/jfairfaxball-348/pascal-minus-one/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Pascal Minus-One GCD", "description": "A Lean 4 formalization of p-adic valuations for the restricted binomial gcd G(N;m) = gcd{C(N,k) : 0 < k < N, m divides k}. The principal submitted result gives the complete valuation formula for primes p congruent to -1 modulo m, including the mixed-parity digit-sum branch and the exceptional p = m - 1 valuation-2 case. The repository also proves the complementary p congruent to 1 branch, a prime-power scaling reduction, and complete prime-by-prime formulas for m = 3, 4, and 6.", "authors": [], "maintainers": [ "John Fairfax-Ball" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Full p ≡ -1 (mod m) valuation classification for the restricted Pascal-row gcd", "id": "", "authors": [ "John Fairfax-Ball" ], "type": "original-proof", "relationship": "other", "endorsement": "" }, { "title": "On the Greatest Common Divisor of Binomial Coefficients C(n,q), C(n,2q), C(n,3q), ...", "id": "DOI:10.4169/amer.math.monthly.124.4.353", "authors": [ "Carl McTague" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Computing the Greatest Common Divisor of Binomial Coefficients C(mn,mk)", "id": "arXiv:2606.20940v2", "authors": [ "Chai Wah Wu" ], "type": "paper", "relationship": "background", "endorsement": "" } ], "related": [ { "id": "https://github.com/MechMath/binomial-gcd", "relationship": "independent", "note": "Guo, Qiu, Cao, Feng and Gao's 2026 Lean 4 formalisation proves the OEIS A080170 criterion for D(k) = gcd_{2 <= q <= k+1} C(qk,k), where the lower index is fixed and the upper row varies. It is related binomial-gcd mathematics but is not a formalisation of McTague's fixed-row arithmetic-progression family or of this project's minus-one theorem; no code or theorem dependency is shared." } ], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11B65" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "OpenAI GPT-5.6 Sol" ], "framework": "ChatGPT" } ], "strongest": "agent", "models": [ "OpenAI GPT-5.6 Sol" ], "spend": "subscription-based; exact API-equivalent spend not tracked", "notes": "The human maintainer selected the research target, theorem scope, validation standards, novelty-claim level, and publication decisions. AI assistance was used extensively for proof development, source comparison, repository work, and packaging. Computational sweeps serve only as regression evidence and are not substitutes for the Lean proofs." }, "scope": "The repository fully proves PascalMinusOne.minus_one_valuation, PascalMinusOne.plus_one_valuation, PascalMinusOne.scaling_valuation, and the complete modulus-three, modulus-four, and modulus-six valuation corollaries. The Palomar Comparator entry deliberately advertises only the principal minus-one theorem. Its Challenge restates the restricted gcd, the parity digit sums, and the piecewise expected valuation using Mathlib-only definitions, and Comparator checks all of those definition values together with the theorem. The deliberate Challenge sorry is not part of the proof development count.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "No mathematical divergence is introduced between the advertised Challenge theorem and PascalMinusOne.minus_one_valuation. The Challenge gives ordinary Mathlib-only definitions of exactly the same restricted gcd, parity digit sums, and piecewise expected valuation; the Solution bridges those definitions to the existing theorem. The Palomar entry is intentionally narrower than the repository's full public theorem layer because it selects the principal minus-one contribution rather than bundling the plus-one, scaling, and small-modulus corollaries into one reviewed result.", "alignment": true, "original": true, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "none" ], "notes": "The theorem layer is Lean-complete and has been subjected to repeated build, regression, finite reference-sweep, zero-sorry/admit/replacement-axiom, and statement-alignment checks. The final pre-paper source-level literature audit through 25 September 2026 is recorded in notes/literature.md and notes/novelty-audit-2026-09.md. Palomar registration is complete as PALOMAR-2026-09-25-000017, version 1. No independent human peer review is claimed; the registry record is distinct from this self-assessment and from later paper/arXiv publication." }, "canonical": { "repo": "jfairfaxball-348/pascal-minus-one", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-25-000017", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-25-000017", "commit": "5f20593464d86ef8c3c34e1c00d84f8a48c0457e", "trust": "high", "theorems": 1, "date": "2026-09-25" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "jfairfaxball-348/TreeStack-Structural-Certificates-for-Stacking-on-Trees/formalization.yaml", "repo": "jfairfaxball-348/TreeStack-Structural-Certificates-for-Stacking-on-Trees", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/jfairfaxball-348/TreeStack-Structural-Certificates-for-Stacking-on-Trees/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Tree stacking equals the Csernák--Soukup estimator on nontrivial finite trees", "description": "A Lean 4 proof of the corrected nontrivial-tree form of the Csernák--Soukup tree-stacking conjecture. For every finite tree with at least two vertices, the exact-size stacking number equals the explicit rooted-distance-and-degree estimator. The one-vertex case is excluded because the source stacking-number convention restricts admissible thresholds to t >= 2 while the displayed estimator is 1 on the one-vertex tree.", "authors": [], "maintainers": [ "John Fairfax-Ball" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Stacking and clearing in graph pebbling", "id": "arXiv:2604.22341v1", "authors": [ "Tamás Csernák", "Lajos Soukup" ], "type": "paper", "relationship": "adapts", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05C05" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "OpenAI GPT-5.6 Sol" ], "framework": "ChatGPT and Codex" } ], "strongest": "agent", "models": [ "OpenAI GPT-5.6 Sol" ], "spend": "subscription-based; exact API-equivalent spend not tracked", "notes": "The human maintainer selected the research target and scope, directed the proof-validation programme, required repeated machine checks, and controls legal/licensing and publication decisions. AI assistance was used extensively throughout the formalization and packaging work." }, "scope": "Proves TreeStack.stack_eq_estim_of_two_le_card: for every finite simple connected acyclic graph T on a finite vertex type V, if 2 <= |V| then stack(T) = estim(T). UniversalStackable uses configurations of exactly the specified mass, and stack is the least exact mass t satisfying both 2 <= t and universal stackability. The one-vertex case is deliberately outside the theorem. The proof development also formalizes the exact rooted branch-message invariant, estimator obstruction, arbitrary-defect upper bound, and global weighted cancellation needed for the equality.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The advertised theorem explicitly assumes at least two vertices. This is a convention correction rather than a change to the pebbling dynamics: the source stacking-number convention requires admissible thresholds t >= 2, while its displayed tree estimator evaluates to 1 on the one-vertex tree. UniversalStackable remains exact-size, not \"at least t\". No other known semantic divergence is introduced in the advertised result.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "The completed mathematical argument is self-assessed by the repository maintainer. The proof has also received private, informal mathematical feedback from an external reader. That feedback was not formally recorded and is not represented here as peer review, professional endorsement, or an independently documented external mathematical review. No reviewer name, affiliation, date, or quotation is asserted. A comprehensive public-record prior-art/originality audit was completed on 2026-09-23. It located no prior published or publicly accessible proof of the Csernák--Soukup tree-stacking estimator conjecture and no equivalent theorem under the alternative formulations searched. The search record and limitations are documented in notes/literature.md. Separate machine validation includes the c" }, "canonical": { "repo": "jfairfaxball-348/treestack-structural-certificates-for-stacking-on-trees", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-25-000010", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-25-000010", "commit": "457b88b785e50a749710108a935a661a9d992348", "trust": "high", "theorems": 1, "date": "2026-09-25" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "jsmorph/vq833a/formalization.yaml", "repo": "jsmorph/vq833a", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/jsmorph/vq833a/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "An 833-qubit secp256k1 discrete-logarithm program", "description": "A Lean verification of a complete semiclassical double-scalar quantum program for secp256k1 ECDLP, parameterized by the public point. The result specifies complex branch amplitudes, workspace restoration, selected-outcome decoding, and an analytic recovery probability greater than 99/100 over 26 independent executions. It proves a fixed allocation of 833 logical qubits and 768 mutable classical bits, with exactly 588551462912 compiled CCZ operations on every path and in expectation. Functional correctness and recovery assume a valid public key Q = dG with 0 < d < q and the specified initializa", "authors": [], "maintainers": [ "js@morphism.com" ], "license": "MIT", "role": "", "substantive": "", "sources": [ { "title": "Windowed Euclidean Arithmetic in an 833-Qubit secp256k1 ECDLP Program", "id": "https://github.com/jsmorph/vq833a", "authors": [], "type": "paper", "relationship": "formalizes", "endorsement": "" }, { "title": "Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer", "id": "https://doi.org/10.1137/S0097539795293172", "authors": [], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Shor's Discrete Logarithm Quantum Algorithm for Elliptic Curves", "id": "https://doi.org/10.26421/QIC3.4-3", "authors": [], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "A new quantum ripple-carry addition circuit", "id": "https://arxiv.org/abs/quant-ph/0410184", "authors": [], "type": "paper", "relationship": "adapts", "endorsement": "" }, { "title": "Improved quantum circuits for elliptic curve discrete logarithms", "id": "https://doi.org/10.1007/978-3-030-44223-1_23", "authors": [], "type": "paper", "relationship": "adapts", "endorsement": "" }, { "title": "Halving the cost of quantum addition", "id": "https://doi.org/10.22331/q-2018-06-18-74", "authors": [], "type": "paper", "relationship": "adapts", "endorsement": "" }, { "title": "Quantum Algorithm for Elliptic Curve Discrete Logarithms with Space-Efficient Point Addition", "id": "https://arxiv.org/abs/2607.13816v2", "authors": [], "type": "paper", "relationship": "adapts", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "quant-ph", "cs.CR", "cs.LO" ], "msc2020": [] }, "automation": { "methods": [ { "method": "agent", "models": [], "framework": "" }, { "method": "other", "models": [], "framework": "" } ], "strongest": "agent", "models": [], "spend": "", "notes": "" }, "scope": "", "sorry_count": null, "sorry_in_definitions": null, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "", "alignment": false, "original": false, "review": { "status": "lean compilation, complete comparator statement and axiom checks, and independent nanoda and lean proof checks passed for the source commit identified in docs/verification.md. this revision updates documentation and citation metadata while preserving the lean sources and checking configuration. human peer review remains outstanding.", "bucket": "other", "reviewers": [], "notes": "" }, "canonical": { "repo": "jsmorph/vq833a", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "kim-em/erdos-unit-distance-comparator/formalization.yaml", "repo": "kim-em/erdos-unit-distance-comparator", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/kim-em/erdos-unit-distance-comparator/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "erdos-unit-distance-comparator", "description": "", "authors": [ "Kim Morrison" ], "maintainers": [ "Kim Morrison" ], "license": "Apache-2.0", "role": "thin-wrapper", "substantive": "kim-em/erdos-unit-distance", "sources": [ { "title": "Integral points on norm-one tori and the Erdős unit-distance exponent", "id": "https://x.com/__alpoge__/status/2064409100562366543", "authors": [ "Levent Alpöge" ], "type": "", "relationship": "formalizes", "endorsement": "" }, { "title": "Integral points on norm-one tori and the Erdős unit-distance exponent", "id": "https://www-cdn.anthropic.com/files/4zrzovbb/website/ca35f196125c899a5ad11f011080202a652aef02.pdf", "authors": [], "type": "", "relationship": "background", "endorsement": "" }, { "title": "On sets of distances of n points", "id": "", "authors": [ "Paul Erdős" ], "type": "", "relationship": "background", "endorsement": "" }, { "title": "Some problems in number theory, combinatorics and combinatorial geometry", "id": "", "authors": [ "Paul Erdős" ], "type": "", "relationship": "background", "endorsement": "" }, { "title": "Some of my favourite problems which recently have been solved", "id": "", "authors": [ "Paul Erdős" ], "type": "", "relationship": "background", "endorsement": "" }, { "title": "Planar Point Sets with Many Unit Distances", "id": "https://cdn.openai.com/pdf/74c24085-19b0-4534-9c90-465b8e29ad73/unit-distance-proof.pdf", "authors": [ "OpenAI" ], "type": "", "relationship": "background", "endorsement": "" } ], "related": [ { "id": "https://github.com/plby/Erdos90", "relationship": "independent", "note": "Boris Alexeev's Lean formalization of OpenAI's 2026 counterexample, announced on the Lean Zulip and verified on lean-eval on 2026-06-26. He reports that Codex produced it over about a month with himself as the one person in the loop. It proves the fixed-power statement, and proves it outright: the class field theory is formalized rather than assumed, the proof path carries no sorries, and the repo" }, { "id": "https://github.com/logical-intelligence/erdos-unit-distance", "relationship": "independent", "note": "Logical Intelligence's Lean 4 formalization of the same result, produced with Aleph Prover, announced on 2026-05-28 at roughly 33,000 lines. It also reaches the fixed-power statement, that {n : 0 < n and nu(n) >= n^(1+delta)} is infinite for some delta > 0, by way of class field towers. It is conditional, deliberately and visibly: `main_theorem` takes the Golod-Shafarevich inequality and Shafarevi" } ], "classification": { "arxiv": [ "math.CO", "math.MG" ], "msc2020": [ "52C10", "11R21" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "claude-fable-5" ], "framework": "Claude Code (Anthropic)" }, { "method": "agent", "models": [], "framework": "Aristotle (Harmonic), CLI v2" }, { "method": "agent", "models": [ "gpt-5.5" ], "framework": "Codex CLI (OpenAI)" } ], "strongest": "agent", "models": [ "claude-fable-5", "gpt-5.5" ], "spend": "", "notes": "The formalization was completed on 2026-06-11 in a single working day, and migrated to Lean v4.31.0-rc2 on 2026-06-17. Per-lemma attribution is in the git history of both repositories: each integration commit names the prover. The figures above are measured from the session and rollout logs after the fact, and supersede those recorded in the proof library's own formalization.yaml. Those were written part way through the session, on 2026-06-11 at 12:11 UTC, so they undercount the directing session, which ran for another day; and they overcount Codex, because that measurement matched rollouts by" }, "scope": "", "sorry_count": null, "sorry_in_definitions": null, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The pinned dependency kim-em/PrimeNumberTheoremAnd contains sorried declarations in Wiener.lean, but they are not in the axiom closure of the compared theorem: the exported axiom set is exactly propext, Quot.sound and Classical.choice. Challenge.lean contains one `sorry`, standing for the proof that Solution.lean supplies; comparator.json declares no definition_names, so nothing in the statement is left open for the solution to instantiate.", "alignment": false, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Kim Morrison" ], "notes": "No independent human review of the mathematics. The submitter, the directing session's operator, and the author of this metadata are the same person. The Lean proof is machine-checked and sorry-free, and Comparator independently certifies that the library proves the Mathlib-only statement in Challenge.lean from the three standard axioms; neither of those is a check on whether the informal argument is correct or the statement is the right formalization of Erdős's conjecture." }, "canonical": { "repo": "kim-em/erdos-unit-distance-comparator", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-08-000001", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-08-000001", "commit": "be6c2ee4c9fb16fd6bed442b4c361fb10369beb5", "trust": "high", "theorems": 1, "date": "2026-08-14" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "kimihiro64/Robin1984/formalization.yaml", "repo": "kimihiro64/Robin1984", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/kimihiro64/Robin1984/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Robin1984", "description": "A Lean 4 formalization of Guy Robin's 1984 equivalence between the Riemann hypothesis and the strict divisor-sum inequality sigma(n) < exp(gamma) n log(log n) for every integer n > 5040. The development also proves the restriction to colossally abundant integers, reconstructs the Nicolas--Landau oscillation argument used for the converse implication, and checks the finite ranges with exact Lean certificates. It also formalizes Lagarias's 2001 elementary harmonic-number criterion and its equivalence to the Riemann hypothesis. This is a formalization of equivalence criteria; it does not prove th", "authors": [ "Jonas Whidden" ], "maintainers": [ "Jonas Whidden" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "Grandes valeurs de la fonction somme des diviseurs et hypothese de Riemann", "id": "MR0774171", "authors": [ "Guy Robin" ], "type": "paper", "relationship": "formalizes", "endorsement": "" }, { "title": "An Elementary Problem Equivalent to the Riemann Hypothesis", "id": "https://doi.org/10.1080/00029890.2002.11919883", "authors": [ "Jeffrey C. Lagarias" ], "type": "paper", "relationship": "formalizes", "endorsement": "" }, { "title": "Petites valeurs de la fonction d'Euler", "id": "https://doi.org/10.1016/0022-314X(83)90055-0", "authors": [ "Jean-Louis Nicolas" ], "type": "paper", "relationship": "formalizes", "endorsement": "" }, { "title": "Uber einen Satz von Tschebyschef", "id": "https://eudml.org/doc/158244", "authors": [ "Edmund Landau" ], "type": "paper", "relationship": "formalizes", "endorsement": "" }, { "title": "On highly composite and similar numbers", "id": "", "authors": [ "Leonidas Alaoglu", "Paul Erdos" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Approximate formulas for some functions of prime numbers", "id": "https://doi.org/10.1215/ijm/1255631807", "authors": [ "J. Barkley Rosser", "Lowell Schoenfeld" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Estimates for the Chebyshev Function psi(x) - theta(x)", "id": "https://doi.org/10.2307/2007805", "authors": [ "N. Costa Pereira" ], "type": "paper", "relationship": "formalizes", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11M26", "11N37", "03B35" ] }, "automation": { "methods": [ { "method": "manual", "models": [], "framework": "Lean 4 and Mathlib" }, { "method": "agent", "models": [ "OpenAI Codex (GPT-5 family)" ], "framework": "Codex desktop" } ], "strongest": "agent", "models": [ "OpenAI Codex (GPT-5 family)" ], "spend": "subscription-based; not tracked", "notes": "The formalization was developed collaboratively by the maintainer and AI coding agents. Lean's kernel, focused builds, full builds, source scans, Comparator, and NanoDa are the mechanical correctness gates; automation is not treated as mathematical review." }, "scope": "The three advertised declarations prove Robin's full 1984 biconditional in Mathlib's RiemannHypothesis formulation, the equivalent restriction to colossally abundant integers above 5040, and Lagarias's elementary criterion. The proof development excludes the deliberate statement holes in Challenge.lean from its counts. It does not establish either side of the biconditionals unconditionally and therefore does not prove the Riemann hypothesis.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "Robin1984.robin_inequality_iff_riemannHypothesis", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Robin1984.riemannHypothesis_iff_colossallyAbundant_robin", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Robin1984.riemann_hypothesis_iff_lagarias_elementary_criterion", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The advertised statements match Robin's criterion and its colossally abundant restriction. The Lean proof is a reconstruction rather than a line-by-line translation: it factors analytic arguments into smaller lemmas, uses Mathlib's definitions of the divisor sum, Euler's constant, and the Riemann hypothesis, and replaces prose or tabular finite checks by exact certificates. Intermediate cutoffs and lemma organization may differ from Robin's paper, but no extra hypothesis appears in either advertised theorem.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "The maintainer has audited statement fidelity, import closure, absence of project-local axioms and proof placeholders, exact dependency pins, and finite-certificate coverage. No independent expert mathematical review is claimed." }, "canonical": { "repo": "kimihiro64/robin1984", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-14-000007", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-14-000007", "commit": "acab1a31f31e0499518a4416b63281e0b4838f9c", "trust": "high", "theorems": 3, "date": "2026-09-14" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "langlerangle/NashEmbedding/formalization.yaml", "repo": "langlerangle/NashEmbedding", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/langlerangle/NashEmbedding/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Nash's isometric embedding theorem for closed manifolds (Günther's proof)", "description": "A Lean 4 / Mathlib formalization of Nash's C^∞ isometric embedding theorem for closed manifolds: every compact smooth manifold without boundary, equipped with a smooth Riemannian metric, admits a smooth injective map into some Euclidean space ℝ^q whose differential pulls the Euclidean inner product back to the given metric. The proof formalized is Günther's (1989), which replaces the Nash-Moser iteration by an elliptic fixed-point argument, following the presentation in A. J. Wassermann's Cambridge Part III lecture notes as the primary source. The development has two layers. The first proves t", "authors": [ "David Wiygul" ], "maintainers": [ "David Wiygul" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "The imbedding problem for Riemannian manifolds", "id": "", "authors": [ "John Nash" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "On the perturbation problem associated to isometric embeddings of Riemannian manifolds", "id": "", "authors": [ "Matthias Günther" ], "type": "paper", "relationship": "adapts", "endorsement": "" }, { "title": "The Atiyah-Singer Index Theorem, Part III lecture notes (Cambridge, Lent 2010): Sobolev spaces on the torus, Nash's embedding theorem for the N-torus, the reduction of the compact case", "id": "", "authors": [ "A. J. Wassermann" ], "type": "other", "relationship": "formalizes", "endorsement": "" }, { "title": "Günther's proof of Nash's isometric embedding theorem", "id": "arXiv:math/9807169", "authors": [ "Deane Yang" ], "type": "other", "relationship": "background", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.DG", "math.AP" ], "msc2020": [ "53C42", "53C21", "58J05", "46E35" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Claude Opus 4.7 (Anthropic)" ], "framework": "Claude Code" }, { "method": "agent", "models": [ "Claude Fable 5 (Anthropic)", "Claude Opus 4.7 (Anthropic)" ], "framework": "Claude Code" }, { "method": "agent", "models": [], "framework": "Aristotle (Harmonic), CLI 2.1.0, pinned to Lean v4.28.0 and Mathlib v4.28.0" }, { "method": "agent", "models": [ "Claude Fable 5 (Anthropic)", "Claude Opus 4.7 (Anthropic)" ], "framework": "Claude Code" } ], "strongest": "agent", "models": [ "Claude Fable 5 (Anthropic)", "Claude Opus 4.7 (Anthropic)" ], "spend": "", "notes": "project.authors lists the human project author, per the Palomar policy reserving that field for humans; material AI contributions are credited in automation.methods above and in the per-file provenance record ([PROVENANCE.md](PROVENANCE.md)). The Lean file headers separately list the systems that wrote Lean and identify David Wiygul as the requester and human design contributor; the initial commit message records the division of labor and credits Aristotle-Harmonic. The human author proposed the project, selected Wassermann's notes as the source, and took part in the early design decisions; th" }, "scope": "Complete for closed manifolds: every Hausdorff compact smooth manifold without boundary (Mathlib's [IsManifold I ∞ M] with a boundaryless model with corners, [T2Space M], [CompactSpace M]) and every smooth Riemannian metric in Mathlib's sense (ContMDiffRiemannianMetric I ∞ E (TangentSpace I)). Manifolds with boundary and non-compact manifolds are not treated. The embedding dimension q is existentially quantified and no bound on it is stated; the construction's dimension depends on the chosen bump covering and is far from optimal. The conclusion is a smooth injective map pulling the Euclidean i", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "NashEmbeddingTheorem.nash_isometric_embedding", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The compared statement is Nash's theorem for closed manifolds in its standard form for manifolds modelled on a boundaryless model with corners (I.Boundaryless), Hausdorff and compact, with no further restriction. The proof follows Günther's method as presented in Wassermann's notes; the notes are unpublished lecture notes, so no line-by-line correspondence is claimed, and several estimates were re-derived with different constants or stronger regularity indices where the formalization required it. The flat-torus layer is stated on ℝⁿ with 2πℤⁿ-periodic data rather than on a quotient manifold, which is what the reduction step consumes. The reduction from closed manifolds to the torus uses Mathlib's bump-covering embedding into a Euclidean space rather than a tubular-neighbourhood argument. Lemma names differ from the informal text throughout; the README records the correspondence of the main steps.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "David Wiygul", "GPT-5.6 Sol (OpenAI), statement, provenance and editorial audit" ], "notes": "No independent human review of the mathematics or of the Lean source. Every theorem is checked by Lean; scripts/audit.sh checks the trust conditions (no forbidden tokens, exactly one deliberate sorry in Challenge.lean, standard axioms for each principal result listed in scripts/axioms.lean); each August 2026 Aristotle contribution was separately verified by two Claude sessions before being merged, while April–May 2026 Aristotle contributions were reviewed against the informal blueprints by Claude Opus 4.7 and by the author at the statement level, and were Lean-checked on return (full build, standard-axiom footprint); further scrutiny came from the later `NashEmbeddingTest/` witness tests and from the audit passes recorded in the commit history. The compared statement in Challenge.lean was " }, "canonical": { "repo": "langlerangle/nashembedding", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-02-000004", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-02-000004", "commit": "515a8245c51c952db59d81ef840a146ed853e4ea", "trust": "high", "theorems": 1, "date": "2026-09-02" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "leanprover/con-leche/formalization.yaml", "repo": "leanprover/con-leche", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/leanprover/con-leche/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "ConLeche", "description": "A checker for Lean's declaration export format, written and verified in Lean. The main theorem: if the checker in its default `--verified` mode accepts a stream of declarations, the resulting environment contains no constant of type `False`.", "authors": [ "Joachim Breitner" ], "maintainers": [ "Joachim Breitner" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Soundness of the ConLeche checker", "id": "", "authors": [], "type": "original-proof", "relationship": "other", "endorsement": "" }, { "title": "The Type Theory of Lean", "id": "https://github.com/digama0/lean-type-theory", "authors": [ "Mario Carneiro" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "lean4lean", "id": "https://github.com/digama0/lean4lean", "authors": [ "Mario Carneiro" ], "type": "other", "relationship": "background", "endorsement": "" } ], "related": [ { "id": "https://github.com/digama0/lean4lean", "relationship": "independent", "note": "" } ], "classification": { "arxiv": [ "cs.LO" ], "msc2020": [ "03B35", "68V15" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Claude (Anthropic)" ], "framework": "Claude Code" }, { "method": "manual", "models": [], "framework": "" } ], "strongest": "agent", "models": [ "Claude (Anthropic)" ], "spend": "", "notes": "Implemented and proved by Claude agents. The human maintainer intensively discussed design and steps with the agents. Gates in the repository pin the axioms of the main theorem and fail the build on any compiler escape outside a justified allowlist." }, "scope": "Covered: `ConLeche.model_exists` is stated about `ConLeche.Cached.checkDecls .verified`, the function the `con-leche` binary runs on the parsed export: every environment it accepts has a model (`ConLeche.Model`, over the term reading `ConLeche.Denotes`, both in `ConLeche/Denotes.lean`) — one set per constant under which every stored constant is a member of its type and `False` is empty. `ConLeche.no_False_declaration` is its main corollary, stated about the CHUNKS the binary reads: chunks whose bytes declare a theorem of type `False` — four lines of the export format, with any bytes before, be", "sorry_count": 0, "sorry_in_definitions": null, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "ConLeche.model_exists", "file": "ConLeche/MainTheorem.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": null, "comparator": true, "literature": [] }, { "declaration": "ConLeche.no_False_declaration", "file": "ConLeche/MainTheorem.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": null, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The checker differs from the official Lean kernel in a few places (see README.md); it declines on unsupported features, and the theorem speaks only about acceptance.", "alignment": false, "original": true, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "" }, "canonical": { "repo": "leanprover/con-leche", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "lennrt/palomar-formalizations/zenodo-21890733/palomar/formalization.yaml", "repo": "lennrt/palomar-formalizations", "path": "zenodo-21890733/palomar/formalization.yaml", "directory": "zenodo-21890733/palomar", "origins": [ "palomar" ], "url": "https://github.com/lennrt/palomar-formalizations/blob/HEAD/zenodo-21890733/palomar/formalization.yaml", "version": "v0.4", "missing": [], "name": "Concrete Barnette graph certificate and prescribed-cycle obstruction", "description": "A kernel-checked certificate for the finite mathematical core of the paper's explicit 16-vertex counterexample. The compared declarations prove cubicity, bipartiteness, connectivity after every deletion of at most two vertices, Hamiltonicity of the displayed cycle, its exact complementary perfect matching, the displayed oriented facial incidence and dual data, Euler's equality, properness and uniqueness up to global permutation of the face coloring, the induced edge-color classes, their unavoidable intersections with the complementary matching, and the instantiated final obstruction. The topol", "authors": [ "Lennart Rudolph", "Sol", "Fable" ], "maintainers": [ "Lennart Rudolph" ], "license": "MIT", "role": "", "substantive": "", "sources": [ { "title": "A Counterexample to Prescribed-Cycle Recovery in Barnette Graphs", "id": "doi:10.5281/zenodo.21890733", "authors": [ "Lennart Rudolph", "Sol", "Fable" ], "type": "paper", "relationship": "formalizes", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.CO", "cs.CG" ], "msc2020": [ "05C10", "05C45", "68R10" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "OpenAI Codex (Sol)", "Anthropic Claude (Fable)" ], "framework": "AI-assisted formalization and adversarial review" }, { "method": "manual", "models": [], "framework": "n/a" } ], "strongest": "agent", "models": [ "Anthropic Claude (Fable)", "OpenAI Codex (Sol)" ], "spend": "not tracked", "notes": "The three named authors contributed to the formalization, mathematical checks, and editorial work for the submitted result." }, "scope": "Concrete 16-vertex graph, deletion-connectivity, Hamiltonian cycle, complementary matching, oriented face/dual incidence, Euler, face-coloring uniqueness, induced edge colors, and the conditional Property-1 obstruction. The incidence-to-sphere theorem and Property 1 itself are not formalized.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The graph, cycle, facial boundaries, dual, and base coloring are transcribed finite data. Lean checks their complete finite certificate chain, including standard graph connectivity after deletion of at most two vertices. It does not prove that the oriented facial-incidence certificate determines a cellular embedding of the sphere, nor does it formalize or prove Property 1 of the published algorithm. The final compared theorem therefore assumes that every edge of an admissible green class is contained in the algorithm's output and proves the obstruction for the concrete prescribed cycle.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Anthropic Claude (Fable)", "OpenAI Codex (Sol)" ], "notes": "Anthropic Claude (Fable), acting as a co-author, performed an adversarial self-review of the archived preprint and its original Lean companion before release. After the six-theorem Palomar selection was introduced, OpenAI Codex (Sol), also a co-author, performed a later read-only editorial self-review on 2026-08-20 covering the preprint, compared declarations, their Solution proof cone, Comparator selection, and metadata. No independent human peer review has been performed." }, "canonical": { "repo": "lennrt/palomar-formalizations", "directory": "zenodo-21890733/palomar" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-21-000002", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-21-000002", "commit": "090b3b300ba8ec76fdba09a17f086953d13f2653", "trust": "high", "theorems": 6, "date": "2026-09-03" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "lennrt/palomar-formalizations/zenodo-21892986/palomar/formalization.yaml", "repo": "lennrt/palomar-formalizations", "path": "zenodo-21892986/palomar/formalization.yaml", "directory": "zenodo-21892986/palomar", "origins": [ "palomar" ], "url": "https://github.com/lennrt/palomar-formalizations/blob/HEAD/zenodo-21892986/palomar/formalization.yaml", "version": "v0.4", "missing": [], "name": "Exact ordered enumeration of fixed-endpoint vertex-regular paths", "description": "Lean formalization of the paper's finite skew-symmetric path instance, exact prefix-deletion completion oracle, deterministic oracle-pruned DFS, correctness and duplicate-freedom, and an executable DFS event trace whose extracted first-output, inter-output, post-output, and no-output work intervals satisfy the paper-level depth/LCP bounds.", "authors": [ "Lennart Rudolph", "Sol", "Fable" ], "maintainers": [ "Lennart Rudolph" ], "license": "MIT", "role": "", "substantive": "", "sources": [ { "title": "Polynomial-Delay Enumeration of Fixed-Endpoint Vertex-Regular Paths in Skew-Symmetric Digraphs", "id": "doi:10.5281/zenodo.21892986", "authors": [ "Lennart Rudolph", "Sol", "Fable" ], "type": "paper", "relationship": "formalizes", "endorsement": "participated" } ], "related": [], "classification": { "arxiv": [ "cs.DS", "math.CO" ], "msc2020": [ "05C20", "05C38", "68Q25", "68R10" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "OpenAI Codex (Sol)", "Anthropic Claude (Fable)" ], "framework": "OpenAI Codex and Anthropic Claude" }, { "method": "manual", "models": [], "framework": "n/a" } ], "strongest": "agent", "models": [ "Anthropic Claude (Fable)", "OpenAI Codex (Sol)" ], "spend": "subscription-based; not tracked", "notes": "The three named authors contributed to proof development, statement checking, and release review." }, "scope": "Formalizes a finite skew-symmetric Boolean adjacency representation with fixed successor order and degree bound; genuine directed vertex-regular fixed-endpoint paths; exact regular completion after deletion of a prefix and its complementary vertices; the regularity/deletion join; and a bounded-fuel ordered exact-oracle-pruned DFS. The selected results prove the oracle semantics, soundness, completeness, duplicate-freedom, and the output depth bound. They also define an executable streaming DFS trace with frame-enter, individual query, frame-resume, output, and exit events. The trace's output p", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "Lean proves exact ordered enumeration and an operational combinatorial event-count certificate. Each exact reachability query is one charged event. The executable Goldberg--Karzanov implementation and its O(|V|+|E|) per-query analysis remain external, so no wall-clock runtime or space theorem is claimed.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Anthropic Claude (Fable)", "OpenAI Codex (Sol)" ], "notes": "The earlier Fable co-author review covered the prior four-result selection. A later Sol co-author review found that its fourth result bounded a path-pair proxy rather than actual evaluator intervals and required an executable event trace. The present trace/profile repair responds to that finding. Sol then re-audited the repaired result on 2026-08-20 for evaluator semantics, statement alignment, and definition fidelity. These were co-author self-reviews; no independent human peer review has been performed." }, "canonical": { "repo": "lennrt/palomar-formalizations", "directory": "zenodo-21892986/palomar" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-21-000008", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-21-000008", "commit": "d259d35ad32d0e7c8c60ddd5a574aa987956e1d2", "trust": "high", "theorems": 4, "date": "2026-09-03" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "lennrt/palomar-formalizations/zenodo-21892995/palomar/formalization.yaml", "repo": "lennrt/palomar-formalizations", "path": "zenodo-21892995/palomar/formalization.yaml", "directory": "zenodo-21892995/palomar", "origins": [ "palomar" ], "url": "https://github.com/lennrt/palomar-formalizations/blob/HEAD/zenodo-21892995/palomar/formalization.yaml", "version": "v0.4", "missing": [], "name": "An explicit periodic signing of C_n(1,2) below the conjectured minimum", "description": "The paper's disproof of Suvagiya's signed-circulant minimum conjecture on an infinite arithmetic progression, formalized at the matrix level. For every n divisible by 8, Lean defines the paper's period-8 triangle-flux signing of the circulant graph C_n(1,2) as an explicit signed adjacency matrix, proves it is a plus-minus-one signing of mathlib's circulant graph, and proves by exact rational sum-of-squares window certificates that every eigenvalue has absolute value at most 1397/500. For every n >= 32 this rational separator lies strictly below the conjectured minimum rho_-(n) = 2*sqrt(cos^2(p", "authors": [ "Lennart Rudolph" ], "maintainers": [ "Lennart Rudolph" ], "license": "MIT", "role": "", "substantive": "", "sources": [ { "title": "Periodic Signings of C_n(1,2): An Exact Band Edge and Short-Period Classification", "id": "10.5281/zenodo.21892995", "authors": [ "Lennart Rudolph" ], "type": "paper", "relationship": "formalizes", "endorsement": "participated" }, { "title": "Signed circulants at the Ramanujan bound", "id": "arXiv:2607.18334", "authors": [ "Vaibhav Suvagiya" ], "type": "paper", "relationship": "background", "endorsement": "none" } ], "related": [], "classification": { "arxiv": [ "math.CO", "math.SP" ], "msc2020": [ "05C22", "05C50", "68R10", "15A18" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "OpenAI Codex (Sol)", "Anthropic Claude (Fable)" ], "framework": "Codex and Claude" }, { "method": "manual", "models": [], "framework": "n/a" } ], "strongest": "agent", "models": [ "Anthropic Claude (Fable)", "OpenAI Codex (Sol)" ], "spend": "subscription-based; exact total not tracked", "notes": "Lennart Rudolph made the editorial decisions and is responsible for the formalization. Sol and Fable contributed proof development and mathematical checking as automated assistants." }, "scope": "The new layer defines the period-8 flux word, the induced step-2 signs on ZMod n through the reduction ZMod n -> ZMod 8, and the signed adjacency matrix of C_n(1,2) with positive step-1 edges. Compared theorems: the matrix is a plus-minus-one signing of mathlib's circulantGraph {1,2}; the Rayleigh quotient bound |v^T A v| <= (1397/500)|v|^2 for every real vector, proved by two exact rational sum-of-squares window certificates (period-8 local quadratic forms summing telescopically to (1397/500)I -/+ A); the eigenvalue corollary |mu| <= 1397/500 for every real eigenpair; the strict comparison 13", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The paper proves the family bound through the Bloch determinant, Sturm certificates, and phase monotonicity, and obtains the exact common radius sqrt(t0); the formalization proves the slightly weaker rational bound 1397/500 (the same separator Proposition 2.2 uses) directly at the level of finite quadratic forms, by period-8 sum-of-squares window certificates valid for every quotient order at once. The conjectured value rho_-(n) is defined by its closed trigonometric formula; the comparison uses the paper's nested-radical bounds together with cosine monotonicity rather than a monotonicity claim about rho_- itself. The eigenvalue statements quantify over real eigenpairs of the real symmetric matrix. For such a matrix that is the whole spectrum, but the formalization does not import the spectral theorem and does not assert it: the bound is proved directly for real eigenpairs, and no complex-spectrum or spectral-radius statement is selected. The uniform statement is the Rayleigh bound over all real vectors, from which the eigenvalue bound is derived.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "OpenAI Codex (Sol)", "Anthropic Claude (Fable)" ], "notes": "Anthropic Claude (Fable), an automated assistant, developed and adversarially reviewed the new matrix-level disproof layer, checking the signing, the certificate identities, and the radical comparison against the preprint. OpenAI Codex (Sol), an automated assistant, previously checked the Bloch determinant layer term by term against the rendered preprint. No independent human peer review has been performed." }, "canonical": { "repo": "lennrt/palomar-formalizations", "directory": "zenodo-21892995/palomar" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-02-000008", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-02-000008", "commit": "a7cc03716e7683cfc0c66cfc0d3b30e00a7a2923", "trust": "high", "theorems": 9, "date": "2026-09-02" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "lennrt/palomar-formalizations/zenodo-21892997/palomar/formalization.yaml", "repo": "lennrt/palomar-formalizations", "path": "zenodo-21892997/palomar/formalization.yaml", "directory": "zenodo-21892997/palomar", "origins": [ "palomar" ], "url": "https://github.com/lennrt/palomar-formalizations/blob/HEAD/zenodo-21892997/palomar/formalization.yaml", "version": "v0.4", "missing": [], "name": "Actual F₂ flag homology and the dense extremal counterexample family", "description": "A finite-chain formalization of the F₂ content of Theorem 1.1 and Proposition 3.1 in the source paper. Lean defines the literal flag-chain groups of a finite graph using unordered edges and three-vertex clique faces, their d1 and d2 maps, and actual H1 as ker(d1)/range(d2). It proves that the paper's family G_n has F₂ first Betti number exactly two for every n at least seven, proves the universal F₂ upper bound beta1 <= 2 for every n-vertex graph with choose(n,2)-n edges, and bundles these facts with the exact density and no-complete-bipartite-spanning obstruction. The literal d1-d2 identity i", "authors": [ "Lennart Rudolph" ], "maintainers": [ "Lennart Rudolph" ], "license": "MIT", "role": "", "substantive": "", "sources": [ { "title": "An Infinite Dense Counterexample Family for Extremal First Betti Numbers of Flag Complexes", "id": "10.5281/zenodo.21892997", "authors": [ "Lennart Rudolph" ], "type": "paper", "relationship": "formalizes", "endorsement": "participated" } ], "related": [], "classification": { "arxiv": [ "math.CO", "math.AT" ], "msc2020": [ "05C35", "05C69", "55U10" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "OpenAI Codex (Sol)", "Anthropic Claude (Fable)" ], "framework": "OpenAI Codex and Anthropic Claude" }, { "method": "manual", "models": [], "framework": "n/a" } ], "strongest": "agent", "models": [ "Anthropic Claude (Fable)", "OpenAI Codex (Sol)" ], "spend": "subscription-based; not tracked", "notes": "Lennart Rudolph selected the scope and statements, reviewed the proofs, and approved the release. Sol and Fable contributed proof development and adversarial review as automated assistants." }, "scope": "Four declarations are compared: the structural d1-d2 identity and three paper-level results forming the main results group. The project formalizes actual degree-one simplicial homology over F₂ for finite flag complexes. Its transparent chain model uses graph edges as Sym2 subtypes, triangular clique finsets as 2-faces, and the quotient of the d1 kernel by the d2 range. The family proof performs the paper's leaf reductions directly in this quotient and identifies the K_(2,3) core. Proposition 3.1 is proved for every finite graph at the exact edge count: isolated vertices give cones, leaves redu", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The paper presents the family reduction and Proposition 3.1 using standard independence-complex homotopy tools, including leaf deletion, complements of cycles, and joins. Lean proves the required degree-one consequences internally in a concrete F₂ chain quotient: explicit cone and leaf normal forms, literal cycle calculations, an actual joined-chain quotient equivalence, and finite-graph component decompositions. This changes the proof vehicle but not the selected F₂ statements. The formalization does not promote the result to arbitrary coefficients or integral homology and does not formalize the computational census reported elsewhere in the paper.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "OpenAI Codex (Sol)" ], "notes": "The current Sol automated self-review covers the upgraded actual F₂ quotient, family beta-one computation, and Proposition 3.1 proof cone. The older Fable automated self-review applied to the earlier graph-certificate/ Boolean-core selection and does not cover this upgrade. No independent human peer review has been performed." }, "canonical": { "repo": "lennrt/palomar-formalizations", "directory": "zenodo-21892997/palomar" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-02-000002", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-02-000002", "commit": "a7cc03716e7683cfc0c66cfc0d3b30e00a7a2923", "trust": "high", "theorems": 4, "date": "2026-09-02" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "lennrt/palomar-formalizations/zenodo-21925458/palomar/formalization.yaml", "repo": "lennrt/palomar-formalizations", "path": "zenodo-21925458/palomar/formalization.yaml", "directory": "zenodo-21925458/palomar", "origins": [ "palomar" ], "url": "https://github.com/lennrt/palomar-formalizations/blob/HEAD/zenodo-21925458/palomar/formalization.yaml", "version": "v0.4", "missing": [], "name": "A three-landmark multiset resolving set for even cylindrical graphs", "description": "A concrete formalization of the infinite-family construction in the source: cycle and Cartesian-product distances on P_m □ C_(6m), the displayed landmarks (0,0), (0,2m), and (m-1,0), the four-piece cycle distance-pair locus, and injectivity of the resulting unordered multiset code for every even m at least 2. This establishes the three-landmark resolving upper bound. It does not formalize the separate general lower bound needed to conclude that the set is a basis or that the multiset dimension equals 3.", "authors": [ "Lennart Rudolph", "Sol", "Fable" ], "maintainers": [ "Lennart Rudolph" ], "license": "MIT", "role": "", "substantive": "", "sources": [ { "title": "Multiset Dimension of Cylindrical Graphs: An Infinite Family and a Certified Census", "id": "10.5281/zenodo.21925458", "authors": [ "Lennart Rudolph", "Sol", "Fable" ], "type": "paper", "relationship": "formalizes", "endorsement": "participated" } ], "related": [], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05C12", "05C76", "68R10" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "OpenAI Codex (Sol)", "Anthropic Claude (Fable)" ], "framework": "Codex and Claude" }, { "method": "manual", "models": [], "framework": "n/a" } ], "strongest": "agent", "models": [ "Anthropic Claude (Fable)", "OpenAI Codex (Sol)" ], "spend": "subscription-based; exact total not tracked", "notes": "The three named authors collaborated on the scope, statements, review, and responsibility for the formalization." }, "scope": "The project defines the concrete vertex type Fin m × Fin (6m), the standard cycle and Cartesian-product distance formulas, the paper's three displayed landmarks, unordered multiset codes, and the resolving property. It proves the complete four-piece cycle distance-pair locus and its injectivity, then proves that the displayed landmarks resolve P_m □ C_(6m) for every even m ≥ 2. It does not define or prove the minimum over all landmark sets, and therefore does not claim basis minimality or dim_m(P_m □ C_(6m)) = 3.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The Lean development uses the paper's explicit coordinate distance formula rather than Mathlib's SimpleGraph Cartesian-product distance. It formalizes the resolving-set construction and cycle-locus lemma, but not the external theorem that a connected non-path graph has multiset dimension at least 3; consequently it does not formalize basis minimality or exact dimension.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "OpenAI Codex (Sol)", "Anthropic Claude (Fable)" ], "notes": "OpenAI Codex (Sol), acting as a co-author, performed a line-by-line source-fidelity self-audit of the upgraded statements and proof structure against Lemma 1 and Theorem 2. Fable's earlier co-author adversarial self-review covered the arithmetic package. No independent human peer review has been performed." }, "canonical": { "repo": "lennrt/palomar-formalizations", "directory": "zenodo-21925458/palomar" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-21-000011", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-21-000011", "commit": "d259d35ad32d0e7c8c60ddd5a574aa987956e1d2", "trust": "high", "theorems": 3, "date": "2026-09-03" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "lennrt/palomar-formalizations/zenodo-21925574/palomar/formalization.yaml", "repo": "lennrt/palomar-formalizations", "path": "zenodo-21925574/palomar/formalization.yaml", "directory": "zenodo-21925574/palomar", "origins": [ "palomar" ], "url": "https://github.com/lennrt/palomar-formalizations/blob/HEAD/zenodo-21925574/palomar/formalization.yaml", "version": "v0.4", "missing": [], "name": "Combinatorial core of the K₃,₃,₁-redundancy theorem for tetrahedral balls", "description": "Alexa's three-dimensional Tutte embedding theorem (Discrete & Computational Geometry, 2025) requires the graph of a tetrahedral mesh to have neither a K₆ nor a K₃,₃,₁ minor and asks whether the second exclusion is necessary. The paper proves that it is redundant for finite simplicial complexes whose realization is a closed topological 3-ball and which satisfy the boundary-triangle condition (BT): if the 1-skeleton G has no K₆ minor, then G is linklessly embeddable and hence has no K₃,₃,₁ minor (Theorem 1.1). This Lean development proves the paper's new combinatorial mathematics over mathlib si", "authors": [ "Lennart Rudolph" ], "maintainers": [ "Lennart Rudolph" ], "license": "MIT", "role": "", "substantive": "", "sources": [ { "title": "Simpler Graph Conditions for Embedding Tetrahedral Meshes", "id": "10.5281/zenodo.21925574", "authors": [ "Lennart Rudolph" ], "type": "paper", "relationship": "formalizes", "endorsement": "participated" }, { "title": "Tutte Embeddings of Tetrahedral Meshes", "id": "10.1007/s00454-023-00494-0", "authors": [ "Marc Alexa" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "The Colin de Verdière graph parameter", "id": "", "authors": [ "Hein van der Holst", "László Lovász", "Alexander Schrijver" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "New bounds on maximal linkless graphs", "id": "10.2140/agt.2023.23.2545", "authors": [ "Ramin Naimi", "Andrei Pavelescu", "Elena Pavelescu" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Extremal graphs for contractions to K₇", "id": "", "authors": [ "L. K. Jørgensen" ], "type": "paper", "relationship": "background", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.CO", "cs.CG" ], "msc2020": [ "05C10", "05C40", "05C83", "55N10", "57Q15", "68U05" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "OpenAI Codex (Sol)", "Anthropic Claude (Fable)" ], "framework": "Codex and Claude" } ], "strongest": "agent", "models": [ "Anthropic Claude (Fable)", "OpenAI Codex (Sol)" ], "spend": "subscription-based; exact total not tracked", "notes": "Lennart Rudolph selected the scope and statements, reviewed the proofs, and is solely responsible for the formalization; Sol and Fable contributed proof development and adversarial review." }, "scope": "Compared: the combinatorial core of Theorem 1.1 (structural_theorem) with Corollary 3.6, the Section 4 relative ball duality, Corollary 2.5, and Lemma 5.1 as explicit hypotheses; Lemma 5.2 and Proposition 5.3 over mathlib SimpleGraph values on an arbitrary vertex type, with linkless embeddability an abstract predicate constrained by exactly those hypotheses; and Lemma 7.1, the coordinate identification, and Lemma 4.3 for every finite simplicial pair on Fin n and every injectively labelled four-clique under (BT). 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Linkless embeddability is not defined inside Lean; it is an abstract predicate constrained by the two hypotheses the paper takes from the literature, so the compared theorems are exactly the paper's conditional mechanism. In the finite layer, relative chains and homology are realized by the four-face/six-edge F₂ coordinate complex, with the proved identification connecting it to the induced simplicial pair. In structural_theorem the skeleton is the SimpleGraph on Fin n whose distinct vertices are adjacent exactly when their edge is a simplex of the ambient complex; the nonempty-complement clause of Theorem 4.4 is the hypothesis that some vertex lies outside the f", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "Lennart Rudolph reviewed the compared statements against the paper. 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Lennart Rudolph directed the work, checked the retained statements and proofs, and approved the release." }, "scope": "The project defines all 28 outer roles in every sector and the exact rational entries of the restricted released outer matrix Q_n. It defines the cyclic sector maps, the explicit equivalence sending {24_s,25_s,29_(s+1)} to one 3-coordinate core, and the extracted zero, upper-coupling, singleton, and core blocks. It proves the source-indexed matrix equals the block normal form, proves the 24n zero block is strictly upper triangular, and proves the complete characteristic polynomial with algebraic multiplicities 24n,n,n,n,n for every n at least 3. 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Sol and Fable contributed formalization and adversarial review as automated assistants." }, "scope": "Theorem 1.1 of the preprint in rank form: for all 345 pair-aligned windows of the pinned OneTwo table at m = 16, the exact t-value is 4 for the nine windows starting at dimensions 1, 7, 31, 49, 67, 195, 225, 243, 373 and 5 otherwise, each certified by full row rank at total 16 - t and a dependent selection at total 17 - t; the distribution 9 and 336; and the dimensions 25 to 28 certificate with every smaller threshold refuted. External: the digital-net rank criterion, the discrepancy consequence, source authentication, the ancillary depth-15 census, and the protected-pair audit.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The formalization assumes the external equivalence between the encoded rank test and the t-value. For the Palomar submission, the release's native exhaustive subset evaluator is replaced by deterministic 16-column XOR row reduction checked by ordinary kernel decide; the packed input rows and threshold claims are unchanged. The Challenge and the census data module store the 692-dimension table as one 256-bit literal per dimension and decode it in place by explicit division and remainder; the decoded rows enter the compared statements only through that decoder.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "Lennart Rudolph reviewed the compared statements against the preprint. Anthropic Claude (Fable), acting as an automated assistant rather than a reviewer of record, ran an adversarial review pass separate from proof development before release. 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It also retains the companion's token-order bridge and exact arithmetic endpoints.", "authors": [ "Lennart Rudolph", "Sol", "Fable" ], "maintainers": [ "Lennart Rudolph" ], "license": "MIT", "role": "", "substantive": "", "sources": [ { "title": "An Explicit Obstruction to Uniform Two-Word π-Representability", "id": "10.5281/zenodo.21986230", "authors": [ "Lennart Rudolph", "Sol", "Fable" ], "type": "paper", "relationship": "formalizes", "endorsement": "participated" } ], "related": [], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05C62", "05C75", "06A07", "68R15" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "OpenAI Codex (Sol)", "Anthropic Claude (Fable)" ], "framework": "Codex and Claude" }, { "method": "manual", "models": [], "framework": "n/a" } ], "strongest": "agent", "models": [ "Anthropic Claude (Fable)", "OpenAI Codex (Sol)" ], "spend": "subscription-based; exact total not tracked", "notes": "The authors are Lennart Rudolph, Sol, and Fable. Lennart Rudolph selected the statements, reviewed the proof boundary, and takes responsibility for the submission." }, "scope": "The project defines finite words, k-uniformity, two-word π-representability, neighbourhood traces, and B20. It proves the general stars-and-bars trace bound and the explicit B20 nonmembership theorem. The paper's separate permutation-graph compaction characterization and its VC-dimension consequences remain outside the formalized scope.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The trace bound is stated in the equivalent finite-family form: every outside family on which the neighbourhood-trace map is injective has the stated cardinality bound. 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For the source's row-shifted tiling lattice <(N,0),(N/4,N)> it proves, for every even m >= 8, that one expli", "authors": [ "Lennart Rudolph" ], "maintainers": [ "Lennart Rudolph" ], "license": "MIT", "role": "", "substantive": "", "sources": [ { "title": "The Even-Order Grünschloß–Keller Permutation Nets Are (0,m,2)-Nets", "id": "doi:10.5281/zenodo.22264675", "authors": [ "Lennart Rudolph" ], "type": "paper", "relationship": "formalizes", "endorsement": "participated" }, { "title": "(t,m,s)-Nets and Maximized Minimum Distance, Part II", "id": "doi:10.1007/978-3-642-04107-5_25", "authors": [ "Leonhard Grünschloß", "Alexander Keller" ], "type": "paper", "relationship": "background", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.NA", "cs.LO" ], "msc2020": [ "11K38", "65C05", "65D18", "68V20" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Anthropic Claude", "OpenAI Codex" ], "framework": "AI-assisted proof drafting, formalization, and adversarial review" }, { "method": "manual", "models": [], "framework": "n/a" } ], "strongest": "agent", "models": [ "Anthropic Claude", "OpenAI Codex" ], "spend": "not tracked", "notes": "Anthropic Claude and OpenAI Codex were used for literature searches, algebraic exploration, proof and checker drafting, the Lean formalization, and adversarial review; the author selected the target and the theorem statements, executed the checks and the Lean build, and accepts full responsibility for the claims; no AI system is an author." }, "scope": "Theorem A symbolic for all even m >= 4; Theorem B upper bound symbolic for all even m >= 8; Theorem B exact at m = 8 (kernel decide over all 32,640 pairs plus symbolic lattice-minimum exactness). Not formalized: the all-m lower bound of Theorem B.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "GKEvenNet.net", "file": "GKEvenNet/Main.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "GKEvenNet.distance_upper", "file": "GKEvenNet/Main.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "GKEvenNet.distance_m8_exact", "file": "GKEvenNet/Main.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The Lean point set uses the closed form 2^r * rev r k for the 32-bit van der Corput term of the header; the identity holds for k < 2^r and the m = 8 point lists agree byte-for-byte with a literal reimplementation of the header. The tiling distance is stated through the lattice <(N,0),(N/4,N)> read off getShiftedTiling; the paper's normalized distance sqrt(241 * 2^(m-8)) / 2^m is the integer-scale value divided by 2^m. Theorem A is stated as a unique-existence statement per dyadic box, which is the definition of a (0,m,2)-net after normalization by 2^m.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "Lennart Rudolph reviewed the statements against the paper and the source header; Anthropic Claude and OpenAI Codex, acting as automated assistants rather than reviewers of record, ran adversarial review passes separate from proof development. Every compared theorem is kernel-checked, and the finite claims are reproduced by three exact programs in the accompanying Zenodo archive. 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No AI system is an author. The human author is responsible for the mathematical claims and submission." }, "scope": "General integer-trade criterion and full-factor Cartesian-product kernel/recovery theorem; the all-order grid lower bound; and, for every n ≥ 6, the complete four-parameter integer grid kernel, exact threshold n−2 for odd n and n−3 for even n, optimal n−2 sensor count throughout 2 ≤ h < threshold, and unique four-parameter description of equal-observation populations.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "ShellTomography.Verified.trade_criterion", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "ShellTomography.Verified.product_kernel", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "ShellTomography.Verified.product_recovery", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "ShellTomography.Verified.grid_lower_bound", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "ShellTomography.Verified.grid_kernel_uniform", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "ShellTomography.Verified.grid_capacity_uniform", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "ShellTomography.Verified.grid_sensor_optimality", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "ShellTomography.Verified.grid_fibre_uniform", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "Finite labelled sensor types allow duplicate positions in the product theorem. Lean also proves three benign endpoint extensions beyond the paper's displayed domains: the trade criterion at h = 0 (the paper states h >= 1), the selected product clauses for an empty sensor-label type (the paper assumes a nonempty placement), and vacuous n < 3 cases of the grid lower bound because Nat subtraction makes S.card < n-2 impossible there (the paper states n >= 3). The substantive grid results retain n >= 6. Actual graph shortest-path distances and literal multiplicity-preserving bags are used. The exactly-h equivalence, grid-distance bridge, extraction inverse, direct minimum/attainer, and alternating-placement facts are proved in the source but are not separately selected. The detector/B-channel/capacity-profile formulations, exact detector-cross converse, two-sensor cycle theorem, rational-rank and rational-kernel clauses, coefficient census and exact norm formulas, rational normalized-fibre and slab claims, adjacent-corner metric-dimension and named capacity-jump statements, and executable decoder correctness, termination, and complexity are not formalized as corresponding paper-level Le", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "Automated statement, axiom, kernel, citation, rendering and reproducibility checks completed. No independent human peer review, online Palomar editorial result, or registry entry is claimed." }, "canonical": { "repo": "lennrt/palomar-formalizations", "directory": "zenodo-22404456/palomar" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-07-000003", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-07-000003", "commit": "71bcd530fd1c2c9a02d9112c9ca05839082983de", "trust": "high", "theorems": 8, "date": "2026-09-07" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "lennrt/palomar-formalizations/zenodo-22667596/palomar/formalization.yaml", "repo": "lennrt/palomar-formalizations", "path": "zenodo-22667596/palomar/formalization.yaml", "directory": "zenodo-22667596/palomar", "origins": [ "palomar" ], "url": "https://github.com/lennrt/palomar-formalizations/blob/HEAD/zenodo-22667596/palomar/formalization.yaml", "version": "v0.4", "missing": [], "name": "Short-Cycle Decompositions and Full Zombie Damage in Cubic Graphs", "description": "For every finite simple connected bridgeless cubic graph G, the survivor can force full zombie damage if and only if G is not isomorphic to K4, K3,3, the triangular prism or the cube. In particular, for order n at least 10 this proves Randy Davila's Conjecture 24, zdmg(G)=n, with an explicit bound of 2n(n+1) survivor moves against every initial zombie position and every legal geodesic reply. The game formalization retains starting and turn order, survivor passes, capture and distinct-vertex damage, and proves the numerical payoff's connection to bounded and eventual strategies. The argument cl", "authors": [ "Lennart Rudolph" ], "maintainers": [ "Lennart Rudolph" ], "license": "MIT", "role": "", "substantive": "", "sources": [ { "title": "Short-Cycle Decompositions and Full Zombie Damage in Cubic Graphs", "id": "doi:10.5281/zenodo.22667596", "authors": [ "Lennart Rudolph" ], "type": "paper", "relationship": "formalizes", "endorsement": "participated" }, { "title": "The Zombie Damage Number of a Graph", "id": "arxiv:2607.16382v1", "authors": [ "Randy Davila" ], "type": "paper", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "Distance-Shell Tomography on Graphs: Integer Trades and Optimal Grid Sensing", "id": "doi:10.5281/zenodo.22404456", "authors": [ "Lennart Rudolph" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Computing Metric Dimension of Two Types of Claw-free Cubic Graphs with Applications", "id": "doi:10.61091/jcmcc119-17", "authors": [ "Muhammad Shoaib Sardar", "Shou-Jun Xu", "Murat Cancan", "Mohammad Reza Farahani", "Mehdi Alaeiyan", "Shobha V. Patil" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "The Graph Curvature Calculator and the curvatures of cubic graphs", "id": "doi:10.1080/10586458.2019.1660740", "authors": [ "D. Cushing", "R. Kangaslampi", "V. Lipiäinen", "S. Liu", "G. W. Stagg" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "The damage number of a graph", "id": "", "authors": [ "Danielle Cox", "Asiyeh Sanaei" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Replacing Cops with Zombies", "id": "", "authors": [ "Fengyi Liu", "Avery Miller" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "On the damage number of graphs", "id": "", "authors": [ "Valentin Gledel", "William B. Kinnersley", "Balázs Patkós", "Miloš Stojaković" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "On the multi-robber damage number", "id": "", "authors": [ "Miloš Stojaković", "Lasse Wulf" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "The damage number of the Cartesian product of graphs", "id": "", "authors": [ "Melissa A. Huggan", "Margaret-Ellen Messinger", "Amanda Porter" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Corrigendum to: The damage number of the Cartesian product of graphs", "id": "", "authors": [ "Melissa A. Huggan", "Margaret-Ellen Messinger", "Amanda Porter" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" } ], "related": [], "classification": { "arxiv": [ "math.CO", "cs.LO" ], "msc2020": [ "05C57", "05C75", "68V20" ] }, "automation": { "methods": [ { "method": "manual", "models": [], "framework": "" }, { "method": "agent", "models": [ "OpenAI ChatGPT", "OpenAI Codex (Astra)", "Anthropic Fable 5.1" ], "framework": "Human-directed AI assistance in mathematics, drafting, literature search, verification and release preparation" } ], "strongest": "agent", "models": [ "Anthropic Fable 5.1", "OpenAI ChatGPT", "OpenAI Codex (Astra)" ], "spend": "not tracked", "notes": "OpenAI Codex (Astra) assisted in formalization, source review, verification and release preparation. Anthropic Fable 5.1 assisted in adversarial reviews, as reported by the author. OpenAI ChatGPT also assisted in the research workflow. No AI is an author. AI-produced third-party computational review material is supplementary evidence in the associated Zenodo artifacts and is not represented as independent human peer review." }, "scope": "Full formalization of Randy Davila’s one-zombie damage game and the complete proof of his Conjecture 24. The game includes shortest-walk geodesic replies, adversarial choices, start and turn order, survivor passes, capture, distinct-vertex damage, bounded and eventual strategies, and an explicit numerical damage value. For every finite simple connected bridgeless cubic graph of order n at least 10, the survivor can force damage to all n vertices within 2n(n+1) survivor moves. The project also proves the four-exception characterization, structural classification, routing, recurrence and related", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "PalomarVerified.full_damage_within_iff", "file": "PalomarWrappers.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "PalomarVerified.full_damage_iff", "file": "PalomarWrappers.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "PalomarVerified.davila_conjecture_24", "file": "PalomarWrappers.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "PalomarVerified.full_damage_of_clean_edge", "file": "PalomarWrappers.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "PalomarVerified.recurrent_coverage_of_clean_edge", "file": "PalomarWrappers.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "ZombieMain.mobius_recurrent_coverage", "file": "ZombieMain/PeriodicCoverage.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "PalomarVerified.four_boundary_edges", "file": "PalomarWrappers.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "PalomarVerified.short_component_expansion", "file": "PalomarWrappers.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "PalomarVerified.four_port_principle", "file": "PalomarWrappers.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "PalomarVerified.cycle_union_classification", "file": "PalomarWrappers.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "ZombieDamage.FullGame.zombieDamageNumber_eq_card_iff", "file": "ZombieMain/GameValue.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "PalomarVerified.davila_conjecture_24_damage_number", "file": "PalomarWrappers.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The graph is the actual finite simple graph; Conjecture 24 assumes connectedness, degree three at every vertex, no bridges and at least ten vertices. No routing, decomposition or classification hypothesis replaces a proved step. Every geodesic zombie reply is adversarial. The zombie chooses first and moves first; the survivor may pass; damage is credited on departure after a survived zombie turn. Entering the occupied zombie vertex is normalized to a pass, with payoff equivalence proved for every damaged-set objective in OccupiedVertexNormalization.lean. The finite numerical value is the maximal forceable score, and the threshold, eventual-horizon and full-damage equivalences are proved. Zero is the default score where no distinct starting pair exists. The formal graph-order upper bound also prevents empty-graph vacuity from creating positive scores. No claim covers every historical theorem in Davila’s paper. The structural classification has an explicit one-port alternative; the proof derives that this alternative cannot occur for a short-edge component in the bridgeless cubic application. The general degree-independent interface theorem and the ring family's structural connection", "alignment": true, "original": false, "review": { "status": "ai-assisted source and mathematical review; not independently human-refereed", "bucket": "other", "reviewers": [], "notes": "The selection is limited to 39 pursuit and short-cycle results; the three separate sensing results remain proved in the library but are not advertised as selected research results. VERIFICATION.md distinguishes the revised-selection checks from the historical 42-result checks. 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Used for normal families, holomorphic inverse maps and related analytic foundations." }, { "id": "dependencies/ComplexDynamics", "relationship": "builds-on", "note": "Supplied source snapshot; actual Fatou normality and component definitions. The short Challenge definitions reproduce the same declarations." }, { "id": "dependencies/ComplexApproximation", "relationship": "builds-on", "note": "Supplied source snapshot; filled continua and planar topology." }, { "id": "dependencies/EremenkosConjecture", "relationship": "builds-on", "note": "Supplied source snapshot; simple connectivity of interior components from absence of bounded complementary components." }, { "id": "https://github.com/alonamaloh/schoenflies-lean/tree/05a43d29cde026618777db3d4e4316204ccca237", "relationship": "builds-on", "note": "Attributed vendored topology dependency within EremenkosConjecture; see its PROVENANCE.md." }, { "id": "https://github.com/TauCetiProject/TauCeti/tree/f441315d5d3c9ccc377a691d7a7da00613010718", "relationship": "builds-on", "note": "138 attributed modules within FunctionTheory; original source authors include the Tau Ceti contributors and Chris Birkbeck. Licence, alignment and compatibility records are retained." }, { "id": "https://github.com/girving/ray/tree/753f7131cf96f4651294de4398368abf136c34de", "relationship": "adapts", "note": "Sphere-atlas and coordinate adaptations in FunctionTheory; licence, review copies and source manifest included." }, { "id": "https://github.com/will1491/no-wandering-domains/tree/0a6497b0cc9ed39a6a705bf013449635894b56d0", "relationship": "builds-on", "note": "Eight attributed modules and the derived finite-chart convergence proof within FunctionTheory. Code author Will (Ziang) Li; associated paper by Ziang Li and Yusheng Luo. Licence and adaptation records included." }, { "id": "https://github.com/deancureton/sphere-six-complex/tree/895c0a0", "relationship": "builds-on", "note": "Secondary source attribution retained in TauCeti/Analysis/Complex/Conformal/Inverse/BoundaryCluster.lean (Apache-2.0); not a separately vendored repository." }, { "id": "https://github.com/urkud/mathlib4/tree/d43061d911b1aeae0788591da437a3b115098962", "relationship": "adapts", "note": "BoundedWanderingDomains/RiemannMappingFull.lean and BoundedWanderingDomains/UnitDiscShift.lean; see THIRD_PARTY_NOTICES.md." }, { "id": "https://github.com/will1491/RiemannDynamics/tree/b3fa37cc0f18a23ea66b654ea3f73eb472129010", "relationship": "adapts", "note": "Selected source ported to the submitted Lean/Mathlib pins. Original headers and Apache 2.0 licence retained. Covering second-countability is generalised; the punctured-plane bridge and final dynamical connection are project additions. See THIRD_PARTY_NOTICES.md." }, { "id": "https://github.com/will1491/RMT4/tree/7e092cd5e347a1ee01afb165dd6ab043082e0f4d", "relationship": "adapts", "note": "The cindex and has_sqrt modules are included with their original attribution and Apache 2.0 licence; compatibility adaptations are recorded." } ], "classification": { "arxiv": [ "math.DS", "math.CV" ], "msc2020": [ "37F10", "30D05", "30F45" ] }, "automation": { "methods": [ { "method": "manual", "models": [], "framework": "Mathematical discussion" }, { "method": "agent", "models": [ "OpenAI ChatGPT/Codex (exact model versions not recorded for every turn)" ], "framework": "OpenAI Codex with Lean 4 and Mathlib" } ], "strongest": "agent", "models": [ "OpenAI ChatGPT/Codex (exact model versions not recorded for every turn)" ], "spend": "", "notes": "Lasse Rempe provided mathematical direction. AI assistance produced and checked the formal development and submission files. Library contributions are credited separately; a successful kernel check is not independent mathematical review." }, "scope": "The entire-function theorem rules out a pairwise disjoint forward orbit of actual Fatou components when one point in the initial component has a bounded forward orbit. No boundedness of the union of Fatou components is assumed. Simple connectivity and eventual intrinsic-disc injectivity are derived for auxiliary trapped components in the entire proof. They remain explicit hypotheses only in the local theorem, alongside compact containment in an analytic neighbourhood. The former classical hyperbolic metric input is now proved from formalised uniformisation, covering theory, cusp estimates, Gre", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "BoundedWanderingDomains.no_local_bounded_wandering_domains", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "BoundedWanderingDomains.no_bounded_wandering_domains_transcendental_entire", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] } ], "comparator": false, "literature_dependencies": 0, "divergences": "There is no remaining metric-existence or area-formula hypothesis. Curvature is minus one and area is divided by 2*pi. The local theorem treats total functions on C required analytic near closure(V); its simply connected trapped-component orbit lies in one compact subset of V and satisfies radius-dependent eventual injectivity. The entire theorem assumes only one bounded point orbit and derives the auxiliary properties needed for its proof. Fatou normality is formulated by locally uniform subsequential convergence into the one-point compactification of C, with its compact Hausdorff uniformity; the map itself is never evaluated at infinity. The short supporting definitions are compared explicitly.", "alignment": true, "original": true, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "Lean builds, transitive axiom audits, independent Challenge/Solution declaration comparison and metadata validation are recorded in VERIFICATION.md. Official Comparator and independent-kernel replay status is recorded there separately. No independent human review, Palomar acceptance or public submission is claimed." }, "canonical": { "repo": "lr-uk/bounded-orbit-wandering-domains", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-25-000001", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-25-000001", "commit": "5c983043c79131f3c9525ab52ce748864d7d8b09", "trust": "high", "theorems": 2, "date": "2026-09-25" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "LR-UK/exp-chaotic/formalization.yaml", "repo": "LR-UK/exp-chaotic", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/LR-UK/exp-chaotic/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "The exponential map is chaotic", "description": "In 1926, Fatou conjectured that the Julia set of the complex exponential map is the entire complex plane; in other words, the dynamical system described by the map is unstable under perturbation at every point in the plane. Misiurewicz proved this conjecture in 1981. In 2015, Shen and Rempe published \"The Exponential Map Is Chaotic: An Invitation to Transcendental Dynamics\", giving an elementary and conceptual proof of Misiurewicz's theorem and deriving its dynamical consequences, without requiring advanced results from complex analysis or dynamics. This project formalises these results in Lea", "authors": [ "Lasse Rempe" ], "maintainers": [ "Lasse Rempe" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "The Exponential Map Is Chaotic: An Invitation to Transcendental Dynamics", "id": "arXiv:1408.1129", "authors": [ "Zhaiming Shen", "Lasse Rempe-Gillen" ], "type": "journal article", "relationship": "formalizes", "endorsement": "participated" }, { "title": "On iterates of e^z", "id": "", "authors": [ "Michał Misiurewicz" ], "type": "journal article", "relationship": "formalizes", "endorsement": "not-contacted" } ], "related": [ { "id": "https://github.com/jrh13/hol-light/blob/master/Examples/misiurewicz.ml", "relationship": "adapts", "note": "John Harrison's BSD-2-Clause HOL Light formalisation supplied the six-lemma architecture and mathematical interface for the initial proof stage." } ], "classification": { "arxiv": [ "math.DS", "math.CV" ], "msc2020": [ "37F10", "30D05" ] }, "automation": { "methods": [ { "method": "manual", "models": [], "framework": "Lean 4 and Mathlib" }, { "method": "copilot", "models": [], "framework": "Microsoft 365 Copilot" }, { "method": "agent", "models": [ "Anthropic Claude Opus 5" ], "framework": "Anthropic Claude" }, { "method": "agent", "models": [ "OpenAI GPT-5.6 Sol", "OpenAI GPT-6 Astra" ], "framework": "OpenAI Codex" } ], "strongest": "agent", "models": [ "Anthropic Claude Opus 5", "OpenAI GPT-5.6 Sol", "OpenAI GPT-6 Astra" ], "spend": "not tracked", "notes": "Lasse Rempe determined the mathematical scope, supplied and corrected the statements, and developed the initial scaffold with assistance from Microsoft 365 Copilot. He also provided substantial mathematical guidance for the initial proof, particularly the observation that density of the iterated preimages of the real axis did not require the full Montel theorem and could instead be proved using Schwarz-lemma arguments. Claude Opus 5 implemented and tested substantial parts of this proof and its supporting complex-analysis and strip-geometry results. ChatGPT subsequently compiled, repaired, reo" }, "scope": "Theorems 1.1 and 1.2 of Shen–Rempe-Gillen are fully represented. The project also proves Theorem 4.3, Corollaries 4.4, 5.5, and 5.6, Observation 5.2, Theorem 6.1, the standard density benchmarks, and Misiurewicz's original statement, which had been conjectured by Fatou. The precise sensitivity condition of Definition 2.3 holds with arbitrary positive constants as in Exercise 8.7. The remaining exercises, appendix/outlook, hyperbolic preliminaries, and precise formulations of Proposition 5.3, Observation 5.4, and Lemma 6.2 are not included. Normality is defined for sequences of maps from a topo", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "ExpChaotic.dense_escapingSet", "file": "ExpChaotic/Results.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "ExpChaotic.dense_denseOrbitSet", "file": "ExpChaotic/Results.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "ExpChaotic.dense_repellingPeriodicSet", "file": "ExpChaotic/Results.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "ExpChaotic.devaney_chaotic", "file": "ExpChaotic/Results.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "ExpChaotic.juliaSet_exp", "file": "ExpChaotic/Results.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "ExpChaotic.not_tendstoLocallyUniformly_sphericalExpIterate", "file": "ExpChaotic/Results.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The proof is not a line-by-line implementation of the paper's hyperbolic-metric argument. Its first part adapts the HOL Light proof architecture; later results use explicit logarithmic branches, quantitative open mapping, contraction, and Baire category. Normality is an abstract sequential compactness condition for maps into a uniform space, with Mathlib's convergence predicate and subtype limit domain. The spherical statements treat iterates as maps from the plane to the sphere and do not extend the exponential to infinity.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "The declarations have been compiled, linted, and audited for axioms by the AI workflow. Executable audits check documentation, permitted axioms, and structural agreement of the challenge and solution interfaces. The external Comparator/NanoDa end-to-end run remains outstanding. Lasse Rempe reviewed the preceding challenge statements and supplied annotations. No independent line-by-line implementation review or external review has taken place." }, "canonical": { "repo": "lr-uk/exp-chaotic", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-16-000001", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-16-000001", "commit": "303b2d3d1121ebc3a86399de8f1e71c17a8b6dd8", "trust": "high", "theorems": 15, "date": "2026-09-16" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "lukasliehr/GrothendieckBessel/formalization.yaml", "repo": "lukasliehr/GrothendieckBessel", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/lukasliehr/GrothendieckBessel/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Grothendieck's theorem for Bessel sequences", "description": "A Lean 4 formalization of the existence assertion in Theorem 1.1 of arXiv:2608.12280v1 with the sharp constant K = 1: for every countable Bessel sequence with bound 1 in a complex Hilbert space, there exist measurable functions on the unit interval, pointwise bounded by 1, whose correlations reproduce the sequence's Gram matrix. This solves an open problem posed by Olevskii.", "authors": [ "Lukas Liehr", "Mitchell A. Taylor", "Peiyang Yu" ], "maintainers": [ "Lukas Liehr" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Grothendieck's theorem for Bessel sequences", "id": "arXiv:2608.12280v1", "authors": [ "Lukas Liehr", "Mitchell A. Taylor", "Peiyang Yu" ], "type": "paper", "relationship": "formalizes", "endorsement": "participated" } ], "related": [], "classification": { "arxiv": [ "math.FA", "math.CA" ], "msc2020": [ "46B15", "46B25", "42C15", "68V20" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "GPT-5.4" ], "framework": "OpenAI GPT-5.4" }, { "method": "agent", "models": [ "GPT-5.5" ], "framework": "Codex" }, { "method": "agent", "models": [], "framework": "Codex" } ], "strongest": "agent", "models": [ "GPT-5.4", "GPT-5.5" ], "spend": "", "notes": "The first two entries reproduce the source paper's disclosure. The third records the later preparation of the repository for Palomar submission." }, "scope": "The compared result is the existence assertion in Theorem 1.1 of arXiv:2608.12280v1 for countable Bessel sequences in complex Hilbert spaces, with K = 1. The separate elementary assertion that K = 1 is optimal is not formalized. The application to Olevskii's extension problem in Theorem 1.2 is also outside the formalization. The sorry count excludes the single intentional placeholder in Showcase.lean.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "GrothendieckBessel.main", "file": "Showcase_WithProofs.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "No known mathematical divergence for the formalized existence assertion. As Appendix 3 explains, the Lean conclusion uses ambient representatives f_j : R -> C that are almost everywhere strongly measurable for Lebesgue measure restricted to [0,1] and pointwise bounded there. Their values outside the interval are irrelevant. The formalization omits the separate optimality assertion in Theorem 1.1 and all of Theorem 1.2.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "Appendix 3, authored as part of the source paper, gives a field-by-field account of the public Lean definitions and theorem statement and reports complete machine-checked proofs in Showcase_WithProofs.lean. The current Comparator configuration has also been checked with Lean and NanoDa. The paper does not state that the internal Lean proof was reviewed line by line by humans, so no such claim is made here. The responsible maintainer should confirm this metadata before submission." }, "canonical": { "repo": "lukasliehr/grothendieckbessel", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-26-000002", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-26-000002", "commit": "f1ad1c7680bce2b1ab4218a02bddd694fb57062a", "trust": "high", "theorems": 1, "date": "2026-08-26" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "macgregor-sudo/albert-frame-triple-invariants/formalization.yaml", "repo": "macgregor-sudo/albert-frame-triple-invariants", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/macgregor-sudo/albert-frame-triple-invariants/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Rational invariants of three frames in the split Albert algebra: a Lean formalization", "description": "Let J be the split Albert algebra over a field k of characteristic zero, the 27-dimensional Jordan algebra of 3 by 3 Hermitian matrices over the split octonions, with trace tr. A frame is an ordered triple of trace-one idempotents summing to 1. Three frames E, F, K carry 54 natural invariants under the group G of all unital k-linear bijections of J preserving its Jordan product: the pair traces tr(eᵢ∘fⱼ), tr(fⱼ∘kₗ), tr(kₗ∘eᵢ) and the triple traces tr(eᵢ∘(fⱼ∘kₗ)). This Lean 4 development proves that these traces do not generate all rational invariants of three frames, and that adjoining one add", "authors": [ "Robert R. MacGregor" ], "maintainers": [ "Robert R. MacGregor" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Unpublished notes on frame-triple invariants", "id": "", "authors": [ "Robert R. MacGregor" ], "type": "unpublished notes", "relationship": "adapts", "endorsement": "participated" }, { "title": "Unpublished notes on comparison algebras", "id": "", "authors": [ "Robert R. MacGregor" ], "type": "unpublished notes", "relationship": "adapts", "endorsement": "participated" }, { "title": "Unpublished notes on rational coordinates", "id": "", "authors": [ "Robert R. MacGregor" ], "type": "unpublished notes", "relationship": "adapts", "endorsement": "participated" }, { "title": "Octonions, Jordan Algebras and Exceptional Groups", "id": "doi:10.1007/978-3-662-12622-6", "authors": [ "Tonny A. Springer", "Ferdinand D. Veldkamp" ], "type": "book", "relationship": "background", "endorsement": "" } ], "related": [ { "id": "https://github.com/TauCetiProject/TauCeti/tree/82f8b094231e7c40e3a91fb586287b6282097279", "relationship": "builds-on", "note": "The pinned split Albert algebra. The selected model theorem identifies the repository's 27 coordinates with it, preserving unit, product and trace." } ], "classification": { "arxiv": [ "math.RA" ], "msc2020": [ "17C40", "17C30" ] }, "automation": { "methods": [ { "method": "agent", "models": [], "framework": "" } ], "strongest": "agent", "models": [], "spend": "", "notes": "The maintainer directed the work and has not independently verified the mathematics. Arguments and reviews produced by AI systems are not human expert review." }, "scope": "Five selected statements: the reduced coordinate ring of three labelled frames is a domain; the coordinates are the split Albert algebra; the field of rational invariants under the full automorphism group is a degree-two extension of the field of the 54 pair and triple traces, generated by an explicit trace-cube; two explicit rational frame triples have equal traces and different trace-cubes; and they are inequivalent under every automorphism combined with every relabelling. Three comparison algebras (degree one) and the identity trace-cube = 3 × cubic norm are supporting results.", "sorry_count": 1, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "FrameInvariants.rawFrameRing_isDomain", "file": "FrameInvariants/Domain.lean", "description": "", "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "FrameInvariants.splitAlbertModel", "file": "FrameInvariants/ModelProof.lean", "description": "", "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "FrameInvariants.oneBit", "file": "FrameInvariants/GenericProof.lean", "description": "", "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "FrameInvariants.rationalWitness", "file": "FrameInvariants/FiniteProofs.lean", "description": "", "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "FrameInvariants.rationalWitness_not_equivalent_under_relabelling", "file": "FrameInvariants/FiniteProofs.lean", "description": "", "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The theorem is about the split Albert algebra over a field of characteristic zero, its full automorphism group, and the field of rational functions of three labelled frames; it identifies subfields of that function field. It does not count rational orbits in every fibre of the trace map, treat non-split real forms, present the polynomial invariant ring, or assert that the 54 traces are algebraically independent. The coordinate ring uses the radical of the frame ideal; radicality of the ideal itself is not asserted. The finite relabelling result concerns two explicit configurations, not a generic unlabelled theorem, and its proof does not assume the trace-cube is invariant under relabelling. The comparison results use the full automorphism group of each comparison algebra. No priority, human review or physical interpretation is claimed.", "alignment": false, "original": false, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [], "notes": "The recorded reviews were performed by AI systems; no independent human mathematical review is claimed. Build, axiom-audit and independent-checker logs identify the exact source commit checked; this file does not establish their outcomes or archive publication." }, "canonical": { "repo": "macgregor-sudo/albert-frame-triple-invariants", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-13-000001", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-13-000001", "commit": "dd06ef5d423a0e4a5d814d2c221e999b98887ee1", "trust": "qualified", "theorems": 5, "date": "2026-09-13" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "mario-ullrich/Lean-Discretization/Palomar/KieferWolfowitz/formalization.yaml", "repo": "mario-ullrich/Lean-Discretization", "path": "Palomar/KieferWolfowitz/formalization.yaml", "directory": "Palomar/KieferWolfowitz", "origins": [ "search" ], "url": "https://github.com/mario-ullrich/Lean-Discretization/blob/HEAD/Palomar/KieferWolfowitz/formalization.yaml", "version": "v0.4", "missing": [], "name": "The theorem of Kiefer and Wolfowitz for sampling projections", "description": "On an n-dimensional space V of bounded functions on an arbitrary set D, and for every epsilon > 0, there is a finitely supported probability measure rho = sum_k w_k delta(x_k) for which the uniform norm on V is dominated by the L_2(rho) norm with the constant sqrt(n + epsilon): |f(y)|^2 <= (n + epsilon) * sum_k w_k |f(x_k)|^2 for every y in D and every f in V. For continuous functions on a compact domain the loss disappears and the constant is the sharp sqrt(n). In both cases at most 2n^2 + 1 points are needed. The measure is one whose Gram matrix has an almost maximal determinant; applying a ", "authors": [ "Mario Ullrich" ], "maintainers": [ "Mario Ullrich" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Sampling projections in the uniform norm", "id": "arXiv:2401.02220", "authors": [ "David Krieg", "Kateryna Pozharska", "Mario Ullrich", "Tino Ullrich" ], "type": "article", "relationship": "formalizes", "endorsement": "participated" }, { "title": "The equivalence of two extremum problems", "id": "", "authors": [ "Jack Kiefer", "Jacob Wolfowitz" ], "type": "article", "relationship": "background", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.NA", "math.ST" ], "msc2020": [ "62K05", "41A17", "15A15" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Claude Opus 5 (Anthropic)", "Claude Fable 5.1 (Anthropic)" ], "framework": "Claude Code" } ], "strongest": "agent", "models": [ "Claude Fable 5.1 (Anthropic)", "Claude Opus 5 (Anthropic)" ], "spend": "subscription-based", "notes": "Most of the Lean proofs were produced with AI assistance. Throughout, the author designed the structure of the development and checked and revised the statements of all definitions and results. The proofs themselves are guaranteed by the Lean kernel." }, "scope": "Formalized and advertised: Discretization.KieferWolfowitz.exists_design_kieferWolfowitz_card_le, the bound with the constant n + epsilon on an arbitrary set, and Discretization.KieferWolfowitz.exists_design_kieferWolfowitz_of_compact_card_le, the sharp constant n for continuous functions on a compact domain. Both give a design of at most 2n^2 + 1 points whose Gram matrix is positive definite. Proved in the repository but not advertised here: the same two theorems without the bound on the number of points (exists_design_kieferWolfowitz, exists_design_kieferWolfowitz_of_compact); the statement f", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "Linear independence of the functions is stated as the separating condition \"for every coefficient vector c, if vanishes for every y then c = 0\", which the development identifies with LinearIndependent over the complex numbers (linearIndependent_iff_forall_star). Boundedness is stated as one constant C bounding every value |a_i(y)|. The design is given as points x_1, ..., x_N with weights w_1, ..., w_N rather than as a measure. The measure form is proved in the repository as exists_probabilityMeasure_kieferWolfowitz, with the identity relating its Gram matrix to the Gram matrix of the points and weights. On a general domain the proof does not use a maximiser. Instead of maximising the determinant over the compact convex hull of the rank-one matrices, it takes a near-maximiser of the supremum of the determinants; the perturbation is computed exactly by the matrix determinant lemma, and the first-order condition is replaced by Bernoulli's inequality with the explicit weight alpha = (t-n)/(2n(t-1)). This avoids both a Jacobi formula and the compactness of a convex hull, neither of which Mathlib has. On a compact domain the maximum is attained and the same computation gives th", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "none" ], "notes": "The statements of all definitions and results were checked and revised by the author; the proof scripts were largely produced with AI assistance and rest on the Lean kernel." }, "canonical": { "repo": "mario-ullrich/lean-discretization", "directory": "Palomar/KieferWolfowitz" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "mario-ullrich/Lean-Discretization/Palomar/Sparsification/formalization.yaml", "repo": "mario-ullrich/Lean-Discretization", "path": "Palomar/Sparsification/formalization.yaml", "directory": "Palomar/Sparsification", "origins": [ "search" ], "url": "https://github.com/mario-ullrich/Lean-Discretization/blob/HEAD/Palomar/Sparsification/formalization.yaml", "version": "v0.4", "missing": [], "name": "A generalized sparsification theorem of Batson, Spielman and Srivastava type", "description": "Given two families of square-integrable functions on a measure space, a finite one a = (a_k) with m elements and a second one b, one can select n points and positive weights, for every n >= m, so that the first family stays a frame from below and the second stays a frame from above, with the constants (1 - sqrt((m-1)/n))^2 and (1 + sqrt((M-1)/n))^2 * Lambda. Here Lambda bounds the Gram matrix J of the second family and M = Tr J / Lambda is its effective dimension. What the number of points depends on is M, not the number of functions in the second family, which is what makes the theorem applic", "authors": [ "Mario Ullrich" ], "maintainers": [ "Mario Ullrich" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Constructive discretization and approximation in reproducing kernel Hilbert spaces", "id": "arXiv:2602.18719", "authors": [ "Abdellah Chkifa", "Matthieu Dolbeault", "David Krieg", "Mario Ullrich" ], "type": "preprint", "relationship": "formalizes", "endorsement": "participated" }, { "title": "Twice-Ramanujan sparsifiers", "id": "arXiv:0808.0163", "authors": [ "Joshua Batson", "Daniel A. Spielman", "Nikhil Srivastava" ], "type": "article", "relationship": "background", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.NA", "math.FA" ], "msc2020": [ "41A17", "42C15", "15A45", "46E22" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Claude Opus 5 (Anthropic)", "Claude Fable 5.1 (Anthropic)" ], "framework": "Claude Code" } ], "strongest": "agent", "models": [ "Claude Fable 5.1 (Anthropic)", "Claude Opus 5 (Anthropic)" ], "spend": "subscription-based", "notes": "Most of the Lean proofs were produced with AI assistance. Throughout, the author designed the structure of the development and checked and revised the statements of all definitions and results. The proofs themselves are guaranteed by the Lean kernel." }, "scope": "Formalized and advertised: Discretization.bss_generalized and Discretization.Infinite.bss_generalized, the frame bounds for a finite and for a countably infinite second family, together with the discretization inequalities Discretization.exists_discretization and Discretization.Infinite.exists_discretization. None of the four carries a side condition beyond n >= m. Proved in the repository but not advertised here: the potential argument under the two side conditions m >= 2 and M >= 1 + 1/n (bss_generalized_of_gram_eq_one), the three edge cases that remove them (bss_generalized_of_unique, bss_g", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The lower frame bound of the source is stated with the factor lambda_min(I) for the Gram matrix I of the first family; the formalization states the equivalent Loewner inequality (1 - sqrt((m-1)/n))^2 * I <= sum_i w_i a(x_i) a(x_i)^*, from which the eigenvalue form follows. Traces of the matrices involved are real, and the formalization takes real parts explicitly with RCLike.re wherever a real number is needed. For a countably infinite second family the source works with a family (b_k) indexed by a countable set. The formalization instead takes a single square-integrable map b into a Hilbert space H together with a given operator J tied to it by the Gram identity Re = integral ||^2 dmu, which is what the averaging step consumes and which avoids constructing J as a Bochner integral in the operator algebra. Since Mathlib has no trace-class theory, the trace is taken along a fixed Hilbert basis e of H (ContinuousLinearMap.traceAlong), and the finite-trace hypothesis is the summability of that series. Positive definiteness of the Gram matrix is replaced by injectivity of J, which is the correct hypothesis in infinite dimension because a positive operator of finite trac", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "none" ], "notes": "The statements of all definitions and results were checked and revised by the author; the proof scripts were largely produced with AI assistance and rest on the Lean kernel." }, "canonical": { "repo": "mario-ullrich/lean-discretization", "directory": "Palomar/Sparsification" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "mario-ullrich/Lean-SNumbers/PalomarChallenges/MaxDifference/formalization.yaml", "repo": "mario-ullrich/Lean-SNumbers", "path": "PalomarChallenges/MaxDifference/formalization.yaml", "directory": "PalomarChallenges/MaxDifference", "origins": [ "search" ], "url": "https://github.com/mario-ullrich/Lean-SNumbers/blob/HEAD/PalomarChallenges/MaxDifference/formalization.yaml", "version": "v0.4", "missing": [], "name": "The maximal difference theorem for s-numbers", "description": "A bound between all s-numbers (in the sense of Pietsch). The approximation numbers a_n are the largest s-number sequence and the Hilbert numbers h_n the smallest, so that h_n(S) <= s_n(S) <= a_n(S) holds for every s-number sequence (s_n) and every bounded linear operator S between Banach spaces. The formalized result gives a bound in the reverse direction: a_n(S) <= ((n+1)^(n+1)/n^n) * h_n(S) <= e * (n+1) * h_n(S), for every bounded linear operator between normed spaces over the reals or the complex numbers. Hence, the same factor bounds the gap between any two s-number sequences. The linear g", "authors": [ "Mario Ullrich" ], "maintainers": [ "Mario Ullrich" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "On bounds between all s-numbers and widths of convex sets", "id": "arXiv:2608.05024", "authors": [ "Mario Ullrich" ], "type": "preprint", "relationship": "formalizes", "endorsement": "participated" }, { "title": "Inequalities between s-numbers", "id": "arXiv:2405.05509", "authors": [ "Mario Ullrich" ], "type": "article", "relationship": "background", "endorsement": "" }, { "title": "s-numbers of operators in Banach spaces", "id": "", "authors": [ "Albrecht Pietsch" ], "type": "article", "relationship": "background", "endorsement": "" }, { "title": "Eigenvalues and s-numbers", "id": "", "authors": [ "Albrecht Pietsch" ], "type": "book", "relationship": "background", "endorsement": "" }, { "title": "Operator ideals", "id": "", "authors": [ "Albrecht Pietsch" ], "type": "book", "relationship": "background", "endorsement": "" }, { "title": "Some contributions to the theory of s-numbers", "id": "", "authors": [ "Bernd Carl", "Albrecht Pietsch" ], "type": "article", "relationship": "background", "endorsement": "" }, { "title": "Hilbert-Zahlen von Operatoren in Banachraeumen", "id": "", "authors": [ "Wolfgang Bauhardt" ], "type": "article", "relationship": "background", "endorsement": "" }, { "title": "Inequalities between n-diameters", "id": "", "authors": [ "Boris S. Mityagin", "Gennadi M. Henkin" ], "type": "article", "relationship": "background", "endorsement": "" }, { "title": "Long-standing open problems of Banach space theory: My personal top ten", "id": "", "authors": [ "Albrecht Pietsch" ], "type": "article", "relationship": "background", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.FA", "math.OA" ], "msc2020": [ "47B06", "47L20", "46B06", "41A46" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Claude Opus 4.6 to 5 (Anthropic)", "Claude Fable 5 (Anthropic)" ], "framework": "Claude Code" } ], "strongest": "agent", "models": [ "Claude Fable 5 (Anthropic)", "Claude Opus 4.6 to 5 (Anthropic)" ], "spend": "subscription-based", "notes": "The project started out written by hand; as it grew, AI assistance was used more and more, and by now most of the Lean proofs were produced that way. Throughout, the author designed the structure of the development and checked and revised the statements of all definitions and results. The proofs themselves are guaranteed by the Lean kernel." }, "scope": "Formalized and advertised: a_n(S) <= ((n+1)^(n+1)/n^n) * h_n(S) and the weakening a_n(S) <= e * (n+1) * h_n(S), for every bounded linear operator S between normed spaces over a field with RCLike (that is, the reals or the complex numbers), and every n in the natural numbers, with 0-based indexing. No completeness of the spaces is assumed. Proved in the repository but not advertised here: that a_n and h_n satisfy the Pietsch axioms; the general form a_n(S) <= e (n+1) s_n(S) for every s-number sequence s (SNumbers.approximationNumber_le_e_mul_sn); the resulting bound max(c_n, d_n) <= e (n+1) b_n", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The advertised statement is the Hilbert-number instance of the source paper's theorem, which is stated there for an arbitrary s-number sequence. The two are equivalent given that the Hilbert numbers are the smallest s-number sequence, and the general form is proved in the repository as SNumbers.approximationNumber_le_e_mul_sn; only the instance is advertised here, so that the Challenge module needs no formalization of the s-number axioms themselves.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "none" ], "notes": "The statements of all definitions and results were checked and revised by the author; the proof scripts were largely produced with AI assistance and rest on the Lean kernel." }, "canonical": { "repo": "mario-ullrich/lean-snumbers", "directory": "PalomarChallenges/MaxDifference" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "mathlib-initiative/formalization.yaml/formalization.yaml", "repo": "mathlib-initiative/formalization.yaml", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/mathlib-initiative/formalization.yaml/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "mathlib-initiative/formalization.yaml", "description": "", "authors": [], "maintainers": [], "license": "", "role": "", "substantive": "", "sources": [ { "title": "", "id": "", "authors": [], "type": "", "relationship": "", "endorsement": "" } ], "related": [], "classification": { "arxiv": [], "msc2020": [] }, "automation": { "methods": [ { "method": "", "models": [], "framework": "" } ], "strongest": "other", "models": [], "spend": "", "notes": "" }, "scope": "", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [], "nonstandard_axioms": [], "results": [ { "declaration": "", "file": "", "description": "", "axioms": [], "sorry_count": 0, "comparator": false, "literature": [ { "statement": "", "source": "" } ] } ], "comparator": false, "literature_dependencies": 1, "divergences": "", "alignment": true, "original": false, "review": { "status": "", "bucket": "not stated", "reviewers": [], "notes": "" }, "canonical": { "repo": "mathlib-initiative/formalization.yaml", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "mattppal/ehlich-wojtas/formalization.yaml", "repo": "mattppal/ehlich-wojtas", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/mattppal/ehlich-wojtas/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "The Ehlich–Wojtas determinant bound", "description": "A Lean 4 formalization of the Ehlich–Wojtas upper bound on the determinant of an n×n matrix with entries in {±1} when n ≡ 2 (mod 4): |det M| ≤ (2n−2)(n−2)^{n/2−1}. The argument follows Wojtas’s determinant-theoretic proof as presented in Browne, Egan, Hegarty and Ó Catháin, Electron. J. Combin. 28 (4) (2021), Theorem 19: the Gram matrix of M is positive definite, its off-diagonal entries are even, and a 4-cycle congruence partitions the index set into two classes on which Fischer’s inequality and Wojtas’s off-diagonal magnitude bound apply. On ℕ one has 0^0 = 1, so the n = 2 case is the classi", "authors": [ "Matt Palmer" ], "maintainers": [ "Matt Palmer" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "A Survey of the Hadamard Maximal Determinant Problem", "id": "arxiv:2104.06756", "authors": [ "Patrick Browne", "Ronan Egan", "Fintan Hegarty", "Padraig Ó Catháin" ], "type": "paper", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "On Hadamard’s inequality for the determinants of order non-divisible by 4", "id": "doi:10.4064/cm-12-1-73-83", "authors": [ "M. Wojtas" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Determinantenabschätzungen für binäre Matrizen", "id": "doi:10.1007/BF01111162", "authors": [ "H. Ehlich" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" } ], "related": [], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05B20", "15B34", "15A15" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Cursor Grok 4.6" ], "framework": "Cursor Cloud Agent" } ], "strongest": "agent", "models": [ "Cursor Grok 4.6" ], "spend": "not tracked", "notes": "The informal statement was checked against Browne et al. Theorem 19 during formalization. No separate human referee reviewed the Lean proof." }, "scope": "Only the Ehlich–Wojtas upper bound for {±1}-matrices of order n ≡ 2 (mod 4) is advertised and proved. Equality-case structure (Browne et al., Theorem 20), constructions attaining the bound, Hadamard’s bound in other congruence classes, and Ehlich’s n ≡ 3 (mod 4) analysis are out of scope. Intermediate lemmas (Hadamard’s inequality for real positive definite matrices, Fischer’s inequality for a 2×2 PD block matrix, and Wojtas’s off-diagonal magnitude bound) are proved in the library because they are absent from mathlib v4.32.0; they are not advertised Comparator declarations.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "EhlichWojtas.ehlich_wojtas_bound", "file": "Challenge.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [ { "statement": "Browne et al. Theorem 19, the Ehlich–Wojtas bound", "source": "A Survey of the Hadamard Maximal Determinant Problem" } ] } ], "comparator": true, "literature_dependencies": 1, "divergences": "Browne et al. write det(M) ≤ (2n−2)(n−2)^{(n−2)/2}. The formal statement uses M.det.natAbs, i.e. |det M|, which is the intended meaning (determinants of {±1}-matrices may be negative). The exponent n/2 − 1 equals (n−2)/2 because n is even. The single-class case is reduced to the same numerical bound by an explicit Nat inequality rather than quoting the survey’s intermediate (3n−2)(n−2)^{n−1} form as a named lemma. Fischer’s and Hadamard’s inequalities are proved for real positive definite matrices instead of being imported from mathlib.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "The advertised Challenge statement was checked against Browne et al. Theorem 19. The proof development compiles with lake build and contains no sorry. Local Comparator and NanoDa accepted the Solution. No independent human referee reviewed the proof." }, "canonical": { "repo": "mattppal/ehlich-wojtas", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "mattppal/hall-first-multiplier/formalization.yaml", "repo": "mattppal/hall-first-multiplier", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/mattppal/hall-first-multiplier/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Hall's First Multiplier Theorem", "description": "A Lean 4 formalization of Hall's First Multiplier Theorem for difference sets (Hall–Ryser 1951, Theorem 3.1). If D is a (v,k,λ) difference set in a finite abelian group of order v and p is a prime dividing n = k−λ with p not dividing v and p > λ, then p is a multiplier of D. The cyclic Hall–Ryser statement is recorded as a corollary. The open multiplier conjecture is not claimed.", "authors": [ "Matt Palmer" ], "maintainers": [ "Matt Palmer" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Cyclic Incidence Matrices", "id": "10.4153/CJM-1951-050-8", "authors": [ "Marshall Hall Jr.", "H. J. Ryser" ], "type": "paper", "relationship": "independently-proves", "endorsement": "n/a" }, { "title": "Design Theory", "id": "", "authors": [ "Thomas Beth", "Dieter Jungnickel", "Hanfried Lenz" ], "type": "book", "relationship": "adapts", "endorsement": "n/a" } ], "related": [ { "id": "https://github.com/leanprover-community/mathlib4/blob/v4.32.0/Mathlib/Combinatorics/Hall/Basic.lean", "relationship": "independent", "note": "mathlib's Hall modules formalize Hall's marriage theorem (systems of distinct representatives). That is a different theorem." }, { "id": "https://arxiv.org/abs/2605.03274", "relationship": "independent", "note": "Singer/Sidon Lean 4 work (also d0d1/singer-theorem-lean) constructs perfect difference sets. It does not prove Hall–Ryser Theorem 3.1." }, { "id": "https://github.com/ericwang2004/combinatorial-designs-formalization", "relationship": "independent", "note": "Lean 4 design-theory development (Bruck–Ryser–Chowla, Fisher, difference-set-to-BIBD construction). No First Multiplier Theorem." } ], "classification": { "arxiv": [ "math.CO", "math.GR" ], "msc2020": [ "05B10", "20D60" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Cursor Grok 4.6" ], "framework": "Cursor Cloud Agent" } ], "strongest": "agent", "models": [ "Cursor Grok 4.6" ], "spend": "subscription-based", "notes": "Agent-authored formalization of a published 1951 theorem. Challenge.lean retains deliberate statement-surface sorry; proof-status counts exclude those. Machine checks are lake build, Palomar Template validate-formalization.rb, and leanprover/comparator with NanoDa." }, "scope": "Formalized: the abelian First Multiplier Theorem with hypotheses p | (k−λ), p ∤ v, and p > λ, plus the cyclic Hall–Ryser corollary on ZMod v. Not formalized: the open multiplier conjecture, the second multiplier theorem, and existence constructions of difference sets.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "Hall–Ryser state the theorem for cyclic groups. The formalized main theorem is the standard abelian strengthening with the same hypotheses and the same elementary proof; the cyclic statement is the corollary hall_ryser_first_multiplier. The structure IsDifferenceSet includes the classical inequality λ ≤ k, which is implicit in writing n = k−λ as a natural number. Challenge.lean owns the statement definitions so its import closure is Mathlib-only, as Palomar requires. The proof library repeats the same definitions because Solution cannot import Challenge (the Challenge theorems are sorry and occupy the compared names).", "alignment": true, "original": true, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "none" ], "notes": "Statement checked against Hall–Ryser 1951 Theorem 3.1 and the usual modern abelian wording. Machine checks: lake build, ruby scripts/validate-formalization.rb, and leanprover/comparator on comparator.json (kernel + NanoDa). Novelty reconfirmed 2026-09-07 (GitHub, mathlib, ericwang, arXiv). See VERIFICATION.md." }, "canonical": { "repo": "mattppal/hall-first-multiplier", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "mattppal/hoffman-ratio/formalization.yaml", "repo": "mattppal/hoffman-ratio", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/mattppal/hoffman-ratio/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Hoffman ratio bound", "description": "A Lean 4 formalization of Hoffman's ratio bound: for a nonempty simple k-regular graph G of order n with least adjacency eigenvalue λ_min ≠ k, the independence number satisfies α(G) ≤ n (−λ_min) / (k − λ_min).", "authors": [ "Matt Palmer" ], "maintainers": [ "Matt Palmer" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "Hoffman's ratio bound", "id": "https://doi.org/10.1016/j.laa.2021.02.010", "authors": [ "Willem H. Haemers" ], "type": "paper", "relationship": "independently-proves", "endorsement": "not-contacted" } ], "related": [], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05C50", "05C69" ] }, "automation": { "methods": [ { "method": "autonomous", "models": [ "Cursor Grok 4.6" ], "framework": "Cursor Cloud Agent" } ], "strongest": "autonomous", "models": [ "Cursor Grok 4.6" ], "spend": "not tracked", "notes": "The Lean development was produced autonomously from the published statement and Mathlib's spectral and graph APIs. A later autonomous pass ran lake, axiom print, Challenge isolation, and Comparator." }, "scope": "Proves Hoffman's ratio bound for finite nonempty simple k-regular graphs in Mathlib's SimpleGraph, with λ_min the infimum of the real Hermitian adjacency eigenvalues and α(G) = indepNum. Does not treat weighted, directed, or infinite graphs, and does not formalize tightness examples or the Delsarte–Hoffman bound for association schemes.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "HoffmanRatio.hoffman_ratio_bound", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The source writes the bound as a ratio. The formal statement adds the explicit hypothesis k ≠ λ_min so the denominator is nonzero, and requires a nonempty finite vertex type so n is a positive cardinal. The least eigenvalue is defined as an infimum over the Hermitian eigenvalue function rather than as λ_n in a separately sorted list.", "alignment": true, "original": true, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [], "notes": "Statement surface audited against Haemers display (1) and the LAA 617 (2021) bibliographic record. Challenge.lean imports Mathlib only. scripts/verify-local.sh succeeds. Comparator accepts the Solution under NanoDa and the Lean default kernel. Challenge.lean keeps a deliberate sorry; Solution.lean proves the matching declaration." }, "canonical": { "repo": "mattppal/hoffman-ratio", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "mattppal/P3Removal/formalization.yaml", "repo": "mattppal/P3Removal", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/mattppal/P3Removal/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "P3-removal for square energies", "description": "Formalizes Zhang, Extremal values for the square energies of graphs, arXiv:2409.15504v2, Theorem 1.10. An induced three-vertex path forces a vertex whose deletion drops the negative square energy by more than 1, and likewise for the positive square energy. The development follows the paper's Lemma 3.1 variational bounds and the 3×3 estimate in Lemma 4.1. Superadditivity (Theorem 1.6) is not formalized here.", "authors": [ "Matt Palmer" ], "maintainers": [ "Matt Palmer" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Extremal values for the square energies of graphs", "id": "arXiv:2409.15504v2", "authors": [ "Shengtong Zhang" ], "type": "article", "relationship": "formalizes", "endorsement": "not-contacted" } ], "related": [ { "id": "https://github.com/ShengtongZhang-alt/Sq", "relationship": "independent", "note": "Formalizes the connected-graph lower bound min(s⁺, s⁻) ≥ n−1 from the same paper. It does not contain Theorem 1.10." }, { "id": "https://github.com/ShengtongZhang-alt/SqOmega", "relationship": "independent", "note": "Formalizes the clique bound √s±(G) ≤ (1 − 1/ω(G)) n from the same paper. It does not contain Theorem 1.10." } ], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05C50", "15A42" ] }, "automation": { "methods": [ { "method": "autonomous", "models": [ "Cursor Grok 4.6" ], "framework": "Cursor Cloud Agent" } ], "strongest": "autonomous", "models": [ "Cursor Grok 4.6" ], "spend": "not tracked", "notes": "Autonomous formalization from the paper statement and Palomar template. The 3×3 numerical check is an algebraic identity, not a calculator call." }, "scope": "Proved: Theorem 1.10 for both signs, Lemma 3.1, and Lemma 4.1, for finite simple undirected graphs. Not formalized: Theorem 1.6 (superadditivity), Corollaries 1.11 and 1.12, and the 1 + 1/16 quantitative refinement.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The paper inspects 16(1−t)⁴ > 6(1−4t²)(1−2t²) on [0, 1/2] numerically. This formalization expands the difference as (1−2t)²(10−24t−8t²)+4t² and estimates signs. The A+M half of Lemma 4.1 is reduced to A−M by conjugating the middle vertex by −1. An induced P₃ is packaged as a structure with three named vertices rather than as an abstract graph isomorphism. Vertex deletion uses the subtype {x // x ≠ u}. None of these change the theorem.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "none" ], "notes": "Statement surface matches the paper quote in Challenge.lean. Comparator declarations are the two signed forms and their conjunction. No independent mathematical review has been performed." }, "canonical": { "repo": "mattppal/p3removal", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "mattppal/paley-conference/formalization.yaml", "repo": "mattppal/paley-conference", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/mattppal/paley-conference/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Paley conference matrix", "description": "Formalizes the Paley construction of a symmetric conference matrix of order q+1 for every finite field of cardinality q ≡ 1 (mod 4). The bordered Jacobsthal matrix C satisfies the standard conference-matrix normalization C Cᵀ = q I, matching Goethals–Seidel, Canad. J. Math. 19 (1967), Theorem 2.1.", "authors": [ "Matt Palmer" ], "maintainers": [ "Matt Palmer" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Orthogonal matrices with zero diagonal", "id": "https://doi.org/10.4153/CJM-1967-091-8", "authors": [ "J. M. Goethals", "J. J. Seidel" ], "type": "paper", "relationship": "independently-proves", "endorsement": "not-contacted" }, { "title": "On orthogonal matrices", "id": "https://doi.org/10.1002/sapm1933121311", "authors": [ "R. E. A. C. Paley" ], "type": "paper", "relationship": "background", "endorsement": "n/a" } ], "related": [ { "id": "https://github.com/l534zhan/my_project", "relationship": "other", "note": "Lean 3 development of Paley constructions of Hadamard matrices, including a Jacobsthal matrix and Paley type I (q ≡ 3 mod 4) and type II (order 2(q+1)) Hadamard matrices. It does not state or prove the conference-matrix identity C Cᵀ = q I, is not Lean 4, and is not Palomar-packaged." } ], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05B20", "15B34", "11T24" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Cursor Grok 4.6" ], "framework": "Cursor Cloud Agent" } ], "strongest": "agent", "models": [ "Cursor Grok 4.6" ], "spend": "not tracked", "notes": "Human-authored task statement and cited sources. Agent searched GitHub, Mathlib, Palomar public pages, and X, then wrote the Lean development and Palomar packaging." }, "scope": "Proved for every finite field F with #F ≡ 1 (mod 4): the bordered Jacobsthal matrix C is a symmetric conference matrix of order #F+1 satisfying C Cᵀ = (#F) • I. Existence of conference matrices at non-Paley orders, including the open order 66, is out of scope. The Goethals–Seidel χ(det) form on PG(1, q) is not constructed separately; the bordered affine form is the representative proved.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "paleyConferenceMatrix_mul_transpose", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "paleyConferenceMatrix_isConference", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "paleyConferenceMatrix_isSymm", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "Goethals–Seidel Theorem 2.1 attaches Paley matrices to PG(1, q) via χ(det) and works up to signed permutation equivalence. Their proof of C Cᵀ = q I evaluates the identity on one representative: the Paley matrix of the vectors x and y + αx (α ∈ GF(q)), which is the bordered Jacobsthal matrix C = [0, 1ᵀ; 1, Q] with Q_{a,b} = χ(a-b), using Jacobsthal's formula. This formalization proves that representative (and the conference-matrix axioms) rather than the full χ(det) family. Finite fields are written as Field + Fintype rather than as an explicit prime power, which is equivalent. The skew case q ≡ 3 (mod 4) is out of scope.", "alignment": true, "original": true, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Matt Palmer" ], "notes": "Statement surface was written to match Goethals–Seidel Theorem 2.1 (zero diagonal, ±1 off-diagonal, C Cᵀ = q I, symmetry when q ≡ 1 mod 4). Proofs were checked by lake build and by pinned Comparator on the Lean kernel. No independent mathematical referee." }, "canonical": { "repo": "mattppal/paley-conference", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "mattppal/sqomega-equality/formalization.yaml", "repo": "mattppal/sqomega-equality", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/mattppal/sqomega-equality/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Equality case of the positive square-energy Turán bound", "description": "Classification of equality in the positive square-energy strengthening of Turán's theorem. For a finite simple graph G on n ≥ 1 vertices, √s⁺(G) = (1 − 1/ω(G)) n if and only if either G is edgeless (equivalently ω(G) = 1) or ω(G) ≥ 2 divides n and G is isomorphic to the complete regular ω(G)-partite graph K_{n/ω,…,n/ω}. The compared Lean theorem is SqOmegaEquality.eq_iff_complete_regular_multipartite. The underlying inequality √s⁺(G) ≤ (1 − 1/ω(G)) n is not re-proved; it is imported from the SqOmega formalization of Liu–Tang–Zhang.", "authors": [ "Matt Palmer" ], "maintainers": [ "Matt Palmer" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "The Equality Case in the Positive Square-Energy Strengthening of Turán's Theorem", "id": "https://arxiv.org/abs/2608.24861", "authors": [ "Abhay Jayarajan", "M. Rajesh Kannan", "Shivaramakrishna Pragada", "Rahul Roy" ], "type": "paper", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "A positive square-energy strengthening of Turán's theorem", "id": "https://arxiv.org/abs/2607.18044", "authors": [ "Yinchen Liu", "Quanyu Tang", "Shengtong Zhang" ], "type": "paper", "relationship": "background", "endorsement": "n/a" } ], "related": [ { "id": "https://github.com/ShengtongZhang-alt/SqOmega/tree/e988319dceceb94ae956890a3547af0062465e3e", "relationship": "builds-on", "note": "SqOmega formalizes the Liu–Tang–Zhang inequality and the Caro–Wei / DNN infrastructure. This project adds the missing equality classification and does not re-prove sqrt_positiveSquareEnergy_le_cliqueNum." } ], "classification": { "arxiv": [ "math.CO", "math.SP" ], "msc2020": [ "05C50", "05C35", "15A42" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Cursor Grok 4.6" ], "framework": "Cursor Cloud Agent" } ], "strongest": "agent", "models": [ "Cursor Grok 4.6" ], "spend": "not tracked", "notes": "Agent-authored Lean development on top of SqOmega and Mathlib, under the direction of Matt Palmer. AI contributions are recorded here and not in project.authors / project.responsible_maintainers." }, "scope": "Formalizes Theorem 2 of arXiv:2608.24861 in full: the compared theorem SqOmegaEquality.eq_iff_complete_regular_multipartite. The inequality of Theorem 1 / SqOmega is imported, not re-proved. Negative square energy is not classified.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "Vertex sets are a finite type V rather than {1,…,n}; n is Fintype.card V. The paper's K_{n/r,…,n/r} is Mathlib's completeEquipartiteGraph r (n/r) up to SimpleGraph.Iso. Edgeless graphs are G = ⊥, which is the paper's complement of K_n. Square energy uses algebraic-multiplicity indexing by V, as in SqOmega. Oriented (double-counted) edge masses from SqOmega are used internally; the compared statement is the paper's unoriented equality. The Lean necessity argument follows the paper's complete r-partite conclusion but obtains equal part sizes from a unique positive eigenvalue plus Cauchy–Schwarz / Rayleigh, not from the paper's secular equation and Jensen step. Sufficiency uses that a positive regular degree is a positive adjacency eigenvalue, so s⁺ meets the imported SqOmega upper bound.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "none" ], "notes": "Statement audited against arXiv:2608.24861 Theorem 2. Mechanical lake build of Solution, scripts/verify.sh axiom print, local Palomar shape checks, and pinned Comparator (NanoDa and Lean kernels both accepted; \"Your solution is okay!\"). No independent human referee." }, "canonical": { "repo": "mattppal/sqomega-equality", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "mattppal/srg-spectrum/formalization.yaml", "repo": "mattppal/srg-spectrum", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/mattppal/srg-spectrum/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Spectrum of a strongly regular graph", "description": "Given a finite simple graph satisfying Mathlib's IsSRGWith n k ℓ μ, the restricted eigenvalues of the real adjacency matrix are the two roots r, s of X² - (ℓ - μ)X - (k - μ). When the graph is nonempty, incomplete, and has μ ≠ 0, the geometric multiplicity of r equals the Haemers formula f and the geometric multiplicity of s equals g.", "authors": [ "Matt Palmer" ], "maintainers": [ "Matt Palmer" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Matrix techniques for strongly regular graphs and related geometries", "id": "http://cage.ugent.be/~fdc/intensivecourse2/haemers2.pdf", "authors": [ "Willem H. Haemers" ], "type": "other", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "Algebraic Graph Theory", "id": "https://doi.org/10.1007/978-1-4613-0163-9", "authors": [ "Chris Godsil", "Gordon Royle" ], "type": "book", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Spectra of Graphs", "id": "https://doi.org/10.1007/978-1-4614-1939-6", "authors": [ "Andries E. Brouwer", "Willem H. Haemers" ], "type": "book", "relationship": "background", "endorsement": "not-contacted" } ], "related": [ { "id": "https://github.com/leanprover-community/mathlib4/blob/v4.33.1/Mathlib/Combinatorics/SimpleGraph/StronglyRegular.lean", "relationship": "builds-on", "note": "Mathlib supplies IsSRGWith, the parameter equation, A² = kI + ℓA + μC, and (on later master) diameter 2 when μ ≠ 0. It does not name r, s, f, g or prove geometric multiplicities." } ], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05C50", "05E30" ] }, "automation": { "methods": [ { "method": "autonomous", "models": [ "Cursor Grok 4.6" ], "framework": "Cursor Cloud Agent" } ], "strongest": "autonomous", "models": [ "Cursor Grok 4.6" ], "spend": "not tracked", "notes": "The algebraic dictionary, Jacobi identity, restricted-spectrum dichotomy, and geometric multiplicities are in SrgSpectrum. Challenge restates four theorems against mathlib only. Solution fills them." }, "scope": "Formalized: the Jacobi form of the SRG matrix equation; that a restricted eigenvalue is r or s; and that dim E_r = f and dim E_s = g for a nonempty, incomplete SRG with μ ≠ 0. Not formalized: the case μ = 0 (a disjoint union of cliques, where r = k and dim E_k = 1 + f); complete and empty graphs, where λ or μ is vacuous; algebraic multiplicity stated separately from geometric multiplicity (they coincide because A is symmetric).", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "srg_matrix_eq_jacobi", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "srg_restricted_spectrum", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "srg_geom_mult_r", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "srg_geom_mult_s", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The source writes algebraic multiplicities from the trace of A and of A². The formal theorems are geometric: finrank of the eigenspace of toLin' of the real adjacency matrix. Equality of algebraic and geometric multiplicity is implicit in the Hermitian diagonalization used in the proof. The μ ≠ 0 hypothesis is extra relative to the textbook display of f and g; it excludes the clique-union case where r coincides with the degree.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "No independent human referee. The check is scripts/verify.sh and scripts/verify-comparator.sh. A later trail review by Claude Fable flagged the local yaml gate, the lean4export tag gap, and authorship provenance. Those notes are in .audit/decisions.tsv." }, "canonical": { "repo": "mattppal/srg-spectrum", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "mattppal/superadd-square-energy/formalization.yaml", "repo": "mattppal/superadd-square-energy", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/mattppal/superadd-square-energy/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Super-additivity of graph square energies", "description": "Lean 4 / Mathlib formalization of Zhang, Extremal values for the square energies of graphs, arXiv:2409.15504, Theorem 1.6. For a finite simple graph G and any partition V = U ⊔ W of its vertex set, both the positive and negative square energies are super-additive: s⁺(G) ≥ s⁺(G[U]) + s⁺(G[W]) and s⁻(G) ≥ s⁻(G[U]) + s⁻(G[W]). Square energy is the sum of squares of the strictly positive (resp. strictly negative) adjacency eigenvalues. The proof follows the paper: Lemma 3.1 writes each energy as a minimum of a squared Frobenius norm over the PSD cone, and a principal-submatrix split of that norm y", "authors": [ "Matt Palmer" ], "maintainers": [ "Matt Palmer" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Extremal values for the square energies of graphs", "id": "arXiv:2409.15504", "authors": [ "Shengtong Zhang" ], "type": "paper", "relationship": "formalizes", "endorsement": "not-contacted" } ], "related": [ { "id": "https://github.com/ShengtongZhang-alt/Sq", "relationship": "independent", "note": "Formalizes min(s⁺, s⁻) ≥ n-1 for connected graphs (SquareEnergy.card_sub_one_le_min_squareEnergy). Different statement. Not a dependency." }, { "id": "https://github.com/ShengtongZhang-alt/SqOmega", "relationship": "independent", "note": "Formalizes √s±(G) ≤ (1 - 1/ω(G)) n (SquareEnergy.sqrt_squareEnergies_le_cliqueNum). SqOmega/Partition.lean is a Caro–Wei edge-partition bound, not Zhang Theorem 1.6. Different statement. Not a dependency." }, { "id": "https://github.com/ShengtongZhang-alt/BN", "relationship": "independent", "note": "Bollobás–Nikiforov formalization by overlapping square-energy authors. Different theorem. Palomar file names were copied from BN; the mathematics was not." } ], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05C50", "15A18", "15A42" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Cursor Grok 4.6" ], "framework": "Cursor" } ], "strongest": "agent", "models": [ "Cursor Grok 4.6" ], "spend": "not tracked", "notes": "Human-directed, agent-executed. Matt Palmer is the responsible maintainer. Correctness of the proofs rests on the Lean kernel and on Comparator/NanoDa. Challenge.lean is the audited statement surface." }, "scope": "Fully formalized: Zhang Theorem 1.6 (Superadd.squareEnergy_superadditive) and the supporting Lemma 3.1 inequalities and equality cases. Not formalized: the paper's corollaries that apply Theorem 1.6 (unicyclic bounds, domination decomposition, the k-part Corollary 1.7).", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "Superadd.squareEnergy_superadditive", "file": "Superadd/Main.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "None in the headline inequality. Presentation differences: (a) eigenvalues are Mathlib's Matrix.IsHermitian.eigenvalues of the real adjacency matrix, indexed by the vertex type with algebraic multiplicity; (b) the paper writes s⁺ = ∑_{λ≥0} λ² and s⁻ = ∑_{λ≤0} λ²; Lean uses strict sign filters, which agree because zeros square to 0; (c) the paper's ‖·‖ is the Frobenius norm, defined here as Superadd.frobeniusSq; (d) a partition is a pair of sets U, W with Disjoint U W and U ∪ W = Set.univ; (e) induced subgraphs are SimpleGraph.induce; (f) DecidableEq / DecidableRel / DecidablePred / Fintype hypotheses are typeclass artifacts of the adjacency matrix and subtype sums. Lemma 3.1 is stated as an inequality for every PSD matrix plus an equality case, which is what the min characterization uses.", "alignment": true, "original": false, "review": { "status": "author-verified", "bucket": "author-verified", "reviewers": [ "Matt Palmer" ], "notes": "Matt Palmer marked VERIFICATION PASS on 2026-09-07 after the lake / verify.sh / Comparator / NanoDa package. Challenge and Solution types match under Comparator. #print axioms for Superadd.squareEnergy_superadditive is propext, Classical.choice, Quot.sound. No independent third-party mathematical review of the formalization has been performed." }, "canonical": { "repo": "mattppal/superadd-square-energy", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "mattppal/van-lint-seidel/formalization.yaml", "repo": "mattppal/van-lint-seidel", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/mattppal/van-lint-seidel/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "van Lint–Seidel conference matrix obstruction", "description": "Lean 4 formalization of the classical necessary conditions for a conference matrix: order n > 1 is even; after row and column sign flips the matrix is symmetric when n ≡ 2 (mod 4) and skew-symmetric when n ≡ 0 (mod 4); and a symmetric conference matrix exists only if n − 1 is a sum of two integer squares. The development matches the van Lint–Seidel statement and does not address open existence questions such as a conference matrix of order 66.", "authors": [ "Matt Palmer", "Cursor Grok 4.6" ], "maintainers": [ "Matt Palmer" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Equilateral point sets in elliptic geometry", "id": "https://doi.org/10.1016/S1385-7258(66)50038-5", "authors": [ "J. H. van Lint", "J. J. Seidel" ], "type": "article", "relationship": "formalizes", "endorsement": "n/a" }, { "title": "Algebraic Combinatorics", "id": "", "authors": [ "Chris Godsil" ], "type": "book", "relationship": "background", "endorsement": "n/a" } ], "related": [], "classification": { "arxiv": [ "math.CO", "math.NT" ], "msc2020": [ "05B20", "15B34", "11E25" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "cursor-grok-4.6" ], "framework": "Cursor Cloud Agent" } ], "strongest": "agent", "models": [ "cursor-grok-4.6" ], "spend": "subscription-based", "notes": "Autonomous Lean development against lake build. Evenness and switching were proved first; the two-squares step uses Householder matching of a Hurwitz frame because mathlib has no abstract Witt cancellation." }, "scope": "Formalized: evenness of order n > 1; existence of a switching equivalent that is symmetric for n ≡ 2 (mod 4) and skew for n ≡ 0 (mod 4); the two-squares obstruction for a symmetric conference matrix; and the concrete exclusions of symmetric orders 22 and 34. Not formalized: existence constructions (Paley, etc.), the open existence problem C(66), and any sufficiency converse.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The source is geometric; the formalization uses the equivalent integer-matrix definition (zero diagonal, ±1 off-diagonal, C Cᵀ = (n−1)I) and proves the arithmetic obstruction over Q by Householder reflections and Lagrange's four-square theorem. The three necessary conditions match the classical statement. No existence claim is added.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "none" ], "notes": "lake build of VanLintSeidel, Challenge, and Solution succeeds. Challenge.lean holds the advertised statements with sorry; Solution.lean imports sorry-free proofs. Comparator verification against Palomar infrastructure is out of scope for this package." }, "canonical": { "repo": "mattppal/van-lint-seidel", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "mattppal/vinh-incidence/formalization.yaml", "repo": "mattppal/vinh-incidence", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/mattppal/vinh-incidence/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Vinh incidence inequality", "description": "A Lean 4 formalization of Vinh's 2011 point-line incidence inequality in the affine plane over a finite field. For a set of points P and a set of lines L in F_q², the number of incidences I(P,L) satisfies |I(P,L) - |P||L|/q| ≤ √(q |P| |L|). The compared declarations also include Vinh's one-sided Theorem 3 from the European Journal of Combinatorics paper.", "authors": [ "Matt Palmer" ], "maintainers": [ "Matt Palmer" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "The Szemerédi–Trotter type theorem and the sum-product estimate in finite fields", "id": "doi:10.1016/j.ejc.2011.06.008", "authors": [ "Le Anh Vinh" ], "type": "paper", "relationship": "adapts", "endorsement": "not-contacted" }, { "title": "Incidence bounds and applications over finite fields", "id": "arXiv:1601.00290", "authors": [ "Doowon Koh", "Sui-Chung Lee", "Thang Pham" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" } ], "related": [], "classification": { "arxiv": [ "math.CO", "math.NT" ], "msc2020": [ "05B25", "51E15", "11T99" ] }, "automation": { "methods": [ { "method": "autonomous", "models": [ "Cursor Grok 4.6" ], "framework": "Cursor Cloud Agent" } ], "strongest": "autonomous", "models": [ "Cursor Grok 4.6" ], "spend": "not tracked", "notes": "An autonomous agent wrote the Lean, Palomar metadata, and verification scripts. Palomar forbids listing an agent as project.authors. Matt Palmer is the human author and responsible maintainer." }, "scope": "Formalizes Vinh's incidence inequality for arbitrary finite fields, including characteristic 2. Lines are the slope lines y = mx + b and the verticals x = c, which enumerate AG(2,q). Does not formalize the sum-product corollary in Vinh's paper, Cilleruelo's Sidon method, or a general expander mixing lemma.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "Vinh's Theorem 3 is one-sided. The main compared theorem is the two-sided form |I - |P||L|/q| ≤ √(q|P||L|), which implies Theorem 3 and is the form quoted in later papers such as Koh–Lee–Pham, arXiv:1601.00290, Theorem 1.1. Vinh's writeup uses the spectrum of a projective incidence graph. This proof uses the affine-plane counting identity and Cauchy-Schwarz. The result is stated for every finite field.", "alignment": true, "original": false, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [ "Cursor Cloud Agent" ], "notes": "lake build, scripts/verify.sh, scripts/validate-formalization.rb, and Palomar Comparator are the mechanical checks. Challenge imports are Mathlib-only. No independent human mathematician has reviewed the submission. review.status is agent-reviewed, not peer-reviewed." }, "canonical": { "repo": "mattppal/vinh-incidence", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "mattppal/vosper-theorem/formalization.yaml", "repo": "mattppal/vosper-theorem", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/mattppal/vosper-theorem/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Vosper's theorem in ZMod p", "description": "Lean 4 formalization of Vosper's 1956 inverse theorem: the equality case of Cauchy-Davenport in the prime field Z/pZ. If A and B are subsets of cardinality at least 2 whose sumset attains the Cauchy-Davenport minimum and is at most p-2, then A and B are arithmetic progressions with the same nonzero common difference. The Cauchy-Davenport inequality itself is taken from Mathlib (`ZMod.cauchy_davenport`); this project proves only the structural equality case.", "authors": [ "Matt Palmer" ], "maintainers": [ "Matt Palmer" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "The Critical Pairs of Subsets of a Group of Prime Order", "id": "doi:10.1112/jlms/s1-31.2.200", "authors": [ "A. G. Vosper" ], "type": "article", "relationship": "independently-proves", "endorsement": "n/a" }, { "title": "Cauchy-Davenport theorem in Mathlib", "id": "https://github.com/leanprover-community/mathlib4", "authors": [ "The mathlib community" ], "type": "formalization", "relationship": "background", "endorsement": "n/a" } ], "related": [ { "id": "https://isa-afp.org/entries/Cauchy_Davenport_Vosper.html", "relationship": "independent", "note": "Isabelle/HOL AFP entry (Ramos, Hulak, de Queiroz, 2026) already formalizes Vosper. Different prover; this Lean development is independently written against Mathlib and does not import or translate Isabelle scripts." }, { "id": "https://github.com/sneed-and-feed/lean-theorems-3/blob/3eaeb3e7841a831a835e242daf349aed141d44f4/Formalization/CauchyDavenport/Vosper.lean", "relationship": "other", "note": "Lean file advertises Vosper but declares `axiom vosper_theorem`. Not a proof. Checked 2026-09-07." }, { "id": "https://github.com/Paul3435/open-math-lab/blob/a451d6f1c4b255d5a817f18a8b5d5725cae15146/proofs/lean-project/ProofLab/Vosper.lean", "relationship": "other", "note": "Lean AP-sumset lemmas only. The file states the namesake inverse is not proved. Checked 2026-09-07." }, { "id": "https://github.com/rjwalters/lean-genius/blob/f9c62750e76180f15c7bd6c5759be320d7feffdc/proofs/Proofs/Erdos476OQ05Problem.lean", "relationship": "other", "note": "Partial Dyson e-transform development. The large-set inductive case remains `axiom vosper_case1_exists_large`. Checked 2026-09-07." } ], "classification": { "arxiv": [ "math.NT", "math.CO" ], "msc2020": [ "11P70", "11B25", "11B30" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Cursor Grok 4.6" ], "framework": "Cursor Cloud Agent" } ], "strongest": "agent", "models": [ "Cursor Grok 4.6" ], "spend": "not tracked", "notes": "The statement, Davenport-transform lemmas, and inductive equality-case proof were written by an autonomous coding agent against Mathlib 4.32, then compiled with `lake build`. Challenge.lean retains a deliberate `sorry` for Comparator; the proof development has none." }, "scope": "Formalized: Vosper's classification of critical pairs A, B ⊆ ZMod p with #A ≥ 2, #B ≥ 2, #(A+B) = #A+#B-1, and #(A+B)+2 ≤ p (equivalently |A+B| ≤ p-2). Not formalized: the residual |A+B|=p-1 configurations; Cauchy-Davenport itself; Vosper over general abelian groups; the polynomial-method proof of the inequality used in the AFP entry.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "Vosper.vosper", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [ { "statement": "Cauchy-Davenport inequality in ZMod p", "source": "Cauchy-Davenport theorem in Mathlib" } ] } ], "comparator": true, "literature_dependencies": 1, "divergences": "The source hypothesis 2 ≤ |G \\\\ (A+B)| is written as #(A+B)+2 ≤ p, which is equivalent for G = ZMod p. Arithmetic progressions are encoded as the explicit image apSegment a d n rather than an abstract predicate in the final statement. The development uses Mathlib's pointwise Finset sumsets and ZMod.cauchy_davenport instead of a polynomial-method inequality. The |A+B|=p-1 residual case of some textbook presentations is omitted, matching the AFP prime-field statement's complement-size hypothesis.", "alignment": true, "original": true, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Matt Palmer" ], "notes": "End-to-end verification pass 2026-09-07: lake build, scripts/verify.sh (sorry audit, Challenge Mathlib-only imports, formalization.yaml v0.4 sentinel check, axiom print). Palomar Comparator 68a0641 + NanoDa 68d5ca9 accepted Vosper.vosper and Vosper.apSegment (\"Your solution is okay!\"). Statement compared against Vosper 1956 and the Isabelle AFP prime-field theorem. No independent external referee." }, "canonical": { "repo": "mattppal/vosper-theorem", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "mattppal/zhang-lemma-2-2/formalization.yaml", "repo": "mattppal/zhang-lemma-2-2", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/mattppal/zhang-lemma-2-2/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Zhang (2024) Lemma 2.2 in Lean", "description": "A Lean 4 / Mathlib formalization of Lemma 2.2 from Shengtong Zhang, On the first two eigenvalues of regular graphs, Linear Algebra Appl. 686 (2024), 102–110, arXiv:2309.08184. The lemma is a two-eigenvalue Motzkin–Straus reduction: if a finite simple graph G satisfies, for some r > 1, the edge-sum inequality (2) on the rank-two matrix X_ij = μ₁ v₁(i) v₁(j) + μ₂ v₂(i) v₂(j), or equivalently the quadratic-form inequality (3) for K_G = ((r−1)/r) J − A_G, then μ₁² + μ₂² ≤ ((r−1)/r) · 2m. Here μ₁, μ₂ are eigenvalues of the adjacency matrix with corresponding orthonormal eigenvectors v₁, v₂, and m i", "authors": [ "Matt Palmer" ], "maintainers": [ "Matt Palmer" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "On the first two eigenvalues of regular graphs", "id": "arXiv:2309.08184", "authors": [ "Shengtong Zhang" ], "type": "paper", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "Maxima for graphs and a new proof of a theorem of Turán", "id": "doi:10.4153/CJM-1965-053-6", "authors": [ "Theodore S. Motzkin", "Ernst G. Straus" ], "type": "paper", "relationship": "background", "endorsement": "n/a" }, { "title": "Cliques and the spectral radius", "id": "doi:10.1016/j.jctb.2006.12.002", "authors": [ "Béla Bollobás", "Vladimir Nikiforov" ], "type": "paper", "relationship": "background", "endorsement": "n/a" } ], "related": [ { "id": "https://github.com/ShengtongZhang-alt/BN", "relationship": "independent", "note": "Lean 4 formalization (Coutinho–Liu–Spier–Tang–Zhang; public 2026-09-07) of the Bollobás–Nikiforov conjecture for all graphs via a planar Gram / complete-positivity argument. Its Challenge names BN.lambda1_sq_add_lambda2_sq_le, BN.weighted, BN.matrix_theorem, BN.gram_le, and BN.chiVec3_eq_cliqueNum. Zhang 2024 is cited there as background for the regular case. Lemma 2.2 is not a named theorem in th" } ], "classification": { "arxiv": [ "math.CO", "math.SP" ], "msc2020": [ "05C50", "15A18", "05C35" ] }, "automation": { "methods": [ { "method": "autonomous", "models": [ "Cursor Grok 4.6" ], "framework": "Cursor Cloud Agent" } ], "strongest": "autonomous", "models": [ "Cursor Grok 4.6" ], "spend": "not tracked", "notes": "Agent-executed formalization. Matt Palmer is the responsible maintainer of this repository. Correctness of proofs rests on the Lean kernel; Challenge.lean is the auditable statement surface." }, "scope": "Formalized: Zhang Lemma 2.2 as Zhang2024.lemma_2_2, the equivalence of the paper’s forms (2) and (3) as Zhang2024.hypEdge_iff_hypKG, and the (3) form as Zhang2024.lemma_2_2_of_hypKG. The paper states the lemma for any graph and any r > 1, given two orthonormal eigenpairs; the Lean statement matches that. Not formalized: Theorem 1.2 (the regular BN bound μ₁²+μ₂² ≤ 2(ω−1)/ω m), the Motzkin–Straus argument that a regular graph of degree r satisfies (2) with that r, and the equality-case classification.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "Zhang2024.lemma_2_2", "file": "ZhangLemma22/Basic.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Zhang2024.hypEdge_iff_hypKG", "file": "ZhangLemma22/Basic.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Zhang2024.lemma_2_2_of_hypKG", "file": "ZhangLemma22/Basic.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "None known in the Lemma 2.2 implication. Presentation differences: (a) the paper writes ∑_{ij ∈ E} and Lean writes ∑_{i,j} A_ij · (−)² with the 0-1 adjacency matrix, which is the same sum; (b) the paper applies the lemma to the two largest eigenvalues of a regular graph, while the Lean statement takes any two orthonormal eigenpairs, which is what the printed lemma uses; (c) K_G is defined entrywise as ((r−1)/r) − A_ij rather than as a named all-ones matrix J. The regularity argument that produces hypothesis (2) from Motzkin–Straus is outside the lemma and is not formalized.", "alignment": true, "original": false, "review": { "status": "unchecked", "bucket": "unchecked", "reviewers": [], "notes": "No independent human review of the formalization has been performed. Proofs are machine-checked by Lean 4.32.0. #print axioms for each compared theorem reports exactly propext, Classical.choice and Quot.sound. The Challenge quote was checked against arXiv:2309.08184v3 Lemma 2.2. A 2026-09-07 verification pass ran lake build, the Palomar-pinned Comparator script, and a novelty search; see VERIFICATION.md. Palomar editorial review has not occurred." }, "canonical": { "repo": "mattppal/zhang-lemma-2-2", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "mattrobball/BridgelandStability/formalization.yaml", "repo": "mattrobball/BridgelandStability", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/mattrobball/BridgelandStability/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "BridgelandStability", "description": "AI-assisted Lean 4 / Mathlib formalization of the main results of Tom Bridgeland's \"Stability conditions on triangulated categories\" (Annals of Mathematics 166 (2007), 317-345). Covers Sections 2-7 of the paper, culminating in Theorem 1.2, that the central charge map is a local homeomorphism on each connected component of Stab(D), and Corollary 1.3, that connected components of Stab_N(D) are finite-dimensional complex manifolds. Both are proved in class-map generality, for an arbitrary surjective class map v : K₀(D) →+ Λ as in Bayer–Macrì–Stellari and Bayer–Lahoz–Macrì–Nuer–Perry–Stellari, the", "authors": [ "Matthew Robert Ballard" ], "maintainers": [ "Matthew Robert Ballard" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "Stability conditions on triangulated categories", "id": "arXiv:math/0212237v3", "authors": [ "Tom Bridgeland" ], "type": "paper", "relationship": "adapts", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.AG", "math.CT" ], "msc2020": [ "18G80", "14F08" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "claude-opus-4-6: formalization", "claude-opus-4-6: review", "gpt-5.4: formalization", "gpt-5.4: review", "claude-sonnet-4-6: translation", "chatgpt-5.2-pro: translation" ], "framework": "custom" } ], "strongest": "agent", "models": [ "chatgpt-5.2-pro: translation", "claude-opus-4-6: formalization", "claude-opus-4-6: review", "claude-sonnet-4-6: translation", "gpt-5.4: formalization", "gpt-5.4: review" ], "spend": "", "notes": "" }, "scope": "Formalization of the main results (Sections 2-7) of Bridgeland's \"Stability conditions on triangulated categories\" (Annals 2007). Theorem 1.2 (central charge is local homeomorphism on connected components) and Corollary 1.3 (connected components of numerical stability conditions are finite-dimensional complex manifolds), both generalized to arbitrary surjective class maps.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "CategoryTheory.Triangulated.NumericalStabilityCondition.existsComplexManifoldOnConnectedComponent", "file": "BridgelandStability/NumericalStabilityManifold.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "Theorem 1.2 and Corollary 1.3 are proved for arbitrary surjective class maps v: K_0(D) ->+ Lambda, generalizing the paper's identity (Theorem 1.2) and numerical quotient (Corollary 1.3) cases.", "alignment": false, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "" }, "canonical": { "repo": "mattrobball/bridgelandstability", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "MaxWellApexLab/BicausalOT-palomar/formalization.yaml", "repo": "MaxWellApexLab/BicausalOT-palomar", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/MaxWellApexLab/BicausalOT-palomar/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "BicausalOT", "description": "A Lean 4 formalization of the Bellman value representation for bicausal optimal transport, together with the descriptive-set-theoretic library it requires. The result registered by this Comparator configuration is the Kuratowski–Ryll-Nardzewski measurable selection theorem: for an arbitrary measurable space (alpha, A) and a Polish space Y, a multifunction Phi : alpha -> Set Y with nonempty closed values that is weakly measurable (the hit set {a | Phi a meets U} lies in A for every open U) admits an A-to-Borel measurable selection f with f a in Phi a for every a. The statement is measure-free a", "authors": [ "KT. Wu" ], "maintainers": [ "KT. Wu" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "A general theorem on selectors", "id": "Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 13 (1965), 397–403", "authors": [ "Kazimierz Kuratowski", "Czesław Ryll-Nardzewski" ], "type": "paper", "relationship": "formalizes", "endorsement": "n/a" }, { "title": "Classical Descriptive Set Theory", "id": "ISBN 978-0-387-94374-9 (Graduate Texts in Mathematics 156, Springer, 1995)", "authors": [ "Alexander S. Kechris" ], "type": "book", "relationship": "adapts", "endorsement": "not-contacted" }, { "title": "A Course on Borel Sets", "id": "ISBN 978-0-387-98412-4 (Graduate Texts in Mathematics 180, Springer, 1998)", "authors": [ "S. M. Srivastava" ], "type": "book", "relationship": "adapts", "endorsement": "not-contacted" }, { "title": "Stochastic Optimal Control: The Discrete Time Case", "id": "ISBN 978-1-886529-03-3 (Athena Scientific, 1996; originally Academic Press, 1978)", "authors": [ "Dimitri P. Bertsekas", "Steven E. Shreve" ], "type": "book", "relationship": "background", "endorsement": "not-contacted" } ], "related": [ { "id": "https://github.com/MaxWellApexLab/BicausalOT/tree/611e49d", "relationship": "other", "note": "Provenance rather than a separate formalization. This repository is a frozen snapshot of that repository at commit 611e49d, made for this submission; the same Lean sources are present in both. Development continues there and not here." }, { "id": "https://github.com/leanprover-community/mathlib4/blob/edc39bf7bcc706ba243ae824adaa60fff00416db/Mathlib/Probability/Decision/BayesEstimator.lean#L46", "relationship": "other", "note": "Not a prior formalization but the evidence of absence. Mathlib at the pinned revision edc39bf7bcc706ba243ae824adaa60fff00416db contains no measurable selection theorem and no API for measurable set-valued maps; the linked comment in Mathlib records this (\"Once Mathlib has measurable selection theorems, we will be able to prove `HasArgminEstimator` under ...\"). Searches of that revision for \"Ryll-N" } ], "classification": { "arxiv": [ "math.LO", "math.PR", "math.GN", "math.OC" ], "msc2020": [ "28B20", "54C65", "54H05", "03E15", "28A20", "68V20" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Anthropic Claude" ], "framework": "Claude Code" } ], "strongest": "agent", "models": [ "Anthropic Claude" ], "spend": "", "notes": "The whole repository is AI-generated Lean produced under human direction; this is stated in the project README as well. Responsibility for the correctness and for the claims made in this file rests with the named author and maintainer, not with the model or the tooling. The mechanical guarantees offered are exactly those the build and the axiom audit provide: the development compiles against the pinned Mathlib revision and the registered theorem depends on no axiom beyond `propext`, `Classical.choice` and `Quot.sound`." }, "scope": "This Comparator configuration registers exactly one theorem, the Kuratowski– Ryll-Nardzewski measurable selection theorem, stated in `Challenge.lean` and proved in `Solution.lean` by delegation to the library declaration `exists_measurable_selection` in `BicausalOT/DescriptiveSetTheory/MeasurableSelection.lean`. That file imports Mathlib and nothing else from this repository, so the proof of the registered result depends on no other part of the development. The wider repository (the bicausal optimal transport Bellman theory and the rest of the descriptive-set-theory library) is present at the ", "sorry_count": 5, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "exists_measurable_selection", "file": "BicausalOT/DescriptiveSetTheory/MeasurableSelection.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "Four points where the Lean statement differs from the cited textbook statements. Item (4) is a genuine weakening and is stated as such. (1) Polish codomain presented as a metric space. Kechris 12.13 and Srivastava 5.2.1 assume the codomain Y is Polish, i.e. separable and completely metrizable, without fixing a metric. The Lean statement instead assumes `[MetricSpace Y]`, `[TopologicalSpace.SeparableSpace Y]` and `[CompleteSpace Y]`, i.e. a compatible complete metric has been chosen. This is equivalent in strength: every Polish space admits such a metric, and both the hypothesis on the multifunction and the conclusion depend only on the topology of Y and on its Borel sigma-algebra, not on the metric. A `[PolishSpace Y]` wrapper is derivable from the registered statement in a few lines using Mathlib's `upgradeIsCompletelyMetrizable`; it is deliberately not part of this registration, because no such wrapper exists in the repository at the pinned commit. (2) Weak measurability quantified over all open sets. The Lean hypothesis `hmeas` ranges over every open `U : Set Y`, exactly as in the textbook statements. The proof only ever applies it to intersections of two open balls centred at p", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "KT. Wu" ], "notes": "No external or peer review has taken place. What was actually run, at the pinned commit: a full `lake build` of the repository against mathlib master edc39bf7bcc706ba243ae824adaa60fff00416db with toolchain leanprover/lean4:v4.33.0-rc1, reported as 8725 jobs with 0 errors and 0 warnings; a repository-wide scan for `sorry` and for user-declared `axiom`, both empty outside the deliberate `sorry` placeholders in the four Comparator challenge modules; and the kernel axiom audit `BicausalOT/AxiomsAudit.lean`, which runs `#print axioms` on 78 headline declarations during the build, every one of them reporting only `[propext, Classical.choice, Quot.sound]`. `AxiomAudit.lean` at the repository root repeats the check for four declarations, including `exists_measurable_selection`, as a standalone ext" }, "canonical": { "repo": "maxwellapexlab/bicausalot-palomar", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-31-000011", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-31-000011", "commit": "3fe500ff64a75d7ab71b474d0c656d5471f62992", "trust": "high", "theorems": 1, "date": "2026-08-31" }, { "id": "PALOMAR-2026-08-31-000010", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-31-000010", "commit": "66fbfd2fcf10f570d43946aafbd8731ef0dc127f", "trust": "high", "theorems": 1, "date": "2026-08-31" }, { "id": "PALOMAR-2026-08-31-000009", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-31-000009", "commit": "66fbfd2fcf10f570d43946aafbd8731ef0dc127f", "trust": "high", "theorems": 2, "date": "2026-08-31" }, { "id": "PALOMAR-2026-08-31-000007", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-31-000007", "commit": "06488da9a8379c66bc8d9065e6279d4d36f29d99", "trust": "high", "theorems": 1, "date": "2026-08-31" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "mayankpandey9973/squarefree_formal_compliant/formalization.yaml", "repo": "mayankpandey9973/squarefree_formal_compliant", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/mayankpandey9973/squarefree_formal_compliant/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Squarefree numbers in short intervals: explicit and formalized", "description": "A substantive Lean 4 formalization in analytic number theory of an explicit power-saving asymptotic for squarefree integers in short intervals. For 0 < epsilon <= 1/90935, X >= exp(10^27/epsilon^2), and H = X^(1/5 - 2/90935 + epsilon), it proves that the squarefree count in [X, X+H] differs from (6/pi^2)H by at most (10^450/epsilon) H X^(-epsilon/10^25). This error is smaller than the main term throughout the stated range, so the same theorem immediately implies that the interval contains a squarefree integer. Relative to the original result, the formalized statement makes the exponent and con", "authors": [ "Mayank Pandey" ], "maintainers": [ "Mayank Pandey" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Squarefree numbers in short intervals: explicit and formalized", "id": "arXiv:2608.06682v1", "authors": [ "Mayank Pandey" ], "type": "paper", "relationship": "formalizes", "endorsement": "participated" }, { "title": "Squarefree numbers in short intervals", "id": "arXiv:2401.13981v3", "authors": [ "Mayank Pandey" ], "type": "paper", "relationship": "adapts", "endorsement": "participated" }, { "title": "The quantitative behaviour of polynomial orbits on nilmanifolds", "id": "doi:10.4007/annals.2012.175.2.2", "authors": [ "Ben Green", "Terence Tao" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "The number of lattice points on a convex curve", "id": "doi:10.1016/0022-314X(74)90051-1", "authors": [ "H. P. F. Swinnerton-Dyer" ], "type": "paper", "relationship": "background", "endorsement": "" } ], "related": [ { "id": "https://github.com/mayankpandey9973/squarefree-explicit-formalized/tree/f0c7baf2d046503f49c8f220690ba1c91e42fe82", "relationship": "other", "note": "Immutable source snapshot from which the complete Lean development in this repository was copied before adding the Palomar packaging." } ], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11N25" ] }, "automation": { "methods": [ { "method": "manual", "models": [], "framework": "n/a" }, { "method": "agent", "models": [ "OpenAI GPT-5.4 Pro", "OpenAI GPT-5.5", "Anthropic Claude Opus 4.8" ], "framework": "Codex and Claude Code" } ], "strongest": "agent", "models": [ "Anthropic Claude Opus 4.8", "OpenAI GPT-5.4 Pro", "OpenAI GPT-5.5" ], "spend": "not tracked", "notes": "Most of the labor, including most Lean code, was carried out with AI assistance. Kernel acceptance is not a substitute for independent mathematical or code review." }, "scope": "The repository contains the full Lean proof development and advertises the explicit squarefree-counting asymptotic. The finite sum uses the squarefree indicator over the integers from ceil(X) to floor(X+H), and the declaration requires 0 < epsilon <= 1/90935. Positivity of the count gives existence at the same threshold as an immediate corollary, which is documented but not compared separately. The paper source and expository files are intentionally not included. The one sorry in Challenge.lean is the deliberate statement hole and is excluded below.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "SquarefreeSubmission.count_short_interval_eps", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The compared declaration matches the explicit asymptotic in arXiv:2608.06682v1. Lean spells out the real interval endpoints with ceil and floor and uses the squarefree indicator in place of mu(n)^2. Relative to arXiv:2401.13981v3, the result makes eta, thresholds, constants, and the power saving explicit. The companion note displays 2/92485 only in intentionally weakened intermediate statements; its headline theorem and the compared Lean declaration use 2/90935. Existence is not compared as a second declaration because the explicit error bound makes the count positive already at the counting theorem's threshold.", "alignment": true, "original": false, "review": { "status": "author-guided and mechanically audited; no independent review", "bucket": "other", "reviewers": [], "notes": "This standalone repository completed a full build under the pinned stable toolchain. Its headline declaration was audited to use only propext, Classical.choice, and Quot.sound, and passed trust-zero elaboration. Palomar's current pinned Comparator and toolchain-matched lean4export accepted the Challenge/Solution comparison; both Lean's kernel and NanoDa accepted the exported solution. The local Comparator run used its development no-sandbox Landrun shim, so sandbox enforcement remains part of Palomar's authoritative workflow. 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It also formalizes the analytic and entire-function lemmas used by that theorem.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "IncompleteGammaApproximant.theorem_A", "file": "BoundaryConstantObstruction/IncompleteGamma/PaperTheorem.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "None known for the submitted main theorem.", "alignment": true, "original": false, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [ "OpenAI GPT-5 (Codex)" ], "notes": "Independent agent lanes audited the statement, hypotheses, quantifiers, definitions, axiom report, and paper correspondence. No human peer review is claimed." }, "canonical": { "repo": "mbaccaro-dev/mathematical-proofs", "directory": "MathematicalProofs/BoundaryConstantObstruction/Lean" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-21-000003", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-21-000003", "commit": "4c8fd5edad3fd7eaf12812832d3caf1935ea35d5", "trust": "high", "theorems": 1, "date": "2026-08-21" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "mbaccaro-dev/mathematical-proofs/MathematicalProofs/MobiusResidualComplexity/Lean/formalization.yaml", "repo": "mbaccaro-dev/mathematical-proofs", "path": "MathematicalProofs/MobiusResidualComplexity/Lean/formalization.yaml", "directory": "MathematicalProofs/MobiusResidualComplexity/Lean", "origins": [ "palomar" ], "url": "https://github.com/mbaccaro-dev/mathematical-proofs/blob/HEAD/MathematicalProofs/MobiusResidualComplexity/Lean/formalization.yaml", "version": "v0.4", "missing": [], "name": "Prime-band residual-or-unmatched barriers", "description": "A Lean 4 proof of a residual-or-unmatched barrier for local Möbius cancellation. At every sufficiently large fixed-band scale, either the permitted edge length reaches the retained-prime product or every member of an explicit squarefree prime-product family is unmatched. Chebyshev estimates give the exact binomial and factorial population bounds. The product scale also converts residual rank to its logarithmic threshold with the exact limiting constant.", "authors": [ "Mayk Loide Baccaro" ], "maintainers": [ "Mayk Loide Baccaro" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "Logarithmic Residual-Complexity Barriers for Local Möbius Cancellation", "id": "10.5281/zenodo.22006119", "authors": [ "Mayk Loide Baccaro" ], "type": "paper", "relationship": "formalizes", "endorsement": "participated" } ], "related": [], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11N37", "11A25" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "OpenAI GPT-5 (Codex)" ], "framework": "Codex" } ], "strongest": "agent", "models": [ "OpenAI GPT-5 (Codex)" ], "spend": "subscription-based", "notes": "The proof uses the pinned Lean and Mathlib environment recorded in this directory. The full assistance disclosure is pinned in Manuscript/paper.tex under \"Assistance disclosure.\"" }, "scope": "The selected results prove the obstruction as an exact dichotomy. If a global matching has fewer unmatched vertices than the prime-product family, its permitted edge length must reach the retained-prime product. At every sufficiently large fixed-band scale, either that edge alternative holds or every prime-band product is individually unmatched, with explicit binomial and factorial lower bounds. Along the product scale, rank is asymptotic to delta times log X divided by log log X, and a normalized residual budget below r is eventually below q times the rank whenever r/delta is less than q.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "RiemannHypothesisProofFactory.MobiusResidualComplexity.chebyshev_globalMatching_residual_budget_barrier", "file": "RiemannHypothesisProofFactory/MobiusResidualComplexity/ResearchBarrierCertificate.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "RiemannHypothesisProofFactory.MobiusResidualComplexity.eventually_octuple_globalMatching_residual_or_unmatched_barrier", "file": "RiemannHypothesisProofFactory/MobiusResidualComplexity/ResearchBarrierCertificate.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "RiemannHypothesisProofFactory.MobiusResidualComplexity.threshold_logarithmic_scale_conversion", "file": "RiemannHypothesisProofFactory/MobiusResidualComplexity/ThresholdAsymptotics.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "RiemannHypothesisProofFactory.MobiusResidualComplexity.eventually_threshold_budget_separation", "file": "RiemannHypothesisProofFactory/MobiusResidualComplexity/ThresholdAsymptotics.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The prime-band supply, exact product population, pointwise global-matching obstruction, finite and all-scale residual-or-unmatched barriers, logarithmic scale limit, and residual-budget separation are formalized. The final sparse-binomial comparison with an arbitrary O(X^beta) unmatched set and the resulting full limsup contradiction remain manuscript-level mathematics.", "alignment": true, "original": false, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [ "OpenAI GPT-5 (Codex)" ], "notes": "Independent checks covered prime supply, squarefreeness, exact cardinality, matching ownership, residual and gap bounds, asymptotic scale conversion, quantifiers, and axiom reports. No human peer review is claimed." }, "canonical": { "repo": "mbaccaro-dev/mathematical-proofs", "directory": "MathematicalProofs/MobiusResidualComplexity/Lean" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-22-000005", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-22-000005", "commit": "11d59ae23fb0fe3d8591a00069bec47a868ac93f", "trust": "high", "theorems": 4, "date": "2026-08-22" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "mbaccaro-dev/mathematical-proofs/MathematicalProofs/PrimeLocalWeilObstruction/Lean/formalization.yaml", "repo": "mbaccaro-dev/mathematical-proofs", "path": "MathematicalProofs/PrimeLocalWeilObstruction/Lean/formalization.yaml", "directory": "MathematicalProofs/PrimeLocalWeilObstruction/Lean", "origins": [ "palomar" ], "url": "https://github.com/mbaccaro-dev/mathematical-proofs/blob/HEAD/MathematicalProofs/PrimeLocalWeilObstruction/Lean/formalization.yaml", "version": "v0.4", "missing": [], "name": "Prime-local Weil obstruction: formal kernel", "description": "Formalizes the prime-local obstruction by proving the sharp threshold for each local block, divergence of the associated von Mangoldt and prime-power budgets, and divergence of every scalar budget satisfying the sharp pointwise lower bound at every natural-number index.", "authors": [ "Mayk Loide Baccaro" ], "maintainers": [ "Mayk Loide Baccaro" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "A Prime-Local Weil Obstruction", "id": "10.5281/zenodo.22005240", "authors": [ "Mayk Loide Baccaro" ], "type": "paper", "relationship": "formalizes", "endorsement": "participated" } ], "related": [], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11M26", "11N05" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "OpenAI GPT-5 (Codex)" ], "framework": "Codex" } ], "strongest": "agent", "models": [ "OpenAI GPT-5 (Codex)" ], "spend": "subscription-based", "notes": "The proof was built with a pinned Lean and Mathlib environment and audited against the manuscript." }, "scope": "The submission proves the local block threshold, divergence of the von-Mangoldt weighted prefix, its exact prime-power-indexed form, and the resulting divergence of every scalar budget satisfying twice the Weil-weight lower bound at every natural-number index. Since the Weil weight vanishes away from prime powers, this hypothesis also requires nonnegative off-support terms. The compactly supported smooth dilation, autocorrelation tail argument, and explicit-formula normalization remain manuscript-level mathematics.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "RiemannHypothesisProofFactory.PrimeLocalWeil.scalarPartialSum_tendsto_atTop", "file": "RiemannHypothesisProofFactory/PrimeLocalWeil/BudgetObstruction.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "RiemannHypothesisProofFactory.PrimeLocalWeil.localBlock_nonnegative_iff", "file": "RiemannHypothesisProofFactory/PrimeLocalWeil/LocalThreshold.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "RiemannHypothesisProofFactory.PrimeLocalWeil.weightedPartialSum_tendsto_atTop", "file": "RiemannHypothesisProofFactory/PrimeLocalWeil/ArithmeticBudget.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "RiemannHypothesisProofFactory.PrimeLocalWeil.primePowerPartialSum_tendsto_atTop", "file": "RiemannHypothesisProofFactory/PrimeLocalWeil/PrimePowerBridge.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The submitted formal surface is a stated subset of the paper and does not claim a kernel proof of its smooth-function or explicit-formula steps.", "alignment": true, "original": false, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [ "OpenAI GPT-5 (Codex)" ], "notes": "Independent agent lanes audited the submitted statement types, definitions, arithmetic indexing, axiom reports, and paper correspondence. No human peer review is claimed." }, "canonical": { "repo": "mbaccaro-dev/mathematical-proofs", "directory": "MathematicalProofs/PrimeLocalWeilObstruction/Lean" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-21-000005", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-21-000005", "commit": "979219b93b9e19344219ffd96ad4c8b84ca89cf5", "trust": "high", "theorems": 4, "date": "2026-08-21" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "mbaccaro-dev/mathematical-proofs/MathematicalProofs/SelbergDivisorFeedback/Lean/formalization.yaml", "repo": "mbaccaro-dev/mathematical-proofs", "path": "MathematicalProofs/SelbergDivisorFeedback/Lean/formalization.yaml", "directory": "MathematicalProofs/SelbergDivisorFeedback/Lean", "origins": [ "palomar" ], "url": "https://github.com/mbaccaro-dev/mathematical-proofs/blob/HEAD/MathematicalProofs/SelbergDivisorFeedback/Lean/formalization.yaml", "version": "v0.4", "missing": [], "name": "Near-linear conditioning of Selberg divisor feedback", "description": "A Lean 4 proof of the sequence-level conditioning construction for Selberg divisor feedback. Two explicit perturbations of the von Mangoldt weight stay in the positive prime-power cone, approximate the divisor data to an arbitrary tolerance through twice the observation scale, separate their summatory outputs at a near-linear power scale, and retain one uniform linear Selberg bound.", "authors": [ "Mayk Loide Baccaro" ], "maintainers": [ "Mayk Loide Baccaro" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "Selberg Feedback and Approximate Divisor Data Do Not Stably Determine Square-Root Prime Error", "id": "10.5281/zenodo.22005767", "authors": [ "Mayk Loide Baccaro" ], "type": "paper", "relationship": "formalizes", "endorsement": "participated" } ], "related": [], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11N05", "11M26" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "OpenAI GPT-5 (Codex)" ], "framework": "Codex" } ], "strongest": "agent", "models": [ "OpenAI GPT-5 (Codex)" ], "spend": "subscription-based", "notes": "The proof uses the pinned Lean and Mathlib environment recorded in this directory. The full assistance disclosure is pinned in Manuscript/paper.tex under \"Assistance disclosure.\"" }, "scope": "The selected theorem uses the paper's explicit infinite perturbation, including the distinguished prime-two correction and constant values on every prime-power chain. Its standard analytic inputs state the weighted prime-number asymptotic, the negligible two-chain, coefficient and Chebyshev bounds, the divisor identity, perturbation moment and harmonic bounds, and the classical Selberg estimate. From these inputs Lean derives the two actual von Mangoldt weights, positive-cone membership, divisor residual tolerance at every index through twice the scale, quantitative summatory separation, and a", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "RiemannHypothesisProofFactory.SelbergConditioning.near_linear_conditioning_witness", "file": "RiemannHypothesisProofFactory/SelbergConditioning/NearLinearConditioning.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The infinite prime-power construction and its conditioning conclusion are formalized. The weighted prime-number limit, negligible two-chain, coefficient, divisor, Chebyshev, moment, harmonic-mass, and classical Selberg estimates are explicit premises. The broader generalized von Mangoldt hierarchy remains manuscript-level.", "alignment": true, "original": false, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [ "OpenAI GPT-5 (Codex)" ], "notes": "The prime-power support, correction term, analytic inputs, positivity, divisor residuals, separation scale, uniform Selberg constant, quantifiers, and axiom reports were checked. No human peer review is claimed." }, "canonical": { "repo": "mbaccaro-dev/mathematical-proofs", "directory": "MathematicalProofs/SelbergDivisorFeedback/Lean" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-22-000002", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-22-000002", "commit": "f0ccae19f2cb5afdca00c42694d677c74dc492f6", "trust": "high", "theorems": 1, "date": "2026-08-22" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "mbaccaro-dev/mathematical-proofs/MathematicalProofs/SubpowerDyadicEnergy/Lean/formalization.yaml", "repo": "mbaccaro-dev/mathematical-proofs", "path": "MathematicalProofs/SubpowerDyadicEnergy/Lean/formalization.yaml", "directory": "MathematicalProofs/SubpowerDyadicEnergy/Lean", "origins": [ "palomar" ], "url": "https://github.com/mbaccaro-dev/mathematical-proofs/blob/HEAD/MathematicalProofs/SubpowerDyadicEnergy/Lean/formalization.yaml", "version": "v0.4", "missing": [], "name": "Eventual dyadic defect and Möbius-logarithm energy", "description": "A Lean 4 proof of the global power-of-two balance connecting dyadic defect, block variation, and positive Möbius-logarithm prefix energy. Eventual nonnegativity on exactly those dyadic blocks gives an explicit all-endpoint cubic energy bound. With the stated weighted-squarefree remainder estimate, the formalization also recovers the sharp 6/pi^2 leading coefficient and retains the complete finite-prefix correction.", "authors": [ "Mayk Loide Baccaro" ], "maintainers": [ "Mayk Loide Baccaro" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "Subpower Energy and Dyadic Defect Criteria for the Riemann Hypothesis", "id": "10.5281/zenodo.22038521", "authors": [ "Mayk Loide Baccaro" ], "type": "paper", "relationship": "formalizes", "endorsement": "participated" } ], "related": [], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11M26", "11N37" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "OpenAI GPT-5 (Codex)" ], "framework": "Codex" } ], "strongest": "agent", "models": [ "OpenAI GPT-5 (Codex)" ], "spend": "subscription-based", "notes": "The proof uses the pinned Lean and Mathlib environment recorded in this directory. The full assistance disclosure is pinned in Manuscript/paper.tex under \"Assistance disclosure.\"" }, "scope": "The selected statements use the paper's exact Möbius-logarithm weights, strict predecessor convention, and half-open power-of-two blocks. They telescope the local identities into the exact global energy balance, including its initial boundary term. Eventual defect nonnegativity only on those blocks gives an explicit cubic logarithmic bound for positive cumulative energy at every integer endpoint, with all earlier scales kept in a finite constant. An explicit weighted-squarefree remainder premise yields the sharp 6/pi^2 leading coefficient at powers of two. 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The converse of the coding theorem is proved in full; the achievability half is stated but left as sorry.", "sorry_count": 1, "sorry_in_definitions": 0, "axioms": [], "nonstandard_axioms": [], "results": [ { "declaration": "LeanP1.Channel.IsAchievable.le_capacity", "file": "LeanP1/InfoTheory/Converse.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "LeanP1.Channel.Code.fano", "file": "LeanP1/InfoTheory/Fano.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "LeanP1.Channel.capacity_pow_le", "file": "LeanP1/InfoTheory/ProductChannel.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "LeanP1.Channel.capacity_bsc", "file": "LeanP1/InfoTheory/Examples/BSC.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "LeanP1.Channel.operationalCapacity_eq_capacity", "file": "LeanP1/InfoTheory/Converse.lean", "description": "", "axioms": [], "sorry_count": 1, "comparator": false, "literature": [] } ], "comparator": false, "literature_dependencies": 0, "divergences": "", "alignment": false, "original": false, "review": { "status": "", "bucket": "not stated", "reviewers": [], "notes": "" }, "canonical": { "repo": "mehrasaahmadip/lean_infotheory", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "mkwatson/unique-sums-palomar/formalization.yaml", "repo": "mkwatson/unique-sums-palomar", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/mkwatson/unique-sums-palomar/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Three-element unique-sum-free sets have three-torsion", "description": "For the unordered-pair unique-sum-free predicate, a normalized three-element set {0,x,y} in an additive commutative group forces 3x = 0. This is the algebraic core of the classification of three-element unique-sum-free sets.", "authors": [ "Mark Watson" ], "maintainers": [ "Mark Watson" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Additive bases and unique sums", "id": "", "authors": [], "type": "original-proof", "relationship": "other", "endorsement": "" }, { "title": "Sumsets with a unique representation", "id": "doi:10.7282/t3-1w2k-jr68", "authors": [ "Daniel Scheinerman" ], "type": "doctoral dissertation", "relationship": "background", "endorsement": "not-contacted" } ], "related": [], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "11B13", "20K01" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "OpenAI Codex GPT-5" ], "framework": "Codex" } ], "strongest": "agent", "models": [ "OpenAI Codex GPT-5" ], "spend": "subscription-based", "notes": "AI-assisted throughout. Disclosure wording is a draft for the responsible maintainer's review and must be rewritten or approved by him before publication." }, "scope": "The compared theorem proves only the normalized three-element torsion implication. It does not formalize the full classification, the finite-Abelian census, any minimum-size formula, or any DRAT-backed computation.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The wrapper renames the namespace and omits surrounding census and descent declarations. The theorem statement and proof obligations are unchanged in mathematical content from Dcyc.normalized_three_USF_has_three_torsion_unordered.", "alignment": true, "original": true, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Mark Watson" ], "notes": "Local kernel and Comparator checks are recorded in OUT-PALOPREP.md. 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OS3 is stated in the complex star formulation (OS 1975 axiom E2) with an equivalent real reformulation (OS3_ReflectionPositivity_real). OS4 is split into a clustering statement and an ergodicity statement, with ergodicity derived from a polynomial-clustering bound (alpha = 6). The dimension-generic master theorem is proved for every d >= 2: the OS3 mixed-representation proper-time Fubini domination runs at boundary-vanishing order d, so no upper bound on the dimension is needed. The registry Challenge (in the wrapper repository) restates the headline existentially, in Mathlib-only terms: there exists a probability measure on S'(R^d) whose generating functional is exp(-1/2 ) with C the proper-time (heat-kernel) integral of the free covariance — the characterization clause pins the measure uniquely (bare existence would be satisfied by the Dirac measure at 0) — and which satisfies the five restated OS predicates; the dimension hypothesis 2 <= d is carried as a Fact ", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "0 sorries and 0 custom axioms in the OSforGFF source (52 on-graph files, ~25,500 lines, plus the off-graph OS/NonTrivial.lean — live non-degeneracy results verified by scripts/check-guardrails.sh — and 6 off-graph Legacy files preserving superseded 4D-era proofs). The registry Challenge/Solution pair, and the single `sorry` that is its challenge hole, live in mrdouglasny/OSforGFF-comparator. External dependencies (bochner, gaussian-field, kolmogorov_extension4) are themselves axiom-free per their own documentation. 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Vilenkin" ], "type": "book", "relationship": "background", "endorsement": "n/a" }, { "title": "Methods of Modern Mathematical Physics, Volume II: Fourier Analysis, Self-Adjointness", "id": "ISBN:978-0-12-585002-5", "authors": [ "Michael Reed", "Barry Simon" ], "type": "book", "relationship": "background", "endorsement": "not-contacted" } ], "related": [ { "id": "https://github.com/or4nge19/OSforGFF", "relationship": "other", "note": "Fork by Matteo Cipollina pursuing an alternative measure-construction pipeline (finite-dimensional Gaussians -> Kolmogorov extension -> pushforward), avoiding the Minlos theorem." } ], "classification": { "arxiv": [ "hep-th", "math-ph" ], "msc2020": [ "81T08", "81S40", "60G60" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "claude-opus-4-6" ], "framework": "Claude Code" }, { "method": "agent", "models": [ "gpt-5.2-codex" ], "framework": "Codex" }, { "method": "other", "models": [ "gemini-3-pro" ], "framework": "Gemini" }, { "method": "agent", "models": [ "claude-fable-5" ], "framework": "Claude Code" } ], "strongest": "agent", "models": [ "claude-fable-5", "claude-opus-4-6", "gemini-3-pro", "gpt-5.2-codex" ], "spend": "", "notes": "Build: `lake build` (Challenge and Solution are the default targets); independent verification: `./scripts/verify-comparator.sh` (Comparator in a Landrun sandbox with NanoDa kernel replay). All dependencies, including OSforGFF itself, are pinned to full commit SHAs via lake-manifest.json. Models per README acknowledgements." }, "scope": "This repository carries the registry submission for the OSforGFF library: a self-contained Mathlib-only restatement of its headline result (Challenge.lean, whose single `sorry` is the challenge hole) and the proof of that restatement from the library (Solution.lean), wired through comparator.json. The theorem: for every dimension d >= 2 and mass m > 0 there exists a probability measure on the tempered distributions S'(R^d), characterized uniquely by its Gaussian generating functional with the proper-time free covariance, satisfying all five Osterwalder-Schrader axioms (OS0 analyticity, OS1 reg", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [], "nonstandard_axioms": [], "results": [ { "declaration": "Challenge.gaussianFreeField_satisfies_OS_axioms", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [ { "statement": "for every d >= 2 and m > 0 the free GFF exists on S'(R^d), is characterized by its Gaussian generating functional with the proper-time covariance, and satisfies OS0-OS4", "source": "10.1007/BF01645738" }, { "statement": "reflection positivity in the complex star formulation (axiom E2)", "source": "10.1007/BF01608978" }, { "statement": "probability-measure formulation of the OS axioms", "source": "ISBN:978-0-387-96477-7" } ] } ], "comparator": true, "literature_dependencies": 3, "divergences": "The OS axioms are formalized in the Glimm-Jaffe probability-measure formulation (predicates on ProbabilityMeasure FieldConfiguration) rather than the original Schwinger-function formulation of Osterwalder-Schrader. OS3 is stated in the complex star formulation (OS 1975 axiom E2) with an equivalent real reformulation (OS3_ReflectionPositivity_real). OS4 is split into a clustering statement and an ergodicity statement, with ergodicity derived from a polynomial-clustering bound (alpha = 6). The dimension-generic master theorem is proved for every d >= 2: the OS3 mixed-representation proper-time Fubini domination runs at boundary-vanishing order d, so no upper bound on the dimension is needed. The registry Challenge restates the headline existentially, in Mathlib-only terms: there exists a probability measure on S'(R^d) whose generating functional is exp(-1/2 ) with C the proper-time (heat-kernel) integral of the free covariance — the characterization clause pins the measure uniquely (bare existence would be satisfied by the Dirac measure at 0) — and which satisfies the five restated OS predicates; the dimension hypothesis 2 <= d is carried as a Fact instance binder. The transfo", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "The single `sorry` in Challenge.lean is the registry challenge hole, discharged by Solution.lean (which contains verbatim copies of the Challenge definitions and the proof; the two files differ only in import, module docstring, and the proof). The OSforGFF source at the pinned commit has 0 sorries and 0 custom axioms; its 4D core was reviewed and merged upstream (mrdouglasny/OSforGFF PR #7). External dependencies (bochner, gaussian-field, kolmogorov_extension4) are themselves axiom-free per their own documentation." }, "canonical": { "repo": "mrdouglasny/osforgff-comparator", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-31-000012", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-31-000012", "commit": "6e56a6f2a0fe01272492ae8c2ae3dd4fc5bc5901", "trust": "high", "theorems": 1, "date": "2026-08-31" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "nicholasbulka/li-criterion-rh-equivalence-lean/formalization.yaml", "repo": "nicholasbulka/li-criterion-rh-equivalence-lean", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/nicholasbulka/li-criterion-rh-equivalence-lean/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Li's criterion for the Riemann Hypothesis", "description": "Li's criterion for the Riemann zeta function, formalized in Lean 4 over Mathlib: Mathlib's `RiemannHypothesis` holds if and only if every Li-Keiper coefficient of the completed xi function has nonnegative real part. The zero-indexed analytic coefficient at n is Li's classical lambda_(n+1), the n-th Taylor coefficient at 0 of the logarithmic derivative of s |-> xi(1/(1-s)). A second compared theorem identifies this analytic coefficient with Li's arithmetic coefficient, the symmetrized sum over the nontrivial zeros counted with multiplicity, and proves that symmetrized family summable so the sum", "authors": [ "Nicholas Bulka" ], "maintainers": [ "Nicholas Bulka" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "The positivity of a sequence of numbers and the Riemann hypothesis", "id": "J. Number Theory 65 (1997), 325-333; doi:10.1006/jnth.1997.2137", "authors": [ "Xian-Jin Li" ], "type": "paper", "relationship": "formalizes", "endorsement": "" }, { "title": "Complements to Li's criterion for the Riemann hypothesis", "id": "J. Number Theory 77 (1999), 274-287; doi:10.1006/jnth.1999.2392", "authors": [ "Enrico Bombieri", "Jeffrey C. Lagarias" ], "type": "paper", "relationship": "adapts", "endorsement": "" }, { "title": "Power series expansions of Riemann's xi function", "id": "Math. Comp. 58 (1992), 765-773; doi:10.2307/2153215", "authors": [ "Jerry B. Keiper" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Lectures on Probability Theory - Li Criterion", "id": "https://www.youtube.com/watch?v=M0lgrrskxMw", "authors": [ "M. Ram Murty" ], "type": "other", "relationship": "background", "endorsement": "" }, { "title": "Mathlib (completed Riemann zeta and RiemannHypothesis)", "id": "https://github.com/leanprover-community/mathlib4", "authors": [ "The mathlib community" ], "type": "other", "relationship": "background", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11M26", "11M06", "30D15" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Claude (Anthropic)" ], "framework": "Claude Code" } ], "strongest": "agent", "models": [ "Claude (Anthropic)" ], "spend": "", "notes": "Interactive agent-assisted development directed and reviewed by the maintainer." }, "scope": "Complete. Li's criterion for the Riemann zeta function as an unconditional biconditional against Mathlib's `RiemannHypothesis`, plus the identification of the analytic Taylor coefficients with Li's arithmetic coefficients (symmetrized sum over nontrivial zeros with multiplicity). The supporting theory - genus-1 Hadamard factorization of xi with multiplicity, genus-1 summability, and the order bound order(xi) <= 1 - is proved in the development, not assumed. The order bound is proved in Lc/LiCriterion/XiGrowth.lean from the Mellin representation of completedRiemannZeta0 and the exponential deca", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "li_criterion", "file": "comparator/Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "li_coefficients_eq_zero_sum", "file": "comparator/Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The compared statements are faithful to Li (1997) and Bombieri-Lagarias (1999) with the following choices of formulation. Li's lambda_n is indexed classically from 1; the formal `taylorCoeff riemannXi n` is zero-indexed, so it is lambda_(n+1). The arithmetic coefficient is stated as the symmetrized sum over nontrivial zeros counted with multiplicity, which is the classical symmetric summation convention; the corresponding unpaired family is not absolutely summable. The second compared theorem is a conjunction: `Summable` of the symmetrized weighted family, and the equality of the Taylor coefficient with half its `tsum`, so the sum cannot be Lean's default value. `Challenge.lean` states the results with deliberate `sorry` holes, as the Comparator convention requires, and `Solution.lean` supplies the proofs by delegating to the library. The Challenge definitions (`riemannXi`, `phi`, `logDeriv`, `taylorCoeff`, `NontrivialZero`) are built directly from Mathlib and mirrored character-for-character in `ChallengeDeps.lean`. No additional hypotheses are assumed; the compared theorems depend on propext, Classical.choice and Quot.sound only.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Nicholas Bulka" ], "notes": "The maintainer directed and reviewed the development as it was produced. No external or independent review has been performed. Every step is checked by Lean; the comparator re-checks the statement equality, the axiom allowlist, and the kernel replay." }, "canonical": { "repo": "nicholasbulka/li-criterion-rh-equivalence-lean", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-05-000005", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-05-000005", "commit": "35df682f3b709ffe5fbcfdd452dfa964bd622b87", "trust": "high", "theorems": 2, "date": "2026-09-05" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "nielstron/langlib/formalization.yaml", "repo": "nielstron/langlib", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/nielstron/langlib/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Langlib", "description": "A Lean 4 formalization of grammar and automaton presentations of the extended Chomsky hierarchy. Starting from unrestricted-grammar rewriting, it defines regular, linear, context-free, context-sensitive, indexed, deterministic context-free, recursive, and recursively enumerable language classes; relates finite, pushdown, linearly bounded, and Mathlib TM0 automata to the corresponding grammar classes; proves strict hierarchy inclusions; and establishes closure or non-closure under thirteen operations for seven major classes.", "authors": [ "Niels Mündler", "Martin Dvořák" ], "maintainers": [ "Niels Mündler" ], "license": "BSD-2-Clause", "role": "", "substantive": "", "sources": [ { "title": "Introduction to Automata Theory, Languages, and Computation, 3rd edition", "id": "ISBN 978-0321455369", "authors": [ "John E. Hopcroft", "Rajeev Motwani", "Jeffrey D. Ullman" ], "type": "book", "relationship": "formalizes", "endorsement": "" }, { "title": "Indexed Grammars—An Extension of Context-Free Grammars", "id": "doi:10.1145/321479.321488", "authors": [ "Alfred V. 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Exact historical usage totals and prompts were not reconstructed; no novelty claim is made from the use of automation." }, "scope": "The selected challenge contains transparent definitions of unrestricted, regular, linear, context-free, context-sensitive, LR, indexed, and automaton-recognized language classes; grammar/automaton equivalences for regular, context-free, deterministic context-free, context-sensitive, and recursively enumerable languages; the strict extended Chomsky hierarchy; incomparability of the linear and deterministic context-free classes; and a 13-operation closure/non-closure matrix for regular, deterministic context-free, context-free, indexed, context-sensitive, recursive, and recursively enumerable la", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "RG_eq_DFA", "file": "src/Langlib/Automata/FiniteState/Equivalence/Regular.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "CF_eq_PDA_Class", "file": "src/Langlib/Automata/Pushdown/Equivalence/ContextFree.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "LR_eq_DCF", "file": "src/Langlib/Grammars/LR/Equivalence.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Linear_incomp_DPDA_of_card", "file": "src/Langlib/Classes/Linear/Inclusion/DeterministicContextFree.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "CS_eq_LBA", "file": "src/Langlib/Automata/LinearBounded/Equivalence/ContextSensitive.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "TM_eq_RE", "file": "src/Langlib/Automata/Turing/Equivalence/RecursivelyEnumerable.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "ChomskyHierarchy.Indexed_strict_subclass_CS", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "ChomskyHierarchy.CS_strict_subclass_Recursive_of_card", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "ChomskyHierarchy.Recursive_strict_subclass_RE_of_card", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "Langlib makes implementation conventions explicit rather than claiming literal identity with every cited presentation. Its context-sensitive grammar permits the distinguished rule S -> epsilon only when S never occurs on a right-hand side; its equivalent LBA uses explicit endmarkers and handles the empty word on a genuine two-cell tape. Strict inclusions record explicit lower bounds on the number of alphabet elements; no strictness claim is made below a theorem's stated bound. A negative closure theorem records failure on its stated finite witness alphabet, not failure on every alphabet. Indexed languages are grammar-defined with no nested-stack automaton model; linear languages likewise lack an automaton characterization. DLBA = NLBA is not asserted, and the recursive class has no selected second tape-model equivalence. These are known scope limits, not inferred equivalences. Novelty has not been established and is not claimed.", "alignment": true, "original": false, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [ "OpenAI Codex" ], "notes": "Submission preparation inventoried every selected declaration, checked theorem axioms and unresolved holes, and compared the Challenge signatures with Langlib. This is not independent human peer review." }, "canonical": { "repo": "nielstron/langlib", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-27-000011", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-27-000011", "commit": "10fb20454652d4b7f0f1710880f79240986021c6", "trust": "high", "theorems": 109, "date": "2026-08-27" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "nihargargava/IntegralMatrices/formalization.yaml", "repo": "nihargargava/IntegralMatrices", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/nihargargava/IntegralMatrices/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Integral Matrices of Fixed Rank over Number Fields: Lean formalization", "description": "A Lean formalization of the fixed-rank asymptotic count for integral matrices over a number field and the convergence of moments of lifts of algebraic codes. The public results preserve the submission PDF's main scale, error terms, echelon-integral limit, and denominator normalization. The intermediate-rank height-counting input attributed to Schmidt is an explicit hypothesis; it is not silently assumed as an axiom.", "authors": [ "Nihar Gargava" ], "maintainers": [ "Nihar Gargava" ], "license": "GPL-3.0", "role": "", "substantive": "", "sources": [ { "title": "Integral matrices of fixed rank over number fields", "id": "arXiv:2510.11673", "authors": [ "Nihar Gargava", "Vlad Serban", "Maryna Viazovska", "Ilaria Viglino" ], "type": "paper", "relationship": "formalizes", "endorsement": "" }, { "title": "On Heights of Algebraic Subspaces and Diophantine Approximations", "id": "doi:10.2307/1970360", "authors": [ "Wolfgang M. 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Katznelson" ], "type": "paper", "relationship": "adapts", "endorsement": "" }, { "title": "An Asymptotic Estimate for Heights of Algebraic Subspaces", "id": "doi:10.2307/2154015", "authors": [ "Jeffrey Lin Thunder" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Moments of the number of points in a bounded set for number field lattices", "id": "arXiv:2308.15275", "authors": [ "Nihar Gargava", "Vlad Serban", "Maryna Viazovska" ], "type": "paper", "relationship": "background", "endorsement": "" } ], "related": [ { "id": "https://github.com/nihargargava/MeanValueIdealLattices", "relationship": "builds-on", "note": "Reuses and ports GPL-3.0 metric and covolume formalization ideas from the author's earlier formalization, with attribution in notes/source-attribution.md and the affected Lean files." } ], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11R04", "11H06" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "GPT-5 / Codex" ], "framework": "Lean 4 and Mathlib" } ], "strongest": "agent", "models": [ "GPT-5 / Codex" ], "spend": "", "notes": "The author directed the formalization and reviewed the statement correspondence, adaptations, citations, and final trust audit." }, "scope": "The formalization proves the two main theorem paths in the cited submission PDF: the fixed-rank count and the higher-moment convergence for lifts of codes. Schmidt's intermediate-rank height count remains an explicit hypothesis. The admissibility/Riemann subsystem uses the author-approved raw Euclidean mixed-space convention, while exact coefficient measures and covolume normalizations are bridged to the submission PDF. The exceptional critical-radius and large-radius Riemann branches are documented local adaptations in this metadata and committed Lean comments. The author-supplied submission ", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The public theorem path is conditional on the explicit Schmidt height-count interface. The admissibility/Riemann implementation uses Mathlib's raw Euclidean mixed-space metric rather than making the submission PDF's discriminant-scaled trace metric a global typeclass; exact main-term normalization is proved separately. The approved critical radius and large-radius Riemann adaptations are recorded in the committed Lean comments and this metadata.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "none" ], "notes": "The formal statements were audited against the cited submission PDF, public theorem paths were built with the pinned toolchain, and the final declarations were checked for project-specific axioms, sorryAx, and other hidden proof placeholders." }, "canonical": { "repo": "nihargargava/integralmatrices", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-15-000002", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-15-000002", "commit": "4124db884ee55ef0da9e899d44d1bec5bfe432ea", "trust": "high", "theorems": 2, "date": "2026-09-15" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "nihargargava/MeanValueIdealLattices/formalization.yaml", "repo": "nihargargava/MeanValueIdealLattices", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/nihargargava/MeanValueIdealLattices/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Mean Value for Random Ideal Lattices: Lean formalization", "description": "A Lean formalization of the geometric, analytic, and arithmetic argument for mean values of lattice-point counts in Haar-random unit-covolume ideal lattices. For cyclotomic fields, it proves the conditional quantitative estimate for every exponent below the assumed subconvexity exponent and its positive power-saving corollary, uniformly over origin-centered balls. The subconvexity, Stark residue, and discriminant-growth inputs remain explicit hypotheses rather than claimed consequences of the formalization.", "authors": [ "Nihar Gargava" ], "maintainers": [ "Nihar Gargava" ], "license": "GPL-3.0", "role": "", "substantive": "", "sources": [ { "title": "Mean Value for Random Ideal Lattices", "id": "arXiv:2411.14973", "authors": [ "Nihar Gargava", "Maryna Viazovska" ], "type": "article", "relationship": "formalizes", "endorsement": "not-contacted" } ], "related": [], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11R04", "11M41", "11R42" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "ChatGPT", "Codex" ], "framework": "Lean 4 and Mathlib" } ], "strongest": "agent", "models": [ "ChatGPT", "Codex" ], "spend": "not tracked", "notes": "Lean code was produced almost completely with ChatGPT and Codex and reviewed by the responsible maintainer." }, "scope": "The formalization proves the Arakelov and Haar constructions, the ideal-lattice realization, the analytic unfolding and contour estimates, cyclotomic zeta continuation, and the complete asymptotic deduction from the three explicit paper inputs. It does not formalize proofs that the subconvexity, Stark, and discriminant-growth hypotheses hold for the cyclotomic family.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The final theorems are conditional: the external analytic and arithmetic hypotheses from the paper are parameters, and their cyclotomic validity is intentionally outside the formalized scope.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "none" ], "notes": "The responsible maintainer reviewed the statement correspondence, theorem quantifiers, and stated limitations." }, "canonical": { "repo": "nihargargava/meanvalueideallattices", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-02-000009", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-02-000009", "commit": "339043190fd8b5468af851a680b7e8737f5467e7", "trust": "high", "theorems": 2, "date": "2026-09-02" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "nimaanari/formalization-beyond-bethe/formalization.yaml", "repo": "nimaanari/formalization-beyond-bethe", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/nimaanari/formalization-beyond-bethe/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Beyond the Bethe Approximation of the Permanent", "description": "Computing or approximating the permanent is a canonical problem in theoretical computer science and in counting and sampling. Before this work, the canonical Bethe approximation gave the best known deterministic guarantee, with approximation factor (sqrt 2)^n. This formalization proves that there is an absolute constant c < sqrt 2 and a deterministic polynomial-time c^n-approximation for every nonnegative rational matrix. It checks the mathematical and algorithmic proof down to Lean's standard logical axioms, including the finite-precision convex-optimization implementation.", "authors": [ "Nima Anari" ], "maintainers": [ "Nima Anari" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Beyond the Bethe Approximation of the Permanent", "id": "", "authors": [ "Nima Anari" ], "type": "paper", "relationship": "formalizes", "endorsement": "participated" }, { "title": "The Bethe Permanent of a Nonnegative Matrix", "id": "doi:10.1109/TIT.2012.2227109", "authors": [ "Pascal O. 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It reconstructs the four external source theorems used by the mathematical argument and formalizes the finite-precision convex-optimization routine. No proof obligation is omitted. The single sorry in Challenge.lean is the deliberate Palomar statement hole and is ex", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "There is no known divergence from Theorem 1 of the accompanying paper. In the Palomar-facing statement, the source-stable Cobham formulation is formally identified in Solution.lean with Complexitylib's Cobham class and its deterministic Turing-machine class FP.", "alignment": true, "original": false, "review": { "status": "author-verified", "bucket": "author-verified", "reviewers": [ "Nima Anari" ], "notes": "The author verified the results. 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The Lean development also proves the arbitrary-spanning-subgraph strengthening and a finite Matrix--Tree theorem. Every definition used by the statements is concrete in both the Challenge and Solution environments.", "alignment": true, "original": false, "review": { "status": "author-verified", "bucket": "author-verified", "reviewers": [ "Jiechen Zhang" ], "notes": "Jiechen Zhang checked the definitions, statements, arguments, and cited results. On 2026-08-22 he completed a paper-to-Lean specification audit of the definitions, hypotheses, intermediate results, conclusions, and equality case. 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A graph is called Brill--Noether general if the possible degrees and ranks of its chip configurations, in the sense of Baker and Norine, match the behavior of a general algebraic curve. Here we use only the existential form of Brill--Noether generality, without computing dimensions. 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Nathan Pflueger supplied the mathematical direction and curated the formalization." }, "scope": "The submission proves seven finite-graph statements. Five concern the nonexistence half of Brill--Noether generality for chains. The first two are parts (1) and (2) of the original CDPR theorem. The next three are a sharp cycle-chain theorem stated in terms of torsion orders, a marked mixed-chain theorem with a one-sided torsion budget, and an unmarked mixed-chain theorem for explicitly specified cycles and evenly marked theta graphs with a minimum torsion budget. 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For metric graphs with positive integer edge lengths, passage to a unit-edge subdivision, together with Luo's theory of rank-determining sets, relates the relevant divisor-rank calculations. This comparison is not established in the repository: metric graphs and their divisor theory have not been formally defined here. The submitted CDPR statements should therefore be regarded as finite, discrete adaptations of the metric results. Chains are defined by joining disjoint factors with bridge edges rather than identifying adjacent marked vertices, to simplify the definitions in the challenge file. The general chain criteria are stated explicitly as Theorem 1.13 and Corollary 6.16 of Pflueger--Solomon. The concrete specializations to chains mixing cycles and evenly marked theta graphs are not stated verbatim; they combine those criteria with Example 1.11 and Theorem 1.12, part (2), equivalently Corollary 4.17. 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The source separately credits Claude for the mathematical construction; Claude did not create this Lean formalization." }, "scope": "Defines an explicit order-668 sign matrix, proves it is Hadamard, and packages the result as a Comparator challenge.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "Hadamard.H_isHadamard", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Hadamard.exists_hadamard_668", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": false, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The proof transforms the announcement's four cyclic sequences in ways that preserve their correlations, then uses a different Wallis-Whiteman block array. 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It disproves the generalized Jacobian conjecture over every nontrivial commutative ring using an explicit determinant-one polynomial self-map in three variables. 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Repository-only attributes, the determinant-minus-two presentation, the open two-variable case, and API tests are omitted. The challenge and solution each include their statement dependencies directly so that they import Mathlib only. The checked theorem and its determinant-one counterexample preserve the substantive generalized disproof.", "alignment": false, "original": false, "review": { "status": "peer-reviewed", "bucket": "peer-reviewed", "reviewers": [ "bkim23", "mo271", "alreadydone", "OpenAI Codex (gpt-5)", "kbuzzard" ], "notes": "The source proof was approved through formal-conjectures pull request reviews; this wrapper was separately checked by building both modules, inspecting axioms and sorries, and running Comparator." }, "canonical": { "repo": "paul-lez/jacobian-conjecture", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-21-000006", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-21-000006", "commit": "b244ea482ed086297e379d73e6724cd26b2d301f", "trust": "high", "theorems": 1, "date": "2026-08-21" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "pengzhang91/Feige/formalization.yaml", "repo": "pengzhang91/Feige", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/pengzhang91/Feige/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Feige's 1/e Conjecture in Lean", "description": "A Lean 4 formalization of the sharp unit-slack (delta = 1) small-deviation inequality for sums of independent nonnegative random variables. For every positive integer n and every independent family X_1, ..., X_n of measurable, integrable, nonnegative random variables satisfying E[X_i] <= 1, writing S = sum_i X_i, it proves the sharp bound P(S < E[S] + 1) >= (n / (n + 1))^n and proves that this constant is optimal for each fixed n. This optimal dimension-dependent bound strengthens and implies Feige's 1/e conjecture. The repository also formalizes the Vlassis-Thomas exact calibration theorem an", "authors": [ "Guanyang Wang", "Peng Zhang" ], "maintainers": [ "Guanyang Wang", "Peng Zhang" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Sharp Small-Deviation Inequalities for Sums of Independent Nonnegative Random Variables", "id": "arXiv:2607.23980", "authors": [ "Weibo Fu", "Yanjun Han", "Guanyang Wang", "Jun Yan", "Peng Zhang", "Zhengqing Zhou" ], "type": "paper", "relationship": "formalizes", "endorsement": "participated" }, { "title": "An Exact Distribution-Free Test for Means of Nonnegative Random Variables", "id": "arXiv:2607.08415", "authors": [ "Nikos Vlassis", "Philip S. Thomas" ], "type": "paper", "relationship": "formalizes", "endorsement": "" }, { "title": "Partitions of Mass-Distributions and of Convex Bodies by Hyperplanes", "id": "doi:10.2140/pjm.1960.10.1257", "authors": [ "Branko Grünbaum" ], "type": "paper", "relationship": "formalizes", "endorsement": "" }, { "title": "On Sums of Independent Random Variables with Unbounded Variance and Estimating the Average Degree in a Graph", "id": "doi:10.1137/S0097539704447304", "authors": [ "Uriel Feige" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "On Feige's Conjecture", "id": "arXiv:2607.24528", "authors": [ "Zipei Nie", "Jiaye Wei" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "A Conditional Proof of Feige's Conjecture with a Sharp Finite-Dimensional Bound", "id": "doi:10.5281/zenodo.21626794", "authors": [ "Mark Stander" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "On Samuels' Conjecture", "id": "arXiv:2608.18392", "authors": [ "Zhi Ling" ], "type": "paper", "relationship": "background", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.PR", "math.CO" ], "msc2020": [ "60E15", "60G50", "52A40" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "gpt-5.6-sol" ], "framework": "Codex" } ], "strongest": "agent", "models": [ "gpt-5.6-sol" ], "spend": "not tracked", "notes": "This records automation used for the Lean formalization: Codex with gpt-5.6-sol at ultra reasoning effort. The cited primary paper separately reports ChatGPT 5.6 Pro in discovery of the mathematical proof. Human formalization authors chose the target, sources, statement interface, and metadata and remain responsible for final verification. Lean elaboration and kernel-axiom audits mechanically check accepted proof terms but do not replace human mathematical review." }, "scope": "The headline theorem covers every positive integer n, probability spaces whose sample type lies in Lean's lowest universe (Omega : Type), and families X : Fin n -> Omega -> Real that are measurable, integrable, independent, pointwise nonnegative, and satisfy E[X_i] <= 1. It proves the strict-event bound P(sum X_i < E[sum X_i] + 1) >= (n / (n + 1))^n and proves that this fixed-dimensional universal constant is optimal. It does not cover n = 0, sample types in arbitrary higher universes, or slack delta != 1; in particular it does not formalize the primary source's complete delta > 0 theorem or t", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "PalomarFeige.sharp_unit_slack_feige_complete", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The headline result is only the delta = 1 specialization of the primary source's theorem for all delta > 0, although it includes the source's fixed-dimensional optimality at delta = 1. Probability spaces are quantified with Omega : Type rather than an arbitrary universe. In the Vlassis-Thomas reconstruction, zero lower-side parameters are handled by finite-dimensional continuity instead of deleting deterministic coordinates; the standalone calibration theorem uses an equivalent independent-exponential representation, with the Dirichlet/simplex identity proved in a separate Feige module; and Remark 4 is represented only by the one-step payoff-independent insertion weights and averaged domination inequality, not by a global random-permutation law. The Grünbaum target covers the convex-body centroid-halfspace theorem needed here, rather than every mass-distribution statement in the 1960 paper. Historical, asymptotic, statistical, and expository material not used by the headline theorem is not represented as separate Lean declarations.", "alignment": true, "original": false, "review": { "status": "author-verified", "bucket": "author-verified", "reviewers": [ "Guanyang Wang", "Peng Zhang" ], "notes": "On 2026-08-29, Guanyang Wang and Peng Zhang reviewed and approved Challenge.lean, Solution.lean, and formalization.yaml. This was author verification of the advertised statement, the solution bridge, and the metadata, not external peer review. Builds, Comparator checks, and kernel-axiom audits are mechanical checks rather than human mathematical review." }, "canonical": { "repo": "pengzhang91/feige", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-30-000002", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-30-000002", "commit": "939c2581c8c97339c7d465bc2c85c31e95255c17", "trust": "high", "theorems": 1, "date": "2026-08-30" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "philipp-svinger/Math_Lean_Palomar_Vanishing-Horizon-Boundary-Reg/formalization.yaml", "repo": "philipp-svinger/Math_Lean_Palomar_Vanishing-Horizon-Boundary-Reg", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/philipp-svinger/Math_Lean_Palomar_Vanishing-Horizon-Boundary-Reg/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Boundary regularity for non-local operators with symmetric kernels and vanishing horizon", "description": "This Lean 4 formalization proves the two main theorems of the paper 'Boundary regularity for non-local operators with symmetric kernels and vanishing horizon' (arXiv:2504.00567v1). For a fractional order s in (1/2,1), the paper studies a nonlocal operator with symmetric kernel whose interaction horizon depends on the distance to the boundary and on a horizon parameter sigma in (0,1], and shrinks near the boundary. The formalization proves that weak solutions u to the Dirichlet problem with zero Dirichlet data and bounded right-hand side are in the Hölder space C^{2s-1} up to the boundary. Furt", "authors": [ "Philipp Svinger" ], "maintainers": [ "Philipp Svinger" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Boundary regularity for non-local operators with symmetric kernels and vanishing horizon", "id": "https://arxiv.org/abs/2504.00567v1", "authors": [ "Philipp Svinger" ], "type": "paper", "relationship": "formalizes", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.AP" ], "msc2020": [ "47G20", "35B65", "35S15", "35R09" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "GPT-5.6 Sol", "ChatGPT 6 Astra" ], "framework": "OpenAI Codex" } ], "strongest": "agent", "models": [ "ChatGPT 6 Astra", "GPT-5.6 Sol" ], "spend": "not tracked", "notes": "The formalization was developed with substantial AI assistance. The human author specified the mathematical goals, organized and supervised the proof-development workflow, and manually checked that the two public Lean theorem statements match the two main theorem statements of the paper. AI agents were used to plan and subdivide the proofs into smaller, achievable tasks, with the explicit instruction to follow the proof strategy of the paper. A Python script was then used to run AI agents iteratively through this plan and to implement, test, and refine the Lean code." }, "scope": "The two main results, Theorems 1.1 and 1.2, of the corresponding paper are proved. Theorem 1.1 shows C^{2s-1} Hölder regularity up to the boundary for weak solutions with zero Dirichlet data and bounded right-hand side. Theorem 1.2 shows Hölder regularity of the quotient of the weak solution u by d_Omega ^{2s-1} up to the boundary. The results are proved for bounded C^{1,1} domains which are characterized by satisfying the interior/exterior ball condition. The submission does not claim to formalize every intermediate lemma, proposition, remark, or open problem in the paper.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The overall proof strategy follows that of the paper for both main theorems. Theorem 1.1 combines a global boundedness estimate, distance-power barriers, weak comparison, and scaled interior regularity. Theorem 1.2 uses profile boundary Harnack, tangential rigidity, half-space Liouville classification, and a normalized blow-up contradiction to obtain the boundary expansion and quotient estimate. PROOF_OUTLINE.md gives the theorem-by-theorem outline with manuscript references and Lean declarations.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "It was manually checked that the two public Lean theorem statements coincide with the two main paper statements. 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The results concern inequalities involving Euler's totient, Dedekind's psi function, and the sum-of-divisors function. The source paper proves the k = 1 instances and discusses k = 2, 3; this development covers the full stated range, including k >= 4, and additionally proves for each inequality that equality holds exactly when n is prime. By supplying the full parameter range and sharp equality cases beyond those treated in the source, the project is relevant to researchers in elementary and anal", "authors": [ "Michael Mazur" ], "maintainers": [ "Michael Mazur" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Lower bounds on expressions depending on the functions phi(n), psi(n) and sigma(n), III", "id": "arXiv:2606.12484v1", "authors": [ "S. I. Dimitrov" ], "type": "paper", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "The generating identity of Cauchy-Schwarz-Bunyakovsky inequality", "id": "https://doi.org/10.4171/EM/78", "authors": [ "A. M. Sommariva" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Generalizations and refinements for Nesbitt's inequality", "id": "https://doi.org/10.7153/jmi-05-02", "authors": [ "Mihaly Bencze", "Ovidiu T. Pop" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" } ], "related": [], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11A25", "26D15" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "OpenAI Codex (GPT-5)" ], "framework": "Codex desktop" }, { "method": "manual", "models": [], "framework": "n/a" } ], "strongest": "agent", "models": [ "OpenAI Codex (GPT-5)" ], "spend": "subscription-based; exact spend not tracked", "notes": "OpenAI Codex selected the conjectures, derived and tested the mathematical arguments, researched prior art, wrote the substantive Lean formalization, ran the reported checks, and drafted the documentation. Michael Mazur directed the objective and is the responsible human maintainer. The results must not be described as solely human-authored proofs." }, "scope": "The Comparator record covers four claims: the exact rational full-range forms of Conjectures 1 and 3 and the two companion equality-if-and-only-if- prime theorems. Challenge.lean's four deliberate sorries are excluded from these counts. The substantive proof development contains no sorry and no custom axiom. The source's other conjectures are outside this project.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "PalomarDimitrov.conjecture1FullRange", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "PalomarDimitrov.conjecture1EqualityIffPrime", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "PalomarDimitrov.conjecture3FullRange", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "PalomarDimitrov.conjecture3EqualityIffPrime", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "No known change to either source conjecture's quantified range or displayed inequality. The theorems are stated over the rationals, which is equivalent to the usual real interpretation for these rational-valued expressions. Dedekind psi is defined by the source's standard prime-factor product; the hypothesis n >= 2 makes the convention at zero irrelevant. The two equality iff prime theorems are additional results proved by this project rather than claims attributed to the source paper.", "alignment": true, "original": false, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [ "OpenAI Codex (separate statement, proof, verification, and prior-art audit runs)" ], "notes": "Separate AI-assisted audits checked the source transcription, proof shape, equality statements, numerical behavior, axioms, and available prior art. No independent human mathematical review and no source-author verification have been completed as of 2026-09-13." }, "canonical": { "repo": "pro20541/dimitrov-conjectures-lean", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-13-000003", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-13-000003", "commit": "95809b59deefd8dfad81e398250aedd7e1aea3b3", "trust": "high", "theorems": 4, "date": "2026-09-13" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "project-numina/kakeya-3d/formalization.yaml", "repo": "project-numina/kakeya-3d", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/project-numina/kakeya-3d/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Kakeya in Dimension Three", "description": "A Lean 4 formalization of the three-dimensional Kakeya conjecture following Guth, Wang, and Zahl. The main theorem carries the three-dimensional Sticky Frostman hypothesis as an explicit argument, and a linking library discharges that hypothesis against a second, independent development, giving an unconditional endpoint.", "authors": [ "Bolton Bailey", "Jia Li", "Jiaqi Wang", "Junqi Liu", "Justin Asher", "Pietro Pellegrino", "Ruichen Qiu", "Wenda Li", "Xiao Ma", "Yichuan Cao", "Yihe Huang", "Yunfei Li", "Yunzhou Xie", "Zekai Zhu", "Zihao Zhou", "Weikun He" ], "maintainers": [ "Project Numina" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "A streamlined proof of the Kakeya set conjecture in ℝ³", "id": "https://arxiv.org/abs/2601.14411", "authors": [ "Larry Guth", "Hong Wang", "Joshua Zahl" ], "type": "article", "relationship": "formalizes", "endorsement": "endorsed" }, { "title": "Volume estimates for unions of convex sets, and the Kakeya set conjecture in three dimensions", "id": "https://arxiv.org/abs/2502.17655", "authors": [ "Hong Wang", "Joshua Zahl" ], "type": "article", "relationship": "background", "endorsement": "endorsed" }, { "title": "The Assouad dimension of Kakeya sets in ℝ³", "id": "https://arxiv.org/abs/2401.12337", "authors": [ "Hong Wang", "Joshua Zahl" ], "type": "article", "relationship": "background", "endorsement": "endorsed" }, { "title": "Sticky Kakeya sets and the sticky Kakeya conjecture", "id": "https://arxiv.org/abs/2210.09581", "authors": [ "Hong Wang", "Joshua Zahl" ], "type": "article", "relationship": "background", "endorsement": "endorsed" } ], "related": [ { "id": "https://github.com/M32026/3d-sticky-kakeya", "relationship": "independent", "note": "A Lean 4 development by Nankai University and the ByteDance Seed AI4Math Team, written independently of this one. It proves Wang-Zahl, The Assouad dimension of Kakeya sets in R^3, Theorem 5.2 unconditionally, supplying the estimate this repository's main theorem assumes. It also states, without proving, the two Main Lemmas that this repository proves. The two developments are complementary and are" } ], "classification": { "arxiv": [ "math.CA", "math.MG" ], "msc2020": [ "28A75", "42B25", "52C17" ] }, "automation": { "methods": [ { "method": "agent", "models": [], "framework": "Project Numina Fuse" } ], "strongest": "agent", "models": [], "spend": "", "notes": "The development was assembled from Project Numina's Kakeya formalization; provenance and source revisions are recorded in ATTRIBUTION.md." }, "scope": "Proves that the three-dimensional Sticky Frostman hypothesis implies the three-dimensional Kakeya set conjecture, and discharges that hypothesis against a second development to obtain the conjecture unconditionally.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "KakeyaDimensionThree", "file": "Kakeya/DimensionThree/KakeyaConjecture.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [ { "statement": "The three-dimensional Sticky Frostman hypothesis", "source": "The Assouad dimension of Kakeya sets in ℝ³" } ] }, { "declaration": "Unconditional.KakeyaDimensionThree", "file": "Unconditional/KakeyaConjecture.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 1, "divergences": "(1) The main theorem explicitly assumes the three-dimensional Sticky Frostman hypothesis, whose proof is not formalized here; it is carried as an explicit argument rather than declared as an axiom. The unconditional endpoint discharges it against a second development, which is not vendored here and must be obtained separately. See PROOF-PATH.md for the exact formal statement and dependency chain. (2) Main Lemma 1 is deliberately not sharp: it yields the Frostman estimate at every exponent strictly above beta, never at beta itself, and the bootstrap pays for this by halving the step of Main Lemma 2. (3) Kakeya.katzTaoEstimateDimensionThree, KakeyaEstimateDimensionThree_pos and KakeyaEstimateDimensionThree are stated for index types in Type 0; Main Lemma 1 and Main Lemma 2 remain polymorphic in the index universe. (4) The ENNReal volume bounds of the sticky predicates use ENNReal.ofReal applied to real powers; no equivalence to variants using ENNReal.ofNNReal is claimed. (5) The every-scale conditions are read on the nodes of one nested cover hierarchy, following Wang-Zahl Definition 2.1, rather than on thickenings of the leaves.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "The repository includes source, build, axiom, and comparator checks. A configured check is not a completed verification result. Completed runs record the tested source identities and logs as described in RELEASING.md. No claim of external peer review is made." }, "canonical": { "repo": "project-numina/kakeya-3d", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "RemyDegenne/alpha-rar/formalization.yaml", "repo": "RemyDegenne/alpha-rar", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/RemyDegenne/alpha-rar/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "alpha-rar", "description": "A Lean 4 / Mathlib formalization of the two main theorems of \"On Response-Adaptive Targeting Strategies for Multi-Treatment Experiments\" (Yagouti, Degenne, Kaufmann, arXiv:2606.17777): Theorem 4.1, strong consistency with log-log rates, and Theorem 4.2, joint asymptotic normality of the allocation proportions and plug-in targets, for the αRTS family of response-adaptive designs. Also formalized: Section 3's three concrete designs (distance-based, ERADE 2025, interpolated D-Tracking); the forced-exploration variant αRTS-FE of Section 5 — Theorem 5.1, Theorem 5.2 and Corollary 5.3, the last unde", "authors": [ "Rémy Degenne" ], "maintainers": [ "Rémy Degenne" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "On Response-Adaptive Targeting Strategies for Multi-Treatment Experiments", "id": "arXiv:2606.17777", "authors": [ "Redouane Yagouti", "Rémy Degenne", "Emilie Kaufmann" ], "type": "preprint", "relationship": "formalizes", "endorsement": "participated" } ], "related": [], "classification": { "arxiv": [ "math.ST", "math.PR" ], "msc2020": [ "62L05", "62E20", "60F05", "60F15", "60G42" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "claude-opus-4-8", "claude-opus-5" ], "framework": "Claude Code" } ], "strongest": "agent", "models": [ "claude-opus-4-8", "claude-opus-5" ], "spend": "subscription-based usage", "notes": "All Lean in this repository was produced with Claude Code under human direction and review, as was the auxiliary blueprint and the choice of decomposition into small lemmas. The two divergences below came out of that collaboration. The agent found the gap in the paper's Lemma 4.5 (divergence 1), explored fixes, and reported that recovering Corollary 5.3 was blocked by condition (ii) of Definition 5.1, h(n) = o(√n). The human author — the paper's co-authors having already suspected that condition was unnecessarily restrictive — directed it to relax the condition by strengthening the proofs rath" }, "scope": "The two main theorems of the paper — Theorem 4.1 (strong consistency and rates) and Theorem 4.2 (asymptotic normality, both parts) — are formalized for the αRTS family, together with the forced-exploration variant αRTS-FE of Section 5 and the three concrete designs of Section 3. Formalized: the model, RAR procedures and the sequential estimator (1) of Sections 2 and 3, with Conditions A and B of Section 4; Definition 3.1 (the αRTS family, throttling condition (3)) and Examples 3.1–3.3 (distance-based, ERADE 2025, interpolated D-Tracking); the auxiliary process U of Section 4.2 and the technica", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "AlphaRAR.aRTS_LLN", "file": "AlphaRAR/YDK2026/ARTSRates.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "AlphaRAR.aRTS_clt_joint", "file": "AlphaRAR/YDK2026/ARTSNormality.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "AlphaRAR.aRTS_prop_dev", "file": "AlphaRAR/YDK2026/PropDevARTS.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "AlphaRAR.aRTS_prop_dev_ae", "file": "AlphaRAR/YDK2026/PropDevLIL.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "AlphaRAR.aRTSFE_sparse_clt_of_contDiffAt", "file": "AlphaRAR/YDK2026/ForcedExploration.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "AlphaRAR.aRTSFE_sparse_rate_of_isARTSFE", "file": "AlphaRAR/YDK2026/ForcedExploration.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "1. Lemma 4.5 (Joint CLT with componentwise scaling) — hypothesis changed. The paper states it, in Section 4.2, under the hypothesis that every treatment is sampled infinitely often (N_{n,k} → ∞ a.s.). That hypothesis is not sufficient: the proof in Appendix A.2 (`arxiv/sections/appendixA.tex`) verifies the self-normalized Lindeberg condition and then applies a martingale CLT to a randomly normalized martingale, which additionally needs a stabilization/regularity hypothesis that is never established, and a law-of-the-iterated- logarithm construction shows the conclusion genuinely fails without it. The formalized statement `AlphaRAR.estimatorError_joint_tendsto_multivariateGaussian` therefore assumes regularity, N_{n,k}/c_{k,n} → 1 for a deterministic sequence c. Theorem 4.1 and Theorem 4.2 are unaffected (regularity holds there for free). Write-up: `notes/regularity-gap.md`. 2. Definition 5.1 (αRTS-FE) — exploration-schedule condition reversed. To recover Corollary 5.3, condition (ii) of Definition 5.1, h(n) = o(√n), is replaced by (⋆) √(n log log n) ≪ h(n) = o(n). This is not a weakening in the paper's direction: it is a different condition, and the two overlap only trivially. It i", "alignment": true, "original": false, "review": { "status": "author-verified", "bucket": "author-verified", "reviewers": [ "Rémy Degenne" ], "notes": "No independent reviewer has audited the formalization. Each main result above additionally has a leanprover/comparator setup (its `comparator_config`): a short self-contained challenge file under `comparator/` restates the theorem with its definitional dependencies inlined — the import closure of every challenge bottoms out in Mathlib alone, with the few LML declarations the statements rest on vendored verbatim into the challenge and checked for identity — and comparator verifies mechanically that the project proves exactly that statement from exactly the three axioms listed, with the proof replayed through the Lean kernel. `scripts/comparator-verify.sh` runs all six; see `comparator/README.md` for the trust model." }, "canonical": { "repo": "remydegenne/alpha-rar", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "RemyDegenne/colt-2026-83/formalization.yaml", "repo": "RemyDegenne/colt-2026-83", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/RemyDegenne/colt-2026-83/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "colt-2026-83", "description": "A Lean 4 / Mathlib formalization of \"On the Power of Adaptivity for ε-Best Arm Identification in Linear Bandits\" (Maiti, Xu, Jamieson, COLT 2026): the fixed-design upper bound O((d log(1/δ) + w(𝒳)²)/ε²) in terms of a Gaussian width term w(𝒳) of the action set, the matching lower bounds Ω(d log(1/δ)/ε²) for every algorithm and Ω(w(𝒳)²/ε²) for non-adaptive fixed designs, the properties of w(𝒳), the structured action sets on which adaptivity gives at most logarithmic gains, the ℓ₂-norm estimation procedure, and the action set on which adaptivity gives a polynomial improvement. The linear bandit m", "authors": [ "Rémy Degenne" ], "maintainers": [ "Rémy Degenne" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "On the Power of Adaptivity for ε-Best Arm Identification in Linear Bandits", "id": "COLT 2026, paper 83", "authors": [ "Arnab Maiti", "Yunbei Xu", "Kevin Jamieson" ], "type": "conference-paper", "relationship": "formalizes", "endorsement": "not-contacted" } ], "related": [], "classification": { "arxiv": [ "cs.LG", "stat.ML", "math.ST" ], "msc2020": [ "62L05", "62K05", "68T05", "60G15" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "claude-fable-5" ], "framework": "Claude Code" } ], "strongest": "agent", "models": [ "claude-fable-5" ], "spend": "subscription-based usage", "notes": "All Lean and blueprint LaTeX in this repository is produced with Claude Code under human direction and review." }, "scope": "Phase 1 (statements). The headline results of the paper are stated in Lean, each as a `sorry`-ed theorem, over definitions written in library generality: Theorem 1 (`exists_isFixedDesign_isPAC`, blueprint `cor:upper_exists`/`thm:upper`): existence of a fixed-design (ε,δ)-PAC algorithm with budget ≤ 600 (w(𝒳)² + d log(2/δ))/ε² + 1. Theorem 2 (`le_budget_of_isPAC`, `thm:lower_adaptive`): every (ε,δ)-PAC algorithm, adaptive or not, has budget ≥ d log(1/δ)/(20000 ε²) when δ < 1/16. Theorem 3 (`le_budget_of_isFixedDesign_of_isPAC`, `thm:lower_nonadaptive`): every fixed-design (ε,δ)-PAC algorithm ha", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "Maiti2026Power.exists_isFixedDesign_isPAC", "file": "Maiti2026Power/MXJ2026/Upper.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Maiti2026Power.le_budget_of_isPAC", "file": "Maiti2026Power/MXJ2026/LowerAdaptive.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Maiti2026Power.le_budget_of_isFixedDesign_of_isPAC", "file": "Maiti2026Power/MXJ2026/LowerNonadaptive.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Maiti2026Power.gw_le_card", "file": "Maiti2026Power/MXJ2026/WidthBounds.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Maiti2026Power.width_separation", "file": "Maiti2026Power/MXJ2026/WidthBounds.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Maiti2026Power.exists_isPAC_of_finite", "file": "Maiti2026Power/MXJ2026/LogGains.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Maiti2026Power.multitaskSet_lt_budget_of_isPAC", "file": "Maiti2026Power/MXJ2026/LogGains.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Maiti2026Power.exists_isPAC_blockBallSet", "file": "Maiti2026Power/MXJ2026/Separation.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Maiti2026Power.exists_isAccurateNormEst", "file": "Maiti2026Power/MXJ2026/NormEstimation.lean", "description": "", "axioms": [], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "Statement-level choices (phase 1): 1. Model. The action set is a nonempty compact subset 𝒳 of a Euclidean space `EuclideanSpace ℝ ι` (d = card ι); the paper's standing spanning assumption `span 𝒳 = ℝ^d` is an explicit hypothesis of the results that need it, so that the non-spanning structured sets (multi-task sets) are covered. An identification algorithm (LML `Learning.IdentAlg Unit 𝒳 ℝ 𝒳`, since 2026-09-13) is an LML `Learning.Algorithm Unit 𝒳 ℝ` (a sequence of policy kernels) with a measurable *stopping rule* (a set of histories of variable length) and an *output kernel* from histories of variable length to recommendations; the number of rounds played is the hitting time (`hittingAfter`) of the stopping rule by the process of histories. A run of the algorithm on a probability space is an LML algorithm-environment sequence (`Learning.IsAlgEnvSeq`) together with an output whose conditional law given the history at the stopping time is the output rule (`IdentAlg.IsRun`); the law of the output is the same for all runs (`IdentAlg.outputMeasure`), and PAC (`IdentAlg.IsPAC`) says that under this law the recommendation has simple regret more than ε with probability at most δ — equivalen", "alignment": true, "original": false, "review": { "status": "in-progress", "bucket": "other", "reviewers": [ "Rémy Degenne" ], "notes": "Phase 1: the statements were written against the blueprint and the paper. Phase 2 (complete): all headline statements — Theorems 1, 2, 3, 5, 6, 7 and 8, Proposition 4 and the five lower bounds of Table 1 — are proved, with no `sorry` in the repository. Every headline statement is verified with leanprover/comparator (all 17 configs pass, 2026-09-06, run with `scripts/comparator-verify.sh --insecure`, i.e. without a landrun sandbox): the `comparator/` challenge files restate the theorems with their definitional dependencies inlined over a Mathlib-only import closure (configs listed as `comparator_config` above), and comparator checks that the project proves exactly the challenged statements with no axioms beyond `propext`, `Classical.choice` and `Quot.sound`. The library layers (`Maiti2026Po" }, "canonical": { "repo": "remydegenne/colt-2026-83", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "RemyDegenne/entropy-mdp/formalization.yaml", "repo": "RemyDegenne/entropy-mdp", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/RemyDegenne/entropy-mdp/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [ "automation", "review" ], "name": "entropy-mdp", "description": "A Lean 4 / Mathlib formalization of \"Tight Sample Complexity Bounds for Entropic Best Policy Identification\" (Amer Essakine, Claire Vernade, COLT 2026, arXiv:2605.13717): best-policy identification in finite-horizon tabular MDPs under the entropic risk measure, in the online (forward) model. The paper's two results are the lower bound Ω((e^{|β| G_max} - 1)² e^{-|β| G_max} S A H log(1/δ) / (e^{|β| ε} - 1)²) on the expected number of episodes of every (ε, δ)-PAC algorithm (Theorem 3) and the matching upper bound for the algorithm Entropic-BPI (KL-based bonuses in the exponential space and an ent", "authors": [ "Rémy Degenne" ], "maintainers": [ "Rémy Degenne" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "Tight Sample Complexity Bounds for Entropic Best Policy Identification", "id": "COLT 2026, arXiv:2605.13717", "authors": [ "Amer Essakine", "Claire Vernade" ], "type": "conference-paper", "relationship": "formalizes", "endorsement": "not-contacted" } ], "related": [], "classification": { "arxiv": [ "cs.LG", "stat.ML" ], "msc2020": [ "68T05", "90C40", "62L05", "62L10" ] }, "automation": { "methods": [], "strongest": "", "models": [], "spend": "", "notes": "" }, "scope": "Phase 1 (statements, 2026-09-16). The headline results of the paper are stated in Lean, each as a `sorry`-ed theorem, over definitions written in library generality (`Essakine2026Tight/LeanMachineLearning/ReinforcementLearning/MDP/`: finite-horizon MDPs on top of LML's `MDP`, entropic values, empirical models; `Essakine2026Tight/EV2026/`: the exploration rates, the Entropic-BPI algorithm, the run quantities and events, the constants): Theorem 3 (`Essakine2026Tight.exists_hardMDP_lowerBound_le_lintegral_stoppingTime`, blueprint `thm:lower_bound`): for `S ≥ 6`, `A ≥ 2`, `H ≥ 3 d`, `β ≠ 0`, `δ ∈ ", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "", "alignment": false, "original": false, "review": { "status": "", "bucket": "not stated", "reviewers": [], "notes": "" }, "canonical": { "repo": "remydegenne/entropy-mdp", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "rkirov/enumerative-chromatic-choosability/formalization.yaml", "repo": "rkirov/enumerative-chromatic-choosability", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/rkirov/enumerative-chromatic-choosability/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "List coloring and enumeratively chromatic-choosable graphs", "description": "A machine-checked development of list colouring in Lean 4. Give each vertex of a finite graph its own list of n permitted colours and count the proper colourings that respect those lists; the graph is enumeratively chromatic-choosable at n when that count is minimized by giving every vertex the same list. Kostochka and Sidorenko raised the question in 1990. The development formalizes Kirov and Naimi, \"List coloring and n-monophilic graphs\" (Ars Combinatoria 124 (2016), 329-340; arXiv:1004.5183), in the generality the paper states: every cycle is enumeratively chromatic-choosable at every n (th", "authors": [ "Radoslav Kirov" ], "maintainers": [ "Radoslav Kirov" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "List coloring and n-monophilic graphs", "id": "arXiv:1004.5183", "authors": [ "Radoslav Kirov", "Ramin Naimi" ], "type": "paper", "relationship": "formalizes", "endorsement": "participated" }, { "title": "Choosability in graphs", "id": "Congressus Numerantium 26 (1980), 125-157", "authors": [ "Paul Erdos", "Arthur L. Rubin", "Herbert Taylor" ], "type": "paper", "relationship": "formalizes", "endorsement": "" }, { "title": "On the number of list-colorings", "id": "Journal of Graph Theory 16 (3) (1992), 239-245", "authors": [ "Q. Donner" ], "type": "paper", "relationship": "formalizes", "endorsement": "" }, { "title": "Problem presented at the problem session, Fourth Czechoslovak Symposium on Combinatorics, Prachatice 1990", "id": "Annals of Discrete Mathematics 51 (1992), 380", "authors": [ "Alexandr V. Kostochka", "Alexander F. Sidorenko" ], "type": "paper", "relationship": "background", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.CO", "cs.LO" ], "msc2020": [ "05C15", "05C31", "05C38", "68V20" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "claude-opus-5", "claude-fable-5" ], "framework": "Claude Code" }, { "method": "manual", "models": [], "framework": "" } ], "strongest": "agent", "models": [ "claude-fable-5", "claude-opus-5" ], "spend": "", "notes": "" }, "scope": "ListColoring/ formalizes the source paper and the results it quotes; Cacti/ proves the cactus classification and Ladder/ the ladder theorem, both importing ListColoring/, never the reverse. OpenProblems.lean is a separate library, imported by nothing, that states the source paper's two open questions and asserts them with sorry: there a sorry records that the question is open, not that a proof is missing. The three sorries counted below are exactly those.", "sorry_count": 3, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "SimpleGraph.eval_chromaticPolynomial", "file": "ListColoring/ChromaticPolynomial.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SimpleGraph.ecc_iff_listColorFunction_eq_eval", "file": "ListColoring/ListColorFunction.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SimpleGraph.ERT.not_choosable", "file": "ListColoring/NotChoosable.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SimpleGraph.ERT.colorable", "file": "ListColoring/NotChoosable.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SimpleGraph.KN5.exists_choosable_not_ecc_of_two_le", "file": "ListColoring/Section5.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SimpleGraph.ecc_of_isChordal", "file": "ListColoring/Dirac.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SimpleGraph.ecc_cycleGraph_of_three_le", "file": "ListColoring/CycleGraph.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SimpleGraph.rubinTheorem", "file": "ListColoring/CoreGraphs.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SimpleGraph.ecc_two_iff", "file": "ListColoring/CoreGraphs.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SimpleGraph.exists_ecc_forall_ge", "file": "ListColoring/Threshold.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "ListColoring.isCactus_ecc_of_three_le", "file": "Cacti/Statements.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "ListColoring.isCactus_ecc_two_iff", "file": "Cacti/Statements.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "ListColoring.isCactus_ecc_iff", "file": "Cacti/Statements.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "ListColoring.ecc_boxProd_pathG_one", "file": "Ladder/Statements.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "ListColoring.colConst_boxProd_pathG_one", "file": "Ladder/Statements.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "ListColoring.ecc_boxProd_pathG_two", "file": "Grid3/Statements.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "ListColoring.colConst_boxProd_pathG_two", "file": "Grid3/Statements.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "ListColoring.ecc_boxProd_pathG_two_of_four", "file": "Grid3/Statements.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "ListColoring.ecc_boxProd_pathG_two_of_three", "file": "Grid3/Three/Main.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "Three deliberate deviations from the source paper, each recorded with a `deviation` field on the entry below: Lemma 3(c) is proved for n >= 3 rather than for all n; Section 5's Lemmas 7, 9 and 10 are proved for n >= 2 rather than the paper's n >= 1, because they are false at n = 1 (see `errata`, with machine-checked counterexamples in the library); and Theorem 1 is stated on Mathlib's cycleGraph as well as on the development's own closePath. Nothing downstream is weakened: Section 5's conclusion still holds at every list size k >= 2. The cactus results of Cacti/ have no source to diverge from.", "alignment": true, "original": false, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [ "Claude (adversarial review agents, Claude Code)" ], "notes": "Each landed proof was re-checked by an agent instructed to refute it, working from a fresh copy and required to reproduce the compile, to run #print axioms on every declaration, and to confirm that the statement proved is the statement claimed. Statements were also checked by brute-force evaluation before being proved, which repeatedly caught indexing and orientation errors that do not surface as type errors; those checks are #guard commands in the library. In addition, leanprover/comparator runs in CI on every commit against comparator/Challenge.lean, checking statement identity, the permitted axioms, and replay of the exported proofs through Lean's kernel (the independent NanoDa kernel is not run: a second sequential replay of the k = 3 certificate does not fit CI's time limit). No indep" }, "canonical": { "repo": "rkirov/enumerative-chromatic-choosability", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "rkirov/jordan_pick/formalization.yaml", "repo": "rkirov/jordan_pick", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/rkirov/jordan_pick/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "jordan_pick — Pick's theorem, the Jordan curve theorem, and Radó's theorem (Lean 4 / Mathlib)", "description": "A Lean 4 / Mathlib development of three classical plane- and surface-geometry theorems, each absent from Mathlib at the pinned revision and each submitted to Palomar as its own entry. Pick's theorem: a simple lattice polygon has area I + B/2 - 1, with area the genuine Lebesgue measure of the enclosed region rather than the shoelace formula taken as a definition. The Jordan curve theorem: a continuous injection of the circle into the plane has a complement with exactly two connected components, proved by Maehara's route with the two-dimensional Brouwer fixed point theorem built from the ground ", "authors": [ "Rado Kirov" ], "maintainers": [ "Rado Kirov" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "Geometrisches zur Zahlenlehre (the original statement of Pick's theorem)", "id": "Sitzungsberichte des deutschen naturwissenschaftlich-medicinischen Vereines für Böhmen \"Lotos\" in Prag, (Neue Folge) 19 (1899), 311-319. No DOI or stable digital copy is known to us; this is the standard bibliographic reference for the theorem's first appearance.", "authors": [ "Georg Alexander Pick" ], "type": "paper", "relationship": "formalizes", "endorsement": "n/a" }, { "title": "Cours d'analyse de l'École Polytechnique, Vol. III (the original statement of the Jordan curve theorem)", "id": "Gauthier-Villars, Paris, 2nd ed. (1887), pp. 587-594", "authors": [ "Camille Jordan" ], "type": "book", "relationship": "formalizes", "endorsement": "n/a" }, { "title": "Theory on plane curves in non-metrical analysis situs (the first rigorous proof of the Jordan curve theorem)", "id": "Trans. Amer. Math. Soc. 6 (1905), 83-98; doi:10.1090/S0002-9947-1905-1500697-4", "authors": [ "Oswald Veblen" ], "type": "paper", "relationship": "background", "endorsement": "n/a" }, { "title": "Über den Begriff der Riemannschen Fläche (the original statement of Radó's theorem)", "id": "Acta Litt. Sci. Reg. Univ. Hung. Francisco-Josephinae, Sect. Sci. Math. (Szeged) 2 (1925), 101-121. No DOI known to us; this is the standard bibliographic reference.", "authors": [ "Tibor Radó" ], "type": "paper", "relationship": "formalizes", "endorsement": "n/a" }, { "title": "Uniformization of Riemann surfaces revisited", "id": "arXiv:2008.12189v2; Ann. Global Anal. Geom. 62 (2022), 603-615; doi:10.1007/s10455-022-09860-2", "authors": [ "Cipriana Anghel", "Rareş Stan" ], "type": "paper", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "The Jordan curve theorem via the Brouwer fixed point theorem", "id": "Amer. Math. Monthly 91(10) (1984), 641-643; doi:10.2307/2323369", "authors": [ "Ryuji Maehara" ], "type": "paper", "relationship": "formalizes", "endorsement": "n/a" }, { "title": "Polygons Have Ears", "id": "Amer. Math. Monthly 82(6) (1975), 648-651; doi:10.2307/2319703", "authors": [ "Gary H. Meisters" ], "type": "paper", "relationship": "background", "endorsement": "n/a" }, { "title": "Lectures on Riemann Surfaces (GTM 81), §22-23", "id": "Springer GTM 81 (1981); doi:10.1007/978-1-4612-5961-9", "authors": [ "Otto Forster" ], "type": "book", "relationship": "background", "endorsement": "n/a" }, { "title": "Riemann Surfaces (Vienna lecture notes)", "id": "https://www.mat.univie.ac.at/~armin/lect/Riemann_surfaces.pdf", "authors": [ "Armin Rainer" ], "type": "other", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Teichmüller theory and applications to geometry, topology, and dynamics, Vol. 1, §1.2-1.3", "id": "Matrix Editions (2006), ISBN 978-0-9715766-2-9", "authors": [ "John Hamal Hubbard" ], "type": "book", "relationship": "background", "endorsement": "n/a" }, { "title": "Formalizing Pick's Theorem, efficiently", "id": "arXiv:2603.23095", "authors": [ "Michael Eisermann" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "lean-eval problem `pick` (the exact formal statement submitted)", "id": "https://lean-lang.org/eval/problems/pick/", "authors": [ "Lean FRO (leanprover/lean-eval)" ], "type": "other", "relationship": "other", "endorsement": "not-contacted" }, { "title": "lean-eval problem `jordan_curve` (the exact formal statement submitted)", "id": "https://lean-lang.org/eval/problems/jordan_curve/", "authors": [ "Lean FRO (leanprover/lean-eval)" ], "type": "other", "relationship": "other", "endorsement": "not-contacted" }, { "title": "lean-eval problem `rado_riemannSurface` (the exact formal statement submitted)", "id": "https://lean-lang.org/eval/problems/rado_riemannSurface/", "authors": [ "Junyan Xu (problem submitter)", "Lean FRO (leanprover/lean-eval)" ], "type": "other", "relationship": "other", "endorsement": "not-contacted" } ], "related": [ { "id": "arXiv:2603.23095 — Formalizing Pick's Theorem, efficiently (Michael Eisermann; Lean 4)", "relationship": "adapts", "note": "The nearest prior Lean work on Pick's theorem. This development adapts its discrete-angle-weight device on the count side (`dang` -> `latWeight`, `Welp` -> `latWeightSum`) but ports no code and proves the per-edge identity by a different decomposition. On our reading of that development, the geometric half — the polygonal Jordan curve theorem and the ear-clipping reduction — is left unproved there" }, { "id": "John Harrison, A formal proof of Pick's theorem, Math. Struct. Comput. Sci. 21(4) (2011), 715-729; doi:10.1017/S0960129511000089 (HOL Light)", "relationship": "independent", "note": "Prior formalization of Pick's theorem in a different system. Consulted as prior art only; no code, definitions, or proof structure were ported, and nothing here depends on it." }, { "id": "arXiv:2405.01793 — Formalizing Pick's Theorem in Isabelle/HOL (Sage Binder, Katherine Kosaian); doi:10.1007/978-3-031-66997-2_7; AFP entry `Picks_Theorem`", "relationship": "independent", "note": "Prior formalization of Pick's theorem in a different system. Consulted as prior art only; no code was ported." }, { "id": "Jordan curve theorem, Mizar Mathematical Library (completed 2005 by a team including Artur Korniłowicz, Yatsuka Nakamura and Andrzej Trybulec) — https://mizar.uwb.edu.pl/", "relationship": "independent", "note": "The Jordan curve theorem has been formalized before, notably in Mizar and (by Harrison) in HOL Light. No novelty is claimed for the theorem or for its formalization in general; the claim made here is narrower — that it is absent from Mathlib at the pinned revision." }, { "id": "Lawrence C. Paulson, `Jordan_Curve` in Isabelle/HOL HOL-Analysis (derived from Harrison's HOL Light development)", "relationship": "independent", "note": "As above: prior art for the Jordan curve theorem in another system; no code ported." }, { "id": "lean-eval `rado_riemannSurface` leaderboard entry, Aristotle (Harmonic), 2026-06-22", "relationship": "independent", "note": "An earlier automated solution of the same eval problem. Its submission repository is private and its proof was never seen; this development is clean-room with respect to it. Recorded so that priority on the eval problem is not implicitly claimed here." }, { "id": "lean-eval `rado_riemannSurface` leaderboard entry, Seed Prover (ByteDance), 2026-06-28", "relationship": "independent", "note": "As above: private repository, proof never seen, clean-room with respect to it." } ], "classification": { "arxiv": [ "math.GT", "math.CV" ], "msc2020": [ "52B20", "52C05", "55M25", "54F65", "54H25", "30F15", "30F20", "54D65" ] }, "automation": { "methods": [ { "method": "autonomous", "models": [ "claude-fable-5", "claude-opus-4-8" ], "framework": "Claude Code" } ], "strongest": "autonomous", "models": [ "claude-fable-5", "claude-opus-4-8" ], "spend": "subscription-based usage ($100/month subscription; no separate API spend)", "notes": "Human involvement is direction, statement selection, architecture review, and acceptance; the Lean text is model-produced. A human did not line-by-line audit the ~33.5k lines of proof, which is why the assurance claimed here rests on the kernel and the Comparator rather than on human reading — see `review`. Size: ~33.5k lines of Lean across 39 files (JordanPick: Pick, the polygonal and continuous JCT, Brouwer; Rado: Radó's theorem), sorry-free, with no custom axioms, no `native_decide` or other compiled-evaluation escape hatch, and no compiler warnings. Rado/ additionally sets `autoImplicit fa" }, "scope": "Three results are submitted, each as its own Palomar entry against its own Comparator configuration: Pick's theorem for simple lattice polygons with area as genuine Lebesgue measure (area = I + B/2 - 1; Freek Wiedijk's Formalizing 100 Theorems #92), the continuous Jordan curve theorem (a continuous injection S^1 -> R^2 has a complement with exactly two connected components), and Radó's theorem (every connected Hausdorff Riemann surface is second countable). The supporting repository additionally proves the polygonal Jordan curve theorem, the 2-dimensional Brouwer fixed point theorem, the Poinc", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "pick", "file": "submission/pick/Challenge.lean (statement); submission/pick/Solution.lean (proof)", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "jordan_curve", "file": "submission/jordan_curve/Challenge.lean (statement); submission/jordan_curve/Solution.lean (proof)", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "rado_riemannSurface", "file": "submission/rado_riemannSurface/Challenge.lean (statement); submission/rado_riemannSurface/Solution.lean (proof)", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Pick.pick", "file": "JordanPick/PicksTheorem/Pick.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "Pick.compl_boundary_atMost_two", "file": "JordanPick/PicksTheorem/Pick.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "JordanCurve.Brouwer.brouwerFPT", "file": "JordanPick/JordanCurve/Brouwer.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "Rado.poincare_volterra", "file": "Rado/Topology/PoincareVolterra.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "Rado.exists_harmonic_extension", "file": "Rado/Complex/Poisson.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "Rado.IsPerronFamily.surfaceHarmonicOn_perronSup", "file": "Rado/Surface/Perron.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "Pick's theorem. Area is the genuine Lebesgue measure of the winding interior, not the shoelace formula taken as a definition; the shoelace identity is a proved theorem (Green: area = the double integral of the winding number = shoelace). The spine of the development is the winding number as a per-edge signed ray-crossing sum — pure integer arithmetic, with no transcendental angles and no general topology — and the area identity is reduced to triangles by ear-clipping induction (Meisters' two-ears theorem, realized via a deepest-contained-vertex diagonal split rather than Meisters' own construction). The count side follows Eisermann's discrete-angle-weight device; the per-edge identity is proved by a column decomposition rather than his four-box partition with a reflection involution. The library theorem Pick.pick assumes positive orientation; the submitted `pick` does not, and the bridge in EvalBridgeMain.lean discharges that hypothesis. Jordan curve theorem. This is a separate development, independent of the polygonal one: it follows Maehara's Brouwer-based proof, with Brouwer itself built from the ground up against Mathlib (covering-space path lifting for the circle, no-retractio", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "Verification here is by Lean's kernel, not by a human or by peer review. Specifically: `#print axioms` on every main result reports exactly [propext, Classical.choice, Quot.sound] with no sorryAx; `lake build` of the submitted libraries (JordanPick/ and Rado/) is green and warning-free; and a repository-wide sweep confirms no `sorry` outside Uniformization/. That unfinished directory is not warning-free — Mathlib v4.33 added style and unused-variable linters that fire there — but it is not submitted and nothing above depends on it. Every commit additionally runs the comparator over all three workspaces in CI with the independent NanoDa kernel forced on, so each Solution is replayed by two kernels rather than one. An earlier revision of the Radó development also passed the lean-eval compara" }, "canonical": { "repo": "rkirov/jordan_pick", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "rkirov/jordan_pick/submission/jordan_curve/formalization.yaml", "repo": "rkirov/jordan_pick", "path": "submission/jordan_curve/formalization.yaml", "directory": "submission/jordan_curve", "origins": [ "palomar", "search" ], "url": "https://github.com/rkirov/jordan_pick/blob/HEAD/submission/jordan_curve/formalization.yaml", "version": "v0.4", "missing": [], "name": "jordan_pick — the Jordan curve theorem (Lean 4 / Mathlib)", "description": "The Jordan curve theorem, formalized in Lean 4 against Mathlib: a continuous injective map from the circle into the plane has a complement with exactly two connected components. The hypothesis is bare continuity and injectivity — no smoothness, piecewise-linearity or rectifiability — which is what makes the theorem hard and what separates it from the polygonal case. The proof follows Maehara's reduction to the Brouwer fixed point theorem, via a crossing lemma for transversal paths in a rectangle and the fact that every component has the curve as its boundary, together with a farthest-pair norm", "authors": [ "Rado Kirov" ], "maintainers": [ "Rado Kirov" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "Cours d'analyse de l'École Polytechnique, Vol. III (the original statement)", "id": "Gauthier-Villars, Paris, 2nd ed. (1887), pp. 587-594", "authors": [ "Camille Jordan" ], "type": "book", "relationship": "formalizes", "endorsement": "n/a" }, { "title": "Theory on plane curves in non-metrical analysis situs (the first rigorous proof)", "id": "Trans. Amer. Math. Soc. 6 (1905), 83-98; doi:10.1090/S0002-9947-1905-1500697-4", "authors": [ "Oswald Veblen" ], "type": "paper", "relationship": "background", "endorsement": "n/a" }, { "title": "The Jordan curve theorem via the Brouwer fixed point theorem", "id": "Amer. Math. Monthly 91(10) (1984), 641-643; doi:10.2307/2323369", "authors": [ "Ryuji Maehara" ], "type": "paper", "relationship": "formalizes", "endorsement": "n/a" }, { "title": "lean-eval problem `jordan_curve` (the exact formal statement submitted)", "id": "https://lean-lang.org/eval/problems/jordan_curve/", "authors": [ "Lean FRO (leanprover/lean-eval)" ], "type": "other", "relationship": "other", "endorsement": "not-contacted" } ], "related": [ { "id": "Jordan curve theorem, Mizar Mathematical Library (completed 2005 by a team including Artur Korniłowicz, Yatsuka Nakamura and Andrzej Trybulec) — https://mizar.uwb.edu.pl/", "relationship": "independent", "note": "The theorem has been formalized before, notably in Mizar and by Harrison in HOL Light. No novelty is claimed for the theorem or for formalizing it in general; the claim made here is narrower — that it is absent from Mathlib at the pinned revision." }, { "id": "Lawrence C. Paulson, `Jordan_Curve` in Isabelle/HOL HOL-Analysis (derived from Harrison's HOL Light development)", "relationship": "independent", "note": "Prior art in another system; no code ported." } ], "classification": { "arxiv": [ "math.GT", "math.GN" ], "msc2020": [ "54F65", "54F15", "54H25", "55M20", "55M25", "54D05" ] }, "automation": { "methods": [ { "method": "autonomous", "models": [ "claude-fable-5", "claude-opus-4-8" ], "framework": "Claude Code" } ], "strongest": "autonomous", "models": [ "claude-fable-5", "claude-opus-4-8" ], "spend": "subscription-based usage ($100/month subscription; no separate API spend)", "notes": "Human involvement is direction, statement selection, architecture review, and acceptance; the Lean text is model-produced. A human did not line-by-line audit the development, which is why the assurance claimed here rests on the kernel and the Comparator rather than on human reading — see `review`. The JordanCurve development is sorry-free, with no custom axioms, no `native_decide` or other compiled-evaluation escape hatch, and no compiler warnings; no file in it or in this workspace contains a `maxHeartbeats` override, so every proof elaborates inside Lean's default 200k budget. The workspace " }, "scope": "The Jordan curve theorem: a continuous injection of the circle into the plane has a complement with exactly two connected components. The hypothesis is bare continuity plus injectivity — no smoothness, piecewise-linearity, or rectifiability. Not present in Mathlib at the pinned revision db584cd6, checked by searching that checkout; neither is the 2-dimensional Brouwer fixed point theorem the proof rests on, which is therefore built from the ground up here. No novelty is claimed for the theorem, which is classical, nor for formalizing it in general: it has been formalized in Mizar, HOL Light an", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "jordan_curve", "file": "Challenge.lean (statement); Solution.lean (proof)", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "JordanCurve.Brouwer.brouwerFPT", "file": "Submission/Brouwer.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "This is a separate development from the repository's polygonal Jordan curve theorem and does not use it. It follows Maehara: the separation statement is reduced to the Brouwer fixed point theorem via a crossing lemma for transversal paths in a rectangle and the fact that each component has the curve as its boundary, plus a farthest-pair normalization and the `l, m, p, q, z₀` construction that pins down exactly one bounded component. Because Mathlib has no Brouwer fixed point theorem at the pin, Brouwer is built from the ground up: covering-space path lifting for the circle (π₁(S¹) ≅ ℤ) → no retraction of the disk onto its boundary → Brouwer on the disk by ray-retraction → the general convex-compact case by nearest-point projection.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "Verification here is by Lean's kernel, not by a human or by peer review: `#print axioms jordan_curve` reports exactly [propext, Classical.choice, Quot.sound] with no sorryAx; `lake build` of the library and of this workspace is green and warning-free; and a repository-wide sweep confirms no `sorry` outside Uniformization/ and the deliberate Challenge hole. Every commit additionally runs Comparator over this workspace in CI with the independent NanoDa kernel forced on, so the Solution is replayed by two kernels rather than one. What none of this establishes is that a mathematician has read the proof: no expert review has taken place, and the informal account here was written by the same process that produced the Lean. Expert review is welcome." }, "canonical": { "repo": "rkirov/jordan_pick", "directory": "submission/jordan_curve" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-19-000001", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-19-000001", "commit": "b3c9b7cf7358bf81a077d78ad67e6e8247869ddd", "trust": "high", "theorems": 1, "date": "2026-08-19" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "rkirov/jordan_pick/submission/pick/formalization.yaml", "repo": "rkirov/jordan_pick", "path": "submission/pick/formalization.yaml", "directory": "submission/pick", "origins": [ "search" ], "url": "https://github.com/rkirov/jordan_pick/blob/HEAD/submission/pick/formalization.yaml", "version": "v0.4", "missing": [], "name": "jordan_pick — Pick's theorem (Lean 4 / Mathlib)", "description": "Pick's theorem, formalized in Lean 4 against Mathlib: a simple polygon whose vertices lie on the integer lattice has area I + B/2 - 1, where I counts the lattice points strictly inside it and B those on its boundary. Area here is the genuine Lebesgue measure of the enclosed region, not the shoelace formula taken as a definition; the shoelace identity is itself a proved theorem, obtained from Green's theorem as area = the double integral of the winding number = shoelace. The spine of the development is the winding number realized as a per-edge signed ray-crossing sum, which is pure integer arit", "authors": [ "Rado Kirov" ], "maintainers": [ "Rado Kirov" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "Geometrisches zur Zahlenlehre (the original statement of Pick's theorem)", "id": "Sitzungsberichte des deutschen naturwissenschaftlich-medicinischen Vereines für Böhmen \"Lotos\" in Prag, (Neue Folge) 19 (1899), 311-319. No DOI or stable digital copy is known to us; this is the standard bibliographic reference for the theorem's first appearance.", "authors": [ "Georg Alexander Pick" ], "type": "paper", "relationship": "formalizes", "endorsement": "n/a" }, { "title": "Polygons Have Ears", "id": "Amer. Math. Monthly 82(6) (1975), 648-651; doi:10.2307/2319703", "authors": [ "Gary H. Meisters" ], "type": "paper", "relationship": "background", "endorsement": "n/a" }, { "title": "Formalizing Pick's Theorem, efficiently", "id": "arXiv:2603.23095", "authors": [ "Michael Eisermann" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "lean-eval problem `pick` (the exact formal statement submitted)", "id": "https://lean-lang.org/eval/problems/pick/", "authors": [ "Lean FRO (leanprover/lean-eval)" ], "type": "other", "relationship": "other", "endorsement": "not-contacted" } ], "related": [ { "id": "arXiv:2603.23095 — Formalizing Pick's Theorem, efficiently (Michael Eisermann; Lean 4)", "relationship": "adapts", "note": "The nearest prior Lean work. This development adapts its discrete-angle-weight device on the count side (`dang` -> `latWeight`, `Welp` -> `latWeightSum`) but ports no code and proves the per-edge identity by a different decomposition. On our reading of that development, the geometric half — the polygonal Jordan curve theorem and the ear-clipping reduction — is left unproved there; that half is wha" }, { "id": "John Harrison, A formal proof of Pick's theorem, Math. Struct. Comput. Sci. 21(4) (2011), 715-729; doi:10.1017/S0960129511000089 (HOL Light)", "relationship": "independent", "note": "Prior formalization in a different system. Consulted as prior art only; no code, definitions, or proof structure were ported, and nothing here depends on it." }, { "id": "arXiv:2405.01793 — Formalizing Pick's Theorem in Isabelle/HOL (Sage Binder, Katherine Kosaian); doi:10.1007/978-3-031-66997-2_7; AFP entry `Picks_Theorem`", "relationship": "independent", "note": "Prior formalization in a different system. Consulted as prior art only; no code ported." } ], "classification": { "arxiv": [ "math.MG", "math.CO" ], "msc2020": [ "52B20", "52C05", "51M25", "11H06", "11P21", "55M25" ] }, "automation": { "methods": [ { "method": "autonomous", "models": [ "claude-fable-5", "claude-opus-4-8" ], "framework": "Claude Code" } ], "strongest": "autonomous", "models": [ "claude-fable-5", "claude-opus-4-8" ], "spend": "subscription-based usage ($100/month subscription; no separate API spend)", "notes": "Human involvement is direction, statement selection, architecture review, and acceptance; the Lean text is model-produced. A human did not line-by-line audit the development, which is why the assurance claimed here rests on the kernel and the Comparator rather than on human reading — see `review`. The Pick development is sorry-free, with no custom axioms, no `native_decide` or other compiled-evaluation escape hatch, and no compiler warnings; no file in it or in this workspace contains a `maxHeartbeats` override, so every proof elaborates inside Lean's default 200k budget. The workspace is mech" }, "scope": "Pick's theorem (Freek Wiedijk's Formalizing 100 Theorems #92): a simple polygon whose vertices lie on the integer lattice has area I + B/2 - 1, where I and B count the interior and boundary lattice points. Area here is the genuine Lebesgue measure of the enclosed region, not the shoelace formula taken as a definition; the shoelace identity is itself a proved theorem. Not present in Mathlib at the pinned revision db584cd6, checked by searching that checkout. No novelty is claimed for the theorem, which is classical, nor for formalizing it in general: it has been formalized in HOL Light and Isab", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "pick", "file": "Challenge.lean (statement); Solution.lean (proof)", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Pick.pick", "file": "Submission/Pick.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "Pick.compl_boundary_atMost_two", "file": "Submission/Pick.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "Area is the genuine Lebesgue measure of the winding interior, not the shoelace formula taken as a definition; the shoelace identity is a proved theorem (Green: area = the double integral of the winding number = shoelace). The spine of the development is the winding number as a per-edge signed ray-crossing sum — pure integer arithmetic, with no transcendental angles and no general topology — and the area identity is reduced to triangles by ear-clipping induction (Meisters' two-ears theorem, realized via a deepest-contained-vertex diagonal split rather than Meisters' own construction). The count side follows Eisermann's discrete-angle-weight device; the per-edge identity is proved by a column decomposition rather than his four-box partition with a reflection involution. The library theorem Pick.pick assumes positive orientation; the submitted `pick` does not, and the bridge discharges that hypothesis by vertex reversal.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "Verification here is by Lean's kernel, not by a human or by peer review: `#print axioms pick` reports exactly [propext, Classical.choice, Quot.sound] with no sorryAx; `lake build` of the library and of this workspace is green and warning-free; and a repository-wide sweep confirms no `sorry` outside Uniformization/ and the deliberate Challenge hole. Every commit additionally runs Comparator over this workspace in CI with the independent NanoDa kernel forced on, so the Solution is replayed by two kernels rather than one. What none of this establishes is that a mathematician has read the proof: no expert review has taken place, and the informal account here was written by the same process that produced the Lean. Expert review is welcome." }, "canonical": { "repo": "rkirov/jordan_pick", "directory": "submission/pick" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "rkirov/jordan_pick/submission/rado_riemannSurface/formalization.yaml", "repo": "rkirov/jordan_pick", "path": "submission/rado_riemannSurface/formalization.yaml", "directory": "submission/rado_riemannSurface", "origins": [ "palomar", "search" ], "url": "https://github.com/rkirov/jordan_pick/blob/HEAD/submission/rado_riemannSurface/formalization.yaml", "version": "v0.4", "missing": [], "name": "jordan_pick — Radó's theorem (Lean 4 / Mathlib)", "description": "Radó's theorem, formalized in Lean 4 against Mathlib: every connected Hausdorff Riemann surface is second countable. Second countability is normally imposed as part of the definition of a Riemann surface; Radó's theorem shows the assumption is redundant, because the one-dimensional complex-analytic structure forces it. The analogous statement fails for real surfaces — the Prüfer surface is a connected Hausdorff real-analytic 2-manifold that is not second countable — so any proof must be genuinely complex-analytic. This one follows the Perron treatment of Anghel and Stan: Perron's method on an ", "authors": [ "Rado Kirov" ], "maintainers": [ "Rado Kirov" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "Über den Begriff der Riemannschen Fläche (the original statement of Radó's theorem)", "id": "Acta Litt. Sci. Reg. Univ. Hung. Francisco-Josephinae, Sect. Sci. Math. (Szeged) 2 (1925), 101-121. No DOI known to us; this is the standard bibliographic reference.", "authors": [ "Tibor Radó" ], "type": "paper", "relationship": "formalizes", "endorsement": "n/a" }, { "title": "Uniformization of Riemann surfaces revisited", "id": "arXiv:2008.12189v2; Ann. Global Anal. Geom. 62 (2022), 603-615; doi:10.1007/s10455-022-09860-2", "authors": [ "Cipriana Anghel", "Rareş Stan" ], "type": "paper", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "Lectures on Riemann Surfaces (GTM 81), §22-23", "id": "Springer GTM 81 (1981); doi:10.1007/978-1-4612-5961-9", "authors": [ "Otto Forster" ], "type": "book", "relationship": "background", "endorsement": "n/a" }, { "title": "Riemann Surfaces (Vienna lecture notes)", "id": "https://www.mat.univie.ac.at/~armin/lect/Riemann_surfaces.pdf", "authors": [ "Armin Rainer" ], "type": "other", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Teichmüller theory and applications to geometry, topology, and dynamics, Vol. 1, §1.2-1.3", "id": "Matrix Editions (2006), ISBN 978-0-9715766-2-9", "authors": [ "John Hamal Hubbard" ], "type": "book", "relationship": "background", "endorsement": "n/a" }, { "title": "lean-eval problem `rado_riemannSurface` (the exact formal statement submitted)", "id": "https://lean-lang.org/eval/problems/rado_riemannSurface/", "authors": [ "Junyan Xu (problem submitter)", "Lean FRO (leanprover/lean-eval)" ], "type": "other", "relationship": "other", "endorsement": "not-contacted" } ], "related": [ { "id": "lean-eval `rado_riemannSurface` leaderboard entry, Aristotle (Harmonic), 2026-06-22", "relationship": "independent", "note": "An earlier automated solution of the same problem statement. Its submission repository is private and its proof was never seen; this development is clean-room with respect to it. Recorded so that priority is not implicitly claimed here." }, { "id": "lean-eval `rado_riemannSurface` leaderboard entry, Seed Prover (ByteDance), 2026-06-28", "relationship": "independent", "note": "As above: private repository, proof never seen, clean-room with respect to it." } ], "classification": { "arxiv": [ "math.CV", "math.GN" ], "msc2020": [ "30F20", "30F15", "30F99", "54D65", "32Q55", "31A05" ] }, "automation": { "methods": [ { "method": "autonomous", "models": [ "claude-fable-5", "claude-opus-4-8" ], "framework": "Claude Code" } ], "strongest": "autonomous", "models": [ "claude-fable-5", "claude-opus-4-8" ], "spend": "subscription-based usage ($100/month subscription; no separate API spend)", "notes": "Human involvement is direction, statement selection, architecture review, and acceptance; the Lean text is model-produced. A human did not line-by-line audit the Rado/ development, which is why the assurance claimed here rests on the kernel and the Comparator rather than on human reading — see `review`. Rado/ is ~13k lines across 13 files, sorry-free, with no custom axioms, no `native_decide` or other compiled-evaluation escape hatch, no compiler warnings, `autoImplicit false` throughout, and `#lint` clean. No file in Rado/ or in this workspace contains a `maxHeartbeats` override, so every pro" }, "scope": "Radó's theorem: every connected Hausdorff Riemann surface is second countable. Second countability is usually imposed as part of the definition of a Riemann surface; Radó's theorem says the assumption is redundant, and that the complex structure alone forces it. The corresponding statement is false for real surfaces — the Prüfer surface is a connected Hausdorff real-analytic 2-manifold that is not second countable — so the proof must use complex analysis, and this one does. Not present in Mathlib at the pinned revision db584cd6, checked by searching that checkout; that absence is why the resul", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "rado_riemannSurface", "file": "Challenge.lean (statement); Solution.lean (proof)", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "isManifold_omega_of_one", "file": "Submission/Surface/HolomorphicCompat.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "Rado.poincare_volterra", "file": "Submission/Topology/PoincareVolterra.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "Rado.exists_harmonic_extension", "file": "Submission/Complex/Poisson.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "Rado.IsPerronFamily.surfaceHarmonicOn_perronSup", "file": "Submission/Surface/Perron.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The proof follows Anghel-Stan's Perron treatment with two formalization-motivated deviations, both recorded in Rado/PLAN.md as route \"A'\": (i) an explicit two-disk configuration with log-barriers — unit disks at ±4 inside the chart ball B(0,8), witnesses at ±4 + 2^(1/4) — in place of general Dirichlet boundary regularity, and (ii) an étale space of exact harmonic-conjugate germs (Re F = u exactly, so sheets differ by imaginary constants) in place of the universal cover, with descent via Poincaré-Volterra. Both deviations change the proof, not the theorem: the statement proved is the standard one.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "Verification here is by Lean's kernel, not by a human or by peer review. Specifically: `#print axioms rado_riemannSurface` reports exactly [propext, Classical.choice, Quot.sound] with no sorryAx; `lake build` of Rado/ and of this workspace is green and warning-free; and a repository-wide sweep confirms no `sorry` outside Uniformization/ and the deliberate Challenge hole. Every commit additionally runs Comparator over this workspace in CI with the independent NanoDa kernel forced on, so the Solution is replayed by two kernels rather than one. An earlier revision passed the lean-eval comparator on 2026-07-10, under that harness's pin (Lean v4.32.2, Mathlib 905b9581) rather than this commit's. What none of this establishes is that a mathematician has read the proof: no expert review has taken" }, "canonical": { "repo": "rkirov/jordan_pick", "directory": "submission/rado_riemannSurface" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-17-000001", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-17-000001", "commit": "ccbcc61444864ceb7990ab6b1abb5238c4fc6243", "trust": "high", "theorems": 1, "date": "2026-08-17" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "robertmartin8/AmericanPutConvexity/palomar/formalization.yaml", "repo": "robertmartin8/AmericanPutConvexity", "path": "palomar/formalization.yaml", "directory": "palomar", "origins": [ "search" ], "url": "https://github.com/robertmartin8/AmericanPutConvexity/blob/HEAD/palomar/formalization.yaml", "version": "v0.4", "missing": [], "name": "Log-convexity of the exercise boundary of an American put option", "description": "In the constant-parameter Black–Scholes model, the logarithm of the American put option exercise boundary is convex in positive time-to-expiry when the strike, risk-free interest rate and volatility are positive and the dividend yield satisfies 0 ≤ q ≤ r. The stock-price exercise boundary is strictly convex. The boundary is defined from a supremum over all stopping times bounded by the horizon almost surely, on a constructed Brownian probability space with its completed usual natural filtration. Model certificates, value bounds, boundary positivity and the payoff-contact characterization are i", "authors": [ "Robert Martin" ], "maintainers": [ "Robert Martin" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Log-convexity of the exercise boundary of an American put option", "id": "", "authors": [ "Robert Martin" ], "type": "paper", "relationship": "formalizes", "endorsement": "participated" }, { "title": "Convexity of the exercise boundary of the American put option on a zero dividend asset", "id": "", "authors": [ "Xinfu Chen", "John Chadam", "Lishang Jiang", "Weian Zheng" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Nonconvexity of the optimal exercise boundary for an American put option on a dividend-paying asset", "id": "", "authors": [ "Xinfu Chen", "Huibin Cheng", "John Chadam" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Convexity of the exercise boundary of the American put option", "id": "", "authors": [ "Hsuan-Ku Liu" ], "type": "paper", "relationship": "background", "endorsement": "" } ], "related": [ { "id": "https://github.com/formal-applied-math/formal-mathfin", "relationship": "builds-on", "note": "Financial and stochastic infrastructure used by the substantive Solution, not imported by the Challenge." }, { "id": "https://github.com/RemyDegenne/brownian-motion", "relationship": "builds-on", "note": "Supplies the constructed Gaussian probability measure and continuous Brownian process, whose properties are separately compared." }, { "id": "https://github.com/leanprover-community/mathlib4", "relationship": "builds-on", "note": "The only Challenge import; supplies standard measure, probability and convexity definitions." } ], "classification": { "arxiv": [ "q-fin.PR", "math.AP" ], "msc2020": [ "91G20", "35R35" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "ChatGPT-6 Pro (astra)", "Fable 5.1", "gpt-6-astra high" ], "framework": "" } ], "strongest": "agent", "models": [ "ChatGPT-6 Pro (astra)", "Fable 5.1", "gpt-6-astra high" ], "spend": "", "notes": "" }, "scope": "", "sorry_count": null, "sorry_in_definitions": null, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "", "alignment": false, "original": false, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [], "notes": "The source development has automated builds and axiom audits and has received agent-assisted mathematical and semantic review. The Palomar adapter passed local Comparator correspondence, NanoDa and fresh Lean-kernel replay checks; see VERIFICATION.md for the exact scope and platform limits. Neither independent expert peer review nor a Palomar acceptance is claimed." }, "canonical": { "repo": "robertmartin8/americanputconvexity", "directory": "palomar" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "roed-math/gq2-lean/formalization.yaml", "repo": "roed-math/gq2-lean", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/roed-math/gq2-lean/blob/HEAD/formalization.yaml", "version": "v0.3", "missing": [], "name": "gq2-lean", "description": "", "authors": [ "David Roe" ], "maintainers": [ "David Roe" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "A Presentation of the Absolute Galois Group of Q_2", "id": "machine-generated preprint (GPT); current writeup at https://roed314.github.io/gq2/; original source snapshot at paper/A_Profinite_Presentation_for_G__Q_2.pdf", "authors": [ "David Turturean" ], "type": "", "relationship": "formalizes", "endorsement": "n/a" } ], "related": [], "classification": { "arxiv": [ "math.NT", "math.GR" ], "msc2020": [ "11S20", "20E18", "20F05", "11S25" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "claude-opus-4-8", "claude-fable-5" ], "framework": "Claude Code" }, { "method": "ai-assisted", "models": [ "gpt-5.6" ], "framework": "OpenAI Codex" } ], "strongest": "agent", "models": [ "claude-fable-5", "claude-opus-4-8", "gpt-5.6" ], "spend": "$600 (three Claude Max 20 subscriptions)", "notes": "Human role was orchestration-level: approving trust-base changes, owner-gated design decisions, and review checkpoints; the Lean code was written by the agent swarm. Commit trailers record the models used per commit." }, "scope": "Complete end-to-end formalization of the paper's main result (numbered Theorem 1.2 in the retained original PDF): the absolute Galois group of Q_2 is isomorphic as a profinite group to an explicit 4-generator 2-relator marked quotient Gamma_A, together with the surjection-count form (for every finite G, the number of continuous surjections G_Q2 ->> G equals the number of admissible marked quadruples in G^4). This metadata records the explicit-input route. The capstone GQ2.main_presentation_literal_of_inputs is sorry-free and takes [LiteratureInputs] as an explicit bundled hypothesis containing", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "GQ2.main_presentation_literal_of_inputs", "file": "GQ2/PresentationLiteralWithInputs.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "GQ2.main_presentation_literal_roe_unconditional_of_inputs", "file": "GQ2/Roe/MainWithInputs.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "Statements track the paper closely; module docstrings carry paper-tag ledgers mapping equations and propositions to declarations. Known deviations: (i) the nine literature inputs are not proved in-repository. The `_of_inputs` capstones take them as fields of an explicit `[LiteratureInputs]` hypothesis. The bundled B7 field uses `M : Type`, which covers every consumer in this development; (ii) a transcription erratum in the paper's H^0 bookkeeping was found and corrected during formalization (docs/erratum-h0-transcription.md); (iii) helper-lemma names follow mathlib conventions rather than paper numbering; (iv) on 2026-07-24 B9 was restated as the relative Stiefel-Whitney identity (relativeStiefelWhitney_dyadic), and the previous composite B9 statement is now a derived theorem (evensKahn_dyadic) proved from it plus B11a; (v) the Gamma_R \"Replacement theorem\" (GQ2/Roe/) is not part of the paper and verifies a different candidate group proposed afterward. Its former extra input, the Labute classification instance, was carried as the explicit theorem binder BLabHypothesis rather than admitted as an axiom, and was discharged in-repository on 2026-07-26 by GQ2.Roe.Labute.bLab. Thus the `", "alignment": true, "original": false, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [], "notes": "No external human peer review yet. Internal review includes a foundations audit (docs/foundations-audit.md), off-path statement checks (docs/off-path-statements.md), kernel-checked builds, zero-sorry enforcement, and transitive environment audits. The Palomar route is checked by GQ2/PalomarAxiomAudit.lean, which imports only SolutionPalomar and checks that both exported solutions use only propext, Classical.choice, and Quot.sound. The two Palomar-facing main results are independently checkable with leanprover/comparator through comparator-config-palomar.json (ChallengePalomar.lean / SolutionPalomar.lean). The post-refactor Lean Compass report identifies 30 project declarations for human semantic review (atlas-audit.md); those declarations have not yet received external human review. The Ga" }, "canonical": { "repo": "roed-math/gq2-lean", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "roos-j/lean-stein-bourgain-palomar/formalization.yaml", "repo": "roos-j/lean-stein-bourgain-palomar", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/roos-j/lean-stein-bourgain-palomar/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Spherical maximal functions: theorems of Stein and Bourgain", "description": "This is a Lean 4 formalization of Stein's spherical maximal theorem and Bourgain's circular maximal theorem. The result concerns the sharp range of Lp boundedness of spherical maximal functions in Euclidean space and was first proved by Stein in dimensions three and greater. The much more subtle two-dimensional theorem was first proved by Bourgain. The proof of Bourgain's theorem in this formalization is via Mockenhaupt-Seeger-Sogge's classical argument via a local smoothing estimate for the wave equation. This required first formalizing various underlying standard results from harmonic analys", "authors": [ "Joris Roos" ], "maintainers": [ "Joris Roos" ], "license": "Apache-2.0", "role": "thin-wrapper", "substantive": "https://github.com/roos-j/lean-spherical", "sources": [ { "title": "Maximal functions: spherical means", "id": "https://doi.org/10.1073/pnas.73.7.2174", "authors": [ "Elias M. Stein" ], "type": "paper", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "Averages in the plane over convex curves and maximal operators", "id": "https://doi.org/10.1007/BF02792533", "authors": [ "Jean Bourgain" ], "type": "paper", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "Wave front sets, local smoothing and Bourgain's circular maximal theorem", "id": "https://doi.org/10.2307/2946549", "authors": [ "Gerd Mockenhaupt", "Andreas Seeger", "Christopher D. Sogge" ], "type": "paper", "relationship": "adapts", "endorsement": "not-contacted" }, { "title": "Problems in harmonic analysis related to curvature and oscillatory integrals", "id": "https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1986.1/ICM1986.1.ocr.pdf", "authors": [ "E. M. Stein" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" } ], "related": [], "classification": { "arxiv": [ "math.CA" ], "msc2020": [ "42B25" ] }, "automation": { "methods": [ { "method": "manual", "models": [], "framework": "Lean 4" }, { "method": "agent", "models": [ "Codex/gpt-5.6-terra-ultra", "Claude/opus-5-high" ], "framework": "Codex and Claude coding agents" } ], "strongest": "agent", "models": [ "Claude/opus-5-high", "Codex/gpt-5.6-terra-ultra" ], "spend": "", "notes": "This metadata describes both the documented automation of the substantive formalization and the generation of this thin wrapper. It does not claim an independent human review of the substantive proofs." }, "scope": "This is a Challenge/Solution wrapper around the main repository. Comparator compares exactly Spherical.eLpNorm_sphericalMaximal_le and Spherical.eLpNorm_sphericalMaximal_eq_top_of_le_criticalExponent. Challenge.lean duplicates only the definitionally faithful spherical-average and spherical-maximal vocabulary required for their types, namely the Euclidean unit sphere, its normalized surface measure, the spherical average, and the spherical maximal function. Solution.lean imports the canonical declarations from LeanSpherical/Theorems.lean in the pinned substantive repository. The substantive re", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "Spherical.eLpNorm_sphericalMaximal_le", "file": "LeanSpherical/Theorems.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Spherical.eLpNorm_sphericalMaximal_eq_top_of_le_criticalExponent", "file": "LeanSpherical/Theorems.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "No divergence is known: the Challenge statements and the five statement-level definitions are copied definitionally from the pinned source revision, and the definition bodies were compared character for character against LeanSpherical/Definitions.lean at that revision. The advertised L^p theorem is the strong-type bound on L^p itself rather than the a priori estimate on Schwartz functions that the substantive repository also proves. The advertised sharpness statement is slightly stronger than the classical remark it formalizes, since it produces one Schwartz function whose maximal function has infinite L^p seminorm rather than only ruling out a uniform constant. The wrapper does not make a claim about results beyond the two selected declarations.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "The wrapper has been checked for source pinning, statement vocabulary, allowed Challenge imports, and permitted axioms, and the Challenge statement text was verified to agree character for character with the pinned substantive declarations. No prior formalization of these results is known to the maintainer, so no related_formalizations entry is given. The substantive proofs have not received an independent human review." }, "canonical": { "repo": "roos-j/lean-stein-bourgain-palomar", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-06-000003", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-06-000003", "commit": "3c29098b6660e97ca52fdf71856be8eecb32e122", "trust": "high", "theorems": 2, "date": "2026-09-06" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "roos-j/lean-superorthogonality/formalization.yaml", "repo": "roos-j/lean-superorthogonality", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/roos-j/lean-superorthogonality/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Formalization of type IV superorthogonality", "description": "A Lean formalization of the main result (Theorem 1) of the paper \"A new type of superorthogonality\" by P. T. Gressman, L. B. Pierce, J. Roos, and P.-L. Yung; arXiv:2212.08956, Proc. Amer. Math. Soc. 152, 665-675, 2024. The theorem provides a simple criterion on a family of functions that implies a square function estimate on Lp for every even integer p ≥ 2. This defines a new type of superorthogonality that is verified by checking a less restrictive criterion than any other type of superorthogonality previously known.", "authors": [ "Joris Roos" ], "maintainers": [ "Joris Roos" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "A new type of superorthogonality", "id": "https://doi.org/10.1090/proc/16631", "authors": [ "Philip T. Gressman", "Lillian B. Pierce", "Joris Roos", "Po-Lam Yung" ], "type": "paper", "relationship": "formalizes", "endorsement": "participated" } ], "related": [], "classification": { "arxiv": [ "math.CA" ], "msc2020": [ "42B25" ] }, "automation": { "methods": [ { "method": "manual", "models": [], "framework": "" }, { "method": "agent", "models": [ "gpt-5.5-xhigh" ], "framework": "Codex" } ], "strongest": "agent", "models": [ "gpt-5.5-xhigh" ], "spend": "", "notes": "Machine-generated proofs are isolated in LeanSuperorthogonality/Codex; see the README's Autoformalization section." }, "scope": "The development formalizes Theorem 1 of the source paper. This Palomar entry compares only its unrestricted final theorem; supporting results are not exposed.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The Lean statement uses representatives of Lp classes, makes almost-everywhere square-sum summability explicit to avoid Lean's tsum junk value, uses two equivalent Lp-norm implementations, and includes a redundant Fact instance for typeclass inference. These are the known implementation-level differences documented in the README.", "alignment": true, "original": false, "review": { "status": "author-verified", "bucket": "author-verified", "reviewers": [], "notes": "The project author reviewed the formalization; the README documents its statement, Lean-specific limitations, and review of generated code." }, "canonical": { "repo": "roos-j/lean-superorthogonality", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-31-000015", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-31-000015", "commit": "d6cebb0d1ec7cf96f7485b839f895f205c3857df", "trust": "high", "theorems": 1, "date": "2026-08-31" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "rwst/Ceiling-Orbits/formalization.yaml", "repo": "rwst/Ceiling-Orbits", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/rwst/Ceiling-Orbits/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Ceiling orbits of rational bases are not P-recursive", "description": "A Lean 4 / Mathlib development of the companion paper of the same name (doi:10.13140/RG.2.2.10727.82081). Fix coprime integers p > q ≥ 2 and an integer x₀ ≥ 1 and iterate the ceiling map of the rational base p/q, xₙ = ⌈p·xₙ₋₁/q⌉, with associated word wₙ = q·xₙ₊₁ − p·xₙ ∈ {0,…,q−1}, the correction that makes p·xₙ divisible by q. Four statements are compared. (A) The word of every such orbit is not P-recursive: it satisfies no nontrivial linear recurrence with polynomial coefficients. (B) At p/q = 3/2 the orbit itself is not P-recursive either; for x₀ = 1 it is the sequence A061419, which is the", "authors": [ "Ralf Stephan" ], "maintainers": [ "Ralf Stephan" ], "license": "CC0-1.0", "role": "", "substantive": "", "sources": [ { "title": "Ceiling orbits of rational bases are not P-recursive", "id": "https://doi.org/10.13140/RG.2.2.10727.82081", "authors": [ "Ralf Stephan" ], "type": "paper", "relationship": "formalizes", "endorsement": "participated" }, { "title": "On integer sequences generated by linear maps", "id": "https://doi.org/10.1017/S0017089508004655", "authors": [ "Arturas Dubickas" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Powers of rationals modulo 1 and rational base number systems", "id": "https://doi.org/10.1007/s11856-008-1056-4", "authors": [ "Shigeki Akiyama", "Christiane Frougny", "Jacques Sakarovitch" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Differentiably finite power series", "id": "https://doi.org/10.1016/S0195-6698(80)80051-5", "authors": [ "Richard P. Stanley" ], "type": "paper", "relationship": "background", "endorsement": "n/a" }, { "title": "Functional iteration and the Josephus problem", "id": "https://doi.org/10.1017/S0017089500008272", "authors": [ "Andrew M. Odlyzko", "Herbert S. Wilf" ], "type": "paper", "relationship": "background", "endorsement": "n/a" }, { "title": "Automatic Sequences: Theory, Applications, Generalizations", "id": "", "authors": [ "Jean-Paul Allouche", "Jeffrey Shallit" ], "type": "book", "relationship": "background", "endorsement": "n/a" } ], "related": [], "classification": { "arxiv": [ "math.NT", "math.CO" ], "msc2020": [ "11B37", "11A63", "05A15", "68R15", "12H05" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "claude-opus-4-8" ], "framework": "Claude Code" }, { "method": "manual", "models": [], "framework": "" } ], "strongest": "agent", "models": [ "claude-opus-4-8" ], "spend": "subscription-based Claude Code usage; per-project spend not recorded separately", "notes": "lake test runs leanprover/comparator over one configuration, comparator/ceiling-orbits.json: the four statements above, permitting only propext, Quot.sound and Classical.choice. It performs the statement match constant by constant against the challenge, the axiom check, and a replay of the exported solution environment through the Lean kernel, and it passes. The configuration is one rather than two from 2026-09-06: until then a second lane certified the two non-algebraicity corollaries under an extra permitted axiom, Stanley.pRecursive_of_isAlgebraic, and the split became empty when that axiom" }, "scope": "Complete, with no unproved step: the paper's Theorems A and B and their two non-algebraicity corollaries (5.5 and 6.4) are proved in full, for every admissible base p > q ≥ 2 coprime and every x₀ ≥ 1 in the case of Theorem A and its corollary, and at p/q = 3/2 in the case of Theorem B and its own. Nothing is assumed from the literature. The aperiodicity input that the engine consumes, which for the base 3/2 is [AFS08] Proposition 26, is reproved from Mathlib as RB.Gen.not_eventually_periodic rather than cited, and Stanley's closure property, which the paper grants as its single citation ([Sta8", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "RB.Gen.not_isPRecursive_word", "file": "RB/GeneralBase.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "RB.not_isPRecursive_orbit", "file": "RB/OrbitSeries.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "RB.Gen.not_isAlgebraic_wordSeries", "file": "RB/GeneralBase.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "RB.not_isAlgebraic_orbitSeries", "file": "RB/OrbitSeries.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The Lean definitions are ℕ-arithmetic normal forms of the paper's. The word is defined as the residue (q − p·xₙ mod q) mod q and the orbit by x(n+1) = (p·xₙ + wₙ)/q, an exact division, so that neither truncated subtraction nor a ceiling occurs in the recursion that every later proof unfolds; the familiar form is recovered once, over ℤ and ℝ, by RB.Gen.x_succ_eq_ceil, for the reader. The challenge file restates these definitions verbatim, so what is certified is the normal form. The paper is one revision behind the repository on its single citation. Its Theorem 1.5 (Stanley) is presented as granted, its Remark after Corollary 5.5 says that corollary is the only statement consuming it, and its Appendix A marks Corollaries 5.5 and 6.4 \"std3 + [Sta80]\" and describes CITED/Stanley.lean as supplying \"the single axiom\". Since 2026-09-06 that statement is a theorem of this repository and the four compared statements are axiom-free beyond the standard three. Nothing else in the paper changes: the mathematics granted is the mathematics now proved. Theorem B and its corollary are stated at the base 3/2 only, as in the paper; no general-base orbit statement is claimed. Corollary 5.5 is compare", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Ralf Stephan" ], "notes": "The author directed the work and reviewed the generated code, and states in the paper's Acknowledgements that he takes full responsibility for the mathematical integrity and final content. There is no independent review: no external referee has read the development or the paper, and the agent that produced the proofs also ran the hygiene passes. What is established mechanically is comparator's three checks — statement match, axiom discipline, kernel replay — over the four declarations named above." }, "canonical": { "repo": "rwst/ceiling-orbits", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-07-000006", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-07-000006", "commit": "f76e02755cf909322015890f568338273d98038f", "trust": "high", "theorems": 4, "date": "2026-09-07" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "rwst/Confinement-Certificates/formalization.yaml", "repo": "rwst/Confinement-Certificates", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/rwst/Confinement-Certificates/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Confinement schemas and the reach of block certificates for powers of rational numbers modulo one", "description": "A Lean 4 / Mathlib development of the second paper on the sets Z_{p/q}(s, s+t) = {ξ > 0 : {ξ(p/q)ⁿ} ∈ [s, s+t) for all n ≥ 0}, the generalized Z-numbers of Flatto, Lagarias and Pollington. The first paper built a corpus of block certificates for individual statements Z_{p/q}(U) = ∅. This paper replaces parts of that corpus by theorems, and then asks what the certificate format can and cannot reach. Thirty certificates in the regime p > q² collapse into one closed-form criterion on the single real parameter ε = {(p−q)s}, settling a set of positions of measure 1 − 2q²/(p(p+q)) at every base and ", "authors": [ "Ralf Stephan" ], "maintainers": [ "Ralf Stephan" ], "license": "CC0-1.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "Confinement schemas and the reach of block certificates for powers of rational numbers modulo one", "id": "", "authors": [ "Ralf Stephan" ], "type": "paper", "relationship": "formalizes", "endorsement": "participated" }, { "title": "Confinement certificates for powers of rational numbers modulo one", "id": "https://doi.org/10.13140/RG.2.2.36190.19520", "authors": [ "Ralf Stephan" ], "type": "paper", "relationship": "background", "endorsement": "participated" }, { "title": "On the range of fractional parts {ξ(p/q)ⁿ}", "id": "", "authors": [ "Leopold Flatto", "Jeffrey C. Lagarias", "Andrew D. Pollington" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Linear mod one transformations and the distribution of fractional parts {ξ(p/q)ⁿ}", "id": "", "authors": [ "Yann Bugeaud" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Powers of a rational number modulo 1 cannot lie in a small interval", "id": "", "authors": [ "Arturas Dubickas" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Integer parts of powers of rational numbers", "id": "", "authors": [ "Arturas Dubickas", "Aivaras Novikas" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "On the powers of 3/2 and other rational numbers", "id": "", "authors": [ "Arturas Dubickas" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Cellular automata and powers of p/q", "id": "", "authors": [ "Jarkko Kari", "Johan Kopra" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "The set of rational cycles for the 3x + 1 problem", "id": "", "authors": [ "Jeffrey C. Lagarias" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Applied Proof Theory: Proof Interpretations and their Use in Mathematics", "id": "", "authors": [ "Ulrich Kohlenbach" ], "type": "book", "relationship": "background", "endorsement": "not-contacted" }, { "title": "On the distance from a rational power to the nearest integer", "id": "", "authors": [ "Arturas Dubickas" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Fractional parts of powers of large rational numbers", "id": "", "authors": [ "Arturas Dubickas" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Progressions arithmétiques généralisées et le problème des (3/2)ⁿ", "id": "", "authors": [ "Andrew D. Pollington" ], "type": "paper", "relationship": "background", "endorsement": "n/a" }, { "title": "Powers of rationals modulo 1 and rational base number systems", "id": "", "authors": [ "Shigeki Akiyama", "Christiane Frougny", "Jacques Sakarovitch" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "On the Diophantine nature of the elements of Cantor sets arising in the dynamics of contracted rotations", "id": "", "authors": [ "Yann Bugeaud", "Dong Han Kim", "Michel Laurent", "Arnaldo Nogueira" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "An unsolved problem on the powers of 3/2", "id": "", "authors": [ "Kurt Mahler" ], "type": "paper", "relationship": "background", "endorsement": "n/a" } ], "related": [], "classification": { "arxiv": [ "math.NT", "math.DS" ], "msc2020": [ "11J71", "11K06", "37E05" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "claude-fable-5" ], "framework": "Claude Code" }, { "method": "agent", "models": [ "claude-opus-5" ], "framework": "Claude Code" }, { "method": "manual", "models": [], "framework": "" } ], "strongest": "agent", "models": [ "claude-fable-5", "claude-opus-5" ], "spend": "subscription-based usage ($100/month subscription; no separate API spend)", "notes": "Two automated systems made substantive contributions, both working as agents in Claude Code under the author's direction: claude-fable-5 in mathematical discovery and in drafting the paper, and claude-opus-5 in writing the Lean formalization and its tooling. They are credited here rather than under project.authors, which Palomar reserves for people; the author directed and reviewed their work, as the manual entry above records. The checks of the first paper still ship with the repository and still apply to the certificates this one reuses: the four independent engines (atlas.c, hold.c, gridcer" }, "scope": "Complete for every numbered statement of the paper except two: Appendix A is a table of every statement with its Lean identifier, its file and its status, and every row is proved at std3 — no sorry, no cited axiom admitted as an axiom, no natively compiled decision procedure. The two exceptions are marked \"not formalized\" in the table itself: Conjecture 7.2 (the period-doubling cascade and its closed form for the entry points c_k) and Remark 7.3, which reads that cascade back off the transient tree. 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"divergences": "The formalization follows the paper's own proofs. Where names and wording part company, the paper records it. Appendix A notes two Lean names that predate the paper's wording: Z32.SchemaCertifiedK — the \"certifiedK\" of the file — is Definition 3.3's \"s escapes at depth K\", and \"two_cell\" throughout Z32/ is the two-interval union [0, 1/5) ∪ [4/5, 1) of Theorem 5.10's first entry. Lemma names follow the development, not the paper's numbering. One entry of Appendix A's tables gives a name that does not resolve: Table 1 row 6 is listed as Z32.BlockCert.escape_union_7083, but the declaration sits after `end Z32.BlockCert` in Z32/UnionRecord.lean and is Z32.escape_union_7083. Challenge2.lean and the development use the real name; comparator2.json no longer names the row at all. §3.3 states the correspondence with the literature and is used nowhere in the development: Definition 3.3 is the finiteness criterion of [FLP95, Thm. 3.4] together with the returning case of [Bug04, Lem. 1]; Proposition 3.4 is the cyclic permutation of [Bug04, Prop. 1]; and the interval (4) is J¹_{K+1}(q/p) in the notation of [Bug04, Lem. 3]. Theorems 3.5 and 3.6 are therefore a machine-checked fragment of [Bug04,", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Ralf Stephan" ], "notes": "The author directed the work and checked the statements against the paper. 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Dubickas (Glasgow Math. J. 48 (2006), 331–336) splits the reals α > 1 in two: α lies in 𝒵 when some real ξ ≠ 0 makes every integral part ⌊ξαⁿ⌋, n ≥ 1, even, and in 𝒮 otherwise. At α = 3/2 this is Mahler's question, and open; his Problem 5 asks whether 𝒮 has an element greater than 2. This formalization decides 𝒮 versus 𝒵 for every quadratic Pisot number: the golden mean is the only one in 𝒮, so none greater than 2 lies there and Problem 5 loses its whole natural class of candidates, though the problem itself st", "authors": [ "Ralf Stephan (direction, review, paper author)", "Claude Code, claude-opus-5 (mathematical discovery, formalization, paper drafting)" ], "maintainers": [ "Ralf Stephan" ], "license": "CC0-1.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "Even integral parts of powers of quadratic Pisot numbers", "id": "https://doi.org/10.13140/RG.2.2.20582.79689", "authors": [ "Ralf Stephan" ], "type": "paper", "relationship": "formalizes", "endorsement": "participated" }, { "title": "Even and odd integral parts of powers of a real number", "id": "https://doi.org/10.1017/S0017089506003090", "authors": [ "Arturas Dubickas" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "On the limit points of the fractional parts of powers of Pisot numbers", "id": "", "authors": [ "Arturas Dubickas" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "An unsolved problem on the powers of 3/2", "id": "", "authors": [ "Kurt Mahler" ], "type": "paper", "relationship": "background", "endorsement": "n/a" }, { "title": "Note on Mahler's 3/2-problem", "id": "", "authors": [ "Robert Tijdeman" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" } ], "related": [], "classification": { "arxiv": [ "math.NT", "math.DS" ], "msc2020": [ "11J71", "11K16", "11R06", "11B37", "11B39", "68V20" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "claude-opus-5" ], "framework": "Claude Code" }, { "method": "manual", "models": [], "framework": "" } ], "strongest": "agent", "models": [ "claude-opus-5" ], "spend": "subscription-based usage ($100/month subscription; no separate API spend)", "notes": "Two independent checks ship with the repository: SZ/checks/d1check2.py recomputes the orbits, the cell bookkeeping and an exhaustive degree-2/3/4 sweep in exact arithmetic, sharing no code with the Lean; and `lake test` runs leanprover/comparator on comparator.json, re-deriving the axiom list from a kernel replay of a fresh export. Cost, wall time and hardware were not recorded." }, "scope": "Complete, with no unproved step: the paper's Theorems A, B, C and D are proved in full, clause by clause, and each conditional claim ships with an instance of its hypothesis so that none is vacuous. Nothing is assumed from the literature — the four cells of the classification that appear in [Dub06EO] are reproved from Mathlib alone, so the classification carries no citation and no cited-literature axiom.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "SZ.quadPisot_notMem_S_of_two_lt", "file": "SZ/Classification.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SZ.isQuadPisot_silver", "file": "SZ/Classification.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SZ.quadPisot_mem_S_iff", "file": "SZ/Classification.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SZ.quadPisot_mem_MahlerZ", "file": "SZ/Classification.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SZ.isQuadPisot_golden", "file": "SZ/Classification.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SZ.floor_silver_pell", "file": "SZ/Silver.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SZ.even_floor_silver", "file": "SZ/Silver.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SZ.silver_mem_MahlerZ", "file": "SZ/Silver.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SZ.dvd_floor_of_neg", "file": "SZ/PellDesign.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SZ.eventually_composite", "file": "SZ/QuadraticDesign.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The paper defines 𝒵 by even integral parts, the second of the two equivalent forms of [Dub06EO] p. 331; SZ.MahlerZ is that form, with the source's quantifier n ≥ 1, although every witness constructed here works from n = 0, which is strictly stronger. A quadratic Pisot number is presented by its symmetric functions (SZ.IsQuadPisot: β ≠ 0, |β| < 1, α + β = a, αβ = −b) rather than by irreducibility; the two agree, but the argument that the degree is then exactly 2 is prose in the docstring, not a formalized lemma. Theorem C is stated with the index shifted, ⌊ξ(1+√2)^(n+1)⌋ = Pₙ − [n odd], to avoid the paper's convention P₋₁ = 1, and SZ.even_floor_silver covers n = 0 separately. Theorem D's hypothesis is the structure SZ.Pair together with β < 0; its \"composite\" is ¬ Prime ⌊ξαⁿ⌋ ∧ 1 < ⌊ξαⁿ⌋ and its \"all but finitely many\" is ∀ᶠ n in Filter.atTop. Lemma names follow the development, not the paper's numbering, and the section 5 cells are proved by elementary arguments that differ from the published ones: an explicit even greedy in place of Tijdeman's nested intervals, and an explicit congruence mod 6 in place of a limit.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Ralf Stephan" ], "notes": "The author directed the work and checked the statements against the source paper. No independent review: the agent that produced the proofs also ran the faithfulness passes, and no external referee has read the development or the paper." }, "canonical": { "repo": "rwst/even-integral", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-20-000006", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-20-000006", "commit": "db8f6025df35517cb9552df04d18345c68d8ac11", "trust": "high", "theorems": 10, "date": "2026-09-03" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "rwst/Square-Roots/formalization.yaml", "repo": "rwst/Square-Roots", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/rwst/Square-Roots/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Even integral parts of powers of square roots", "description": "A Lean 4 / Mathlib development of a paper, which answers a problem by Dubickas. Dubickas (Glasgow Math. J. 48 (2006), 331–336) splits the reals α > 1 in two: α lies in 𝒵 when some real ξ ≠ 0 makes every integral part ⌊ξαⁿ⌋, n ≥ 1, even, and in 𝒮 otherwise. At α = 3/2 this is Mahler's question, and open; his Problem 3 asks whether √3 lies in 𝒮 or in 𝒵. This formalization settles that question and the whole square-root slice: √3 ∈ 𝒵 by an explicit ξ = 1.34160899796…, and for an integer m ≥ 2 one has √m ∈ 𝒮 exactly when m = 2. The cells √5, √6, √7, √8 put four explicit points of 𝒵 in the interval", "authors": [ "Ralf Stephan (direction, review, paper author)", "Claude Code, claude-fable-5 (mathematical discovery, formalization, paper drafting)", "Claude Code, claude-opus-5 (formalization, paper drafting)" ], "maintainers": [ "Ralf Stephan" ], "license": "CC0-1.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "Even integral parts of powers of square roots", "id": "https://doi.org/10.13140/RG.2.2.32215.43682", "authors": [ "Ralf Stephan" ], "type": "paper", "relationship": "formalizes", "endorsement": "participated" }, { "title": "Even and odd integral parts of powers of a real number", "id": "https://doi.org/10.1017/S0017089506003090", "authors": [ "Arturas Dubickas" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "The distance set for the Cantor discontinuum", "id": "", "authors": [ "W. R. Utz" ], "type": "paper", "relationship": "background", "endorsement": "n/a" }, { "title": "An unsolved problem on the powers of 3/2", "id": "", "authors": [ "Kurt Mahler" ], "type": "paper", "relationship": "background", "endorsement": "n/a" }, { "title": "Note on Mahler's 3/2-problem", "id": "", "authors": [ "Robert Tijdeman" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Explicit algebraic numbers all whose integer parts of powers are composite", "id": "arXiv:2608.05309", "authors": [ "Seungki Hahn", "Dan Ismailescu", "Gyumin Kim", "Minjae Kim" ], "type": "preprint", "relationship": "background", "endorsement": "not-contacted" } ], "related": [ { "id": "https://doi.org/10.13140/RG.2.2.20582.79689", "relationship": "builds-on", "note": "The Lean development accompanying the companion paper \"Even integral parts of powers of quadratic Pisot numbers\" (repository directory Even-Integral), which classifies 𝒮 versus 𝒵 for the quadratic Pisot numbers. Same author, same collaboration, same namespace SZ, and two modules shared byte for byte: SZ/Defs.lean (the sets 𝒵 and 𝒮, tail closure, the perfect-square cell) and SZ/Tijdeman.lean (the e" } ], "classification": { "arxiv": [ "math.NT", "math.DS" ], "msc2020": [ "11J71", "11K16", "11A63", "28A80", "11B83" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "claude-fable-5" ], "framework": "Claude Code" }, { "method": "agent", "models": [ "claude-fable-5", "claude-opus-5" ], "framework": "Claude Code" }, { "method": "manual", "models": [], "framework": "" } ], "strongest": "agent", "models": [ "claude-fable-5", "claude-opus-5" ], "spend": "subscription-based usage ($100/month subscription; no separate API spend)", "notes": "Two independent kinds of check ship with the repository. Five Python programs in SZ/checks/ — the files dubD1O5-x3-leanrule.py through dubD1O5-x7-thmC.py, which are the five described in Appendix A.2 of the paper — share no code with the Lean development or with each other and recompute in exact arithmetic: the selection rule at m = 3 with the integer comparator P² ≶ 3Q² in ℤ[√3] (2000 steps, 40 decimals of ξ at three truncation depths, ⌊ξ√3ⁿ⌋ even for n = 1..1000); the general-(p,m) greedy that produced the two printed tables, with the gap conditions verified for all p ≤ 8, m < 200 with U > p" }, "scope": "Complete for every lettered result: the paper's Theorems A, B and C are proved in full, clause by clause, as is Theorem 9.1 with its branching analysis, and each cell of Theorem B ships with its own witness so that none is vacuous. The abstract's shorthand — \"the cases m = 2 (reproof), m = 3, the cases m > 3, the p-divisibility theorem and the transcendentality theorem\" — names Theorems A, B, C and 9.1, and each is certified in its full form rather than in the compressed one: Theorem B as the single iff SZ.sqrt_natCast_mem_S_iff, whose 𝒮 side is the m = 2 cell SZ.sqrtTwo_mem_S; Theorem C with ", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "SZ.sqrtTwo_mem_S", "file": "SZ/SqrtTwo.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SZ.sqrtThree_mem_MahlerZ", "file": "SZ/SqrtThree.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SZ.sqrtThree_notMem_S", "file": "SZ/SqrtThree.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SZ.sqrt_natCast_mem_S_iff", "file": "SZ/Cells.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SZ.mem_MahlerZ_of_three_le", "file": "SZ/Tijdeman.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SZ.sqrt_mem_MahlerZ_of_four_le", "file": "SZ/ThmC.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SZ.exists_dvd_floor_sqrt", "file": "SZ/ThmC.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SZ.exists_eventually_composite_sqrt", "file": "SZ/ThmC.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SZ.continuum_le_mk_witnesses_sqrtThree", "file": "SZ/CoverTree.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SZ.exists_transcendental_witness_sqrtThree", "file": "SZ/CoverTree.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The paper defines 𝒵 by even integral parts, the second of the two equivalent forms of [Dub06EO] p. 331; SZ.MahlerZ is that form, with the source's quantifier n ≥ 1, although every witness constructed here works from n = 0, which is strictly stronger. Theorem A is formalized with two deviations from the text. Lemma 2.1 is proved in the sharper form that computes the integral part exactly, ⌊t·mˢ⌋ = N·mˢ + Σ_{i= 256, the sharp constant is an explicit maximum over a one-parameter cyclic family: it holds in every dimension and cannot be lowered in any dimension at least three.", "authors": [ "Ezzeri Esa" ], "maintainers": [ "Ezzeri Esa" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Hlawka constants for Schatten norms: existence and sharp diagonal bound", "id": "https://github.com/savarin/hlawka-schatten", "authors": [ "Ezzeri Esa" ], "type": "original-proof", "relationship": "other", "endorsement": "participated" }, { "title": "Problems and conjectures in matrix and operator inequalities", "id": "arXiv:1201.5232", "authors": [ "Koenraad M. R. Audenaert", "Fuad Kittaneh" ], "type": "paper", "relationship": "other", "endorsement": "not-contacted" }, { "title": "Sharp uniform convexity and smoothness inequalities for trace norms", "id": "doi:10.1007/BF01231769", "authors": [ "Keith Ball", "Eric A. Carlen", "Elliott H. Lieb" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Sur certaines inégalités qui caractérisent les fonctions convexes", "id": "Zbl:0166.06303", "authors": [ "Tiberiu Popoviciu" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" } ], "related": [], "classification": { "arxiv": [ "math.FA", "math.OA" ], "msc2020": [ "47A30", "47B10", "46B20" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "GPT 5.6 Sol (OpenAI)" ], "framework": "Codex" }, { "method": "agent", "models": [ "GPT 6 Astra (OpenAI)" ], "framework": "Codex" } ], "strongest": "agent", "models": [ "GPT 5.6 Sol (OpenAI)", "GPT 6 Astra (OpenAI)" ], "spend": "", "notes": "Existence: the Lean proof was developed in small kernel-checked layers and audited at the publication boundary for dimensional uniformity and endpoint fidelity. Construction: the argument was developed over two weeks of directed proof search. The human author identified the proof strategy, diagnosed repeated dead ends where agents stalled, and refined the approach through failure analysis at each stage. AI agents executed computational searches, produced candidate lemmas, and formalized the final argument in Lean." }, "scope": "The formalization proves two results about Hlawka constants for Schatten p-norms of finite-dimensional complex operators. Existence: for every real p > 1 a finite constant works for linear maps between all finite-dimensional complex inner product spaces, and no constant exists at p = 1 (complex 2 by 2 operators) or p = infinity (complex 3 by 3 matrices). Construction: for complex diagonal matrices and every real p >= 256, an explicit cyclic maximum is a Hlawka constant in every finite dimension and the smallest one in each dimension at least three. The sharp constant for general operators and ", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "PalomarHlawkaSchatten.dimension_independent_hlawka_constant_for_schatten_norms", "file": "ExistenceSolution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "PalomarHlawkaSchatten.ConstructionDiagonal.diagonal_hlawka_bound", "file": "ConstructionDiagonalSolution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "PalomarHlawkaSchatten.ConstructionDiagonal.diagonal_hlawka_sharp", "file": "ConstructionDiagonalSolution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "PalomarHlawkaSchatten.ConstructionDiagonal.cyclic_maximum_attained", "file": "ConstructionDiagonalSolution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The interior theorem is stated for linear maps between arbitrary finite complex Hilbert spaces, a coordinate-free formulation equivalent to fixing coordinate matrix spaces via choice of orthonormal bases. The proof library constructs explicit constants m_p, M_p from the compactified scalar Bregman-to-Mazur ratio, but the Challenge boundary quantifies existentially over m and M without naming the construction. Infinity is represented by Mathlib's spectral L2 operator norm, since infinity is not a real exponent in schattenPNorm. The diagonal boundary specifies the cyclic constant by a compact supremum on [1/2, 2] and proves attainment, admissibility, and sharpness. It defines the norm through actual singular values. The Lean proof departs from the written proof it formalizes (not included in this repository) only in intermediate choices: an ordered-chord proof of the weighted three-point convexity inequality, a direct sparse-minimizer argument, the rational separator 939/2000, the radial residual constant 300, and looser Hessian coefficients. The exponent cutoff 256 is unchanged; it is set by the localization estimate, not by the curvature margin.", "alignment": true, "original": true, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [ "Ezzeri Esa", "GPT 5.6 Sol (OpenAI)", "Claude Fable 5.1 (Anthropic)" ], "notes": "The existence proof source was independently audited before formalization. The diagonal proof was reviewed against the BLUEPRINT. Both Challenges were checked against their proof sources. Locally validated: full Lake build, both boundary/axiom audits, both Comparator baselines (upstream 8d84e67, macOS development Landrun shim, default Lean kernel), and both negative controls. Protected Landrun/NanoDa validation deferred to Palomar submission." }, "canonical": { "repo": "savarin/hlawka-schatten", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-25-000006", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-25-000006", "commit": "1829591140522193a81ae721330e4d775c81123e", "trust": "high", "theorems": 3, "date": "2026-09-25" }, { "id": "PALOMAR-2026-09-07-000005", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-07-000005", "commit": "97c3363af2722dd6e94ce4d3ef749fab1efab6bf", "trust": "high", "theorems": 1, "date": "2026-09-07" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "savarin/lean-malliavin/formalization.yaml", "repo": "savarin/lean-malliavin", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/savarin/lean-malliavin/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "lean-malliavin", "description": "Malliavin calculus on abstract Gaussian Banach spaces, formalized in Lean 4 against Mathlib. The repository contains two Comparator configurations: one registered (core, via comparator.json, pinned at commit 29c57b3) and one pending (Clark--Ocone, via comparator-clark-ocone.json). The proof library (52 files) builds from Cameron--Martin quasi-invariance through the Wiener chaos decomposition and Itô isometry to the full representation formula.", "authors": [ "Ezzeri Esa" ], "maintainers": [ "Ezzeri Esa" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "The Malliavin Calculus and Related Topics", "id": "10.1007/3-540-28329-3", "authors": [ "David Nualart" ], "type": "book", "relationship": "adapts", "endorsement": "not-contacted" }, { "title": "The Malliavin Calculus and Related Topics", "id": "10.1007/3-540-28329-3", "authors": [ "David Nualart" ], "type": "book", "relationship": "adapts", "endorsement": "not-contacted" } ], "related": [], "classification": { "arxiv": [ "math.PR", "math.FA" ], "msc2020": [ "60H07", "46E35" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Claude Fable (Anthropic)", "GPT 5.6 Sol (OpenAI)" ], "framework": "Claude Code and Codex" } ], "strongest": "agent", "models": [ "Claude Fable (Anthropic)", "GPT 5.6 Sol (OpenAI)" ], "spend": "", "notes": "Challenge statement authored with AI assistance. Proof library developed as a standalone Lean 4 formalization of the Malliavin derivative on abstract Gaussian Banach spaces." }, "scope": "Three compared submissions across two comparator configs, zero definition holes. The Clark--Ocone capstone (PalomarClarkOcone.generated_clark_ocone) is compared via comparator-clark-ocone.json. The registered pair (PALOMAR-2026-08-29-000016, integral_inner_mderiv and mderiv_closable) is compared via comparator.json. No unresolved sorry outside the deliberate Challenge placeholders. No Malliavin calculus or Clark--Ocone material exists in Mathlib at v4.33.0.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "integral_inner_mderiv", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "mderiv_closable", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "PalomarClarkOcone.generated_clark_ocone", "file": "ClarkOconeSolution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "Core (closability): Nualart proves closability for every p ≥ 1 on smooth cylindrical random variables over an isonormal Gaussian process. This formalization proves the p = 2 criterion on a separable Banach space carrying a Gaussian measure, using bounded C¹ Fréchet functionals with uniformly bounded derivative and realizing the Cameron--Martin space as the first chaos. Clark--Ocone: the formalization works on an abstract separable Gaussian Banach space with an IsPreBrownianReal coordinate process (correct finite-dimensional distributions, no path continuity assumed), rather than starting from a path-space Brownian motion. Coordinates are given by continuous linear functionals generating the ambient measurable space, with a totality condition for the Brownian directions in Cameron--Martin space. The theorem operates on the natural (generated, non-augmented) filtration on nonneg time; textbooks state it under the usual conditions on an arbitrary filtration. The Sobolev space D^{1,2} is realized as the closure of a bounded-C¹ graph, not as an a priori defined Sobolev class. The Itô integral is characterized existentially by isometry, zero mean, elementary adapted increments, and marti", "alignment": true, "original": false, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [ "Ezzeri Esa", "Claude Code (Anthropic)", "Codex (OpenAI)" ], "notes": "Core entry (PALOMAR-2026-08-29-000016): integral_inner_mderiv and mderiv_closable verified by Palomar's protected dual-kernel run at commit 29c57b3 via comparator.json. That immutable registered result is unaffected by later commits. Clark--Ocone entry: generated_clark_ocone Comparator-verified locally on 2026-08-31 via comparator-clark-ocone.json using Lean's default kernel and the development fake-Landrun wrapper (not Palomar's protected environment). Negative control confirmed (mutated conclusion correctly rejected). Axiom audit: propext, Classical.choice, Quot.sound only. A protected NanoDa run remains pending. Human review (Ezzeri Esa): verified the Challenge theorem statement against the source (Nualart, §1.3) and the author's dissertation. Lean proof code produced by AI agents under" }, "canonical": { "repo": "savarin/lean-malliavin", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-31-000018", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-31-000018", "commit": "a1a6ccc6eedbdf15fc3aebf9146d1fddfa0c8272", "trust": "high", "theorems": 1, "date": "2026-08-31" }, { "id": "PALOMAR-2026-08-29-000016", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-29-000016", "commit": "29c57b3eba99625e8c7f6b2eaccea7fb1c6cc5f7", "trust": "high", "theorems": 2, "date": "2026-08-29" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "savarin/lean-operator-theory/formalization.yaml", "repo": "savarin/lean-operator-theory", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/savarin/lean-operator-theory/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "lean-operator-theory", "description": "Hilbert-space operator theory from unitary power dilation through von Neumann's polynomial inequality, together with the Crouzeix--Palencia theorem that the closed numerical range is a (1 + square-root-of-2)-polynomial spectral set.", "authors": [ "Ezzeri Esa" ], "maintainers": [ "Ezzeri Esa" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "The numerical range is a (1+sqrt 2)-spectral set", "id": "10.1137/17M1116672", "authors": [ "Michel Crouzeix", "César Palencia" ], "type": "paper", "relationship": "adapts", "endorsement": "not-contacted" }, { "title": "Harmonic Analysis of Operators on Hilbert Space", "id": "10.1007/978-1-4419-6094-8", "authors": [ "Béla Sz.-Nagy", "Ciprian Foias", "Hari Bercovici", "László Kérchy" ], "type": "book", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "Remarks on the Crouzeix--Palencia proof that the numerical range is a (1+sqrt 2)-spectral set", "id": "10.1137/17M1143757", "authors": [ "Thomas Ransford", "Felix Schwenninger" ], "type": "paper", "relationship": "adapts", "endorsement": "not-contacted" } ], "related": [], "classification": { "arxiv": [ "math.FA", "math.OA" ], "msc2020": [ "47A12", "47A20", "47A25", "47A30" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Claude Fable (Anthropic)", "GPT 5.6 Sol (OpenAI)" ], "framework": "Codex, Claude Code, and a local multi-agent coordination harness" } ], "strongest": "agent", "models": [ "Claude Fable (Anthropic)", "GPT 5.6 Sol (OpenAI)" ], "spend": "", "notes": "The development used exact boundary signatures and staged verification. An initially proposed fixed-domain product lemma in the Crouzeix--Palencia route was disproved by a formal counterexample and retired; the final unconditional capstone follows a different valid smooth-support outer-approximation route." }, "scope": "Three compared theorems and zero definition holes. CrouzeixPalenciaChallenge.lean is a 63-line Mathlib-only module with complete definitions of the numerical range, polynomial supremum norm, and polynomial spectral-set predicate, all inside the PalomarCrouzeixPalencia namespace. The proof development contains no unresolved proof placeholders outside the three deliberate Challenge theorem placeholders. No numerical-range or von Neumann inequality material exists in Mathlib at v4.33.0; cross-prover novelty has not been searched.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "PalomarCrouzeixPalencia.exists_unitary_power_dilation", "file": "CrouzeixPalenciaSolution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "PalomarCrouzeixPalencia.vonNeumann_inequality", "file": "CrouzeixPalenciaSolution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "PalomarCrouzeixPalencia.crouzeix_palencia", "file": "CrouzeixPalenciaSolution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The unitary dilation is stated existentially on a larger Hilbert space in the same universe and includes compression of every natural power. The selected von Neumann theorem is for complex polynomials and the closed unit disk. The selected Crouzeix--Palencia theorem concerns polynomial evaluation on the closure of the numerical range; the source article proves a stronger rational/holomorphic spectral-set result.", "alignment": true, "original": false, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [ "Ezzeri Esa", "Claude Code (Anthropic)", "Codex (OpenAI)" ], "notes": "The production source was scanned for unresolved placeholders. All three selected declarations elaborate with exactly propext, Classical.choice, and Quot.sound. On 2026-08-30 the 63-line Mathlib-only Challenge and Solution passed a local Comparator run using Lean's default kernel and a development fake-Landrun wrapper. That run predates the 2026-08-31 namespace-bridge fix (PalomarCrouzeixPalencia): the Challenge and the proof library had independently declared same-named top-level copies of numericalRange, polynomialSupNorm, and IsKPolynomialSpectralSet, which elaborated to non-matching internals and broke the comparator after the 2026-08-30 file renames. The Challenge now declares its boundary inside the PalomarCrouzeixPalencia namespace and the Solution redeclares it as a bridge, delegat" }, "canonical": { "repo": "savarin/lean-operator-theory", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-01-000002", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-01-000002", "commit": "6f90bb7e6b056b54f56005e0416d77d6d7c2a478", "trust": "high", "theorems": 3, "date": "2026-09-01" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "savarin/lean-spectral-theory/formalization.yaml", "repo": "savarin/lean-spectral-theory", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/savarin/lean-spectral-theory/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "lean-spectral-theory", "description": "The spectral theorem for possibly unbounded self-adjoint operators on complex Hilbert spaces, formalized through the Cayley transform and real projection-valued measures. The selected spectral theorem constructs a PVM whose scalar measures give the exact finite-second-moment domain and first-moment quadratic form, with measurable-set uniqueness, and is proved equivalent to equality with the unbounded coordinate spectral integral. The selected Stone theorem proves both that strongly continuous one-parameter unitary groups have self-adjoint generators and that every self-adjoint operator generat", "authors": [ "Ezzeri Esa" ], "maintainers": [ "Ezzeri Esa" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Unbounded Self-adjoint Operators on Hilbert Space", "id": "10.1007/978-94-007-4753-1", "authors": [ "Konrad Schmüdgen" ], "type": "book", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "Methods of Modern Mathematical Physics I: Functional Analysis", "id": "ISBN 978-0-12-585050-6", "authors": [ "Michael Reed", "Barry Simon" ], "type": "book", "relationship": "formalizes", "endorsement": "not-contacted" } ], "related": [], "classification": { "arxiv": [ "math.FA", "math.OA" ], "msc2020": [ "47A10", "47B15" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Claude Fable (Anthropic)", "GPT 5.6 Sol (OpenAI)" ], "framework": "Claude Code and Codex" } ], "strongest": "agent", "models": [ "Claude Fable (Anthropic)", "GPT 5.6 Sol (OpenAI)" ], "spend": "", "notes": "The original implementation-level Challenge was 1,485 lines. A disposable feasibility study reformulated the theorem through scalar spectral measures, formally proved the new predicate equivalent to the constructed unbounded spectral integral. The compact Stone boundary uses an intrinsic difference-quotient generator relation. Both selected theorems were tested against Palomar's pinned Comparator. An adversarial change to the second-moment condition was correctly rejected as a mismatch in PVM.Represents." }, "scope": "Two compared theorems and zero definition holes. SpectralStoneChallenge.lean is a 105-line Mathlib-only module defining a real projection-valued measure, its scalar-measure representation relation, strongly continuous unitary groups, and their intrinsic generator relation. The capstones state the unbounded spectral theorem with measurable-set uniqueness and Stone's theorem in both directions. The proof development contains no unresolved sorry outside the two deliberate Challenge theorem placeholders.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "spectral_theorem_intrinsic", "file": "SpectralStoneSolution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "stone_theorem_intrinsic", "file": "SpectralStoneSolution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The spectral theorem is stated over an arbitrary complete complex inner-product space and does not require separability. The PVM is represented as a function on all sets whose laws are imposed only on Borel-measurable sets; consequently uniqueness is stated on measurable sets rather than as literal structure equality. The operator identity is presented by its exact second-moment domain and diagonal first-moment formula. The library proves this formulation equivalent to equality with the unbounded coordinate spectral integral by measure extensionality and complex polarization. Support on the spectrum is not separately selected in the submitted spectral theorem. The Stone theorem expresses generation by the exact punctured-neighborhood difference quotient, rather than exposing the library's choice-based implementation of its limit.", "alignment": true, "original": false, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [ "Ezzeri Esa", "Claude Code (Anthropic)", "Codex (OpenAI)" ], "notes": "The source was checked for unresolved proof placeholders, the compact boundary was compiled independently, and both theorems were compared with the exact Comparator revision pinned by Palomar using Lean's default kernel and a Lean 4.33-compatible lean4export. Lean's default kernel accepted the solution and Comparator reported success. Local macOS verification used Comparator's development fake-landrun wrapper; Palomar supplies the protected Landrun sandbox during submission. A protected NanoDa run remains pending. Human review (Ezzeri Esa): verified the Challenge theorem statements against the source texts (Schmüdgen and Reed--Simon). Lean proof code produced by AI agents under the author's direction." }, "canonical": { "repo": "savarin/lean-spectral-theory", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-01-000004", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-01-000004", "commit": "d19e279f0ab244657f02ce678e8d67fa69e2f407", "trust": "high", "theorems": 2, "date": "2026-09-01" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "scottnarmstrong/BKARForestFormula/formalization.yaml", "repo": "scottnarmstrong/BKARForestFormula", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/scottnarmstrong/BKARForestFormula/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "BKARForestFormula", "description": "A Lean 4 / Mathlib formalization of the Brydges-Kennedy-Abdesselam-Rivasseau forest interpolation formula: for a finite vertex set V and a C-infinity function rho on the edge-coupling space of the complete graph on V, the value of rho at the all-ones coupling equals the sum, over all acyclic edge subsets (forests) F, of the integral over the unit cube [0,1]^E(F) of the mixed partial derivative of rho in the edge variables of F, evaluated at the path-minimum interpolation point x^F(u); the empty forest contributes rho(0). The flagship theorem is additionally restated Mathlib-only and re-proved ", "authors": [ "Scott Armstrong" ], "maintainers": [ "Scott Armstrong" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Mayer expansions and the Hamilton-Jacobi equation", "id": "doi:10.1007/BF01010398", "authors": [ "David C. Brydges", "Tom Kennedy" ], "type": "journal article", "relationship": "adapts", "endorsement": "" }, { "title": "Trees, forests and jungles: a botanical garden for cluster expansions", "id": "doi:10.1007/3-540-59190-7_20", "authors": [ "Abdelmalek Abdesselam", "Vincent Rivasseau" ], "type": "book chapter", "relationship": "background", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math-ph", "math.PR" ], "msc2020": [ "82B05", "81T15", "05C05" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "gpt-5.5", "claude-opus-4-6", "claude-opus-4-7", "claude-opus-4-8" ], "framework": "Claude Code (Anthropic) and Codex (OpenAI) agentic coding harnesses" }, { "method": "agent", "models": [ "claude-fable-5" ], "framework": "Claude Code (Anthropic) agentic coding harness" } ], "strongest": "agent", "models": [ "claude-fable-5", "claude-opus-4-6", "claude-opus-4-7", "claude-opus-4-8", "gpt-5.5" ], "spend": "not separately tracked (ChatGPT Pro and Claude Max subscriptions)", "notes": "All Lean code in this repository was written by the listed models under the close supervision of the author. Every proof is checked by the Lean kernel and Mathlib; the verified library contains no sorry and no custom axiom. The Mathlib-only comparator challenge file (Audit/ForestFormula/Challenge.lean) contains exactly one intentional statement-level sorry (the theorem body), which is filled by the byte-identical statement proved in Audit/ForestFormula/Solution.lean." }, "scope": "A Lean 4 / Mathlib formalization of the Brydges-Kennedy-Abdesselam-Rivasseau forest interpolation formula: for a finite vertex set V and a C-infinity function rho on the edge-coupling space of the complete graph on V, the value of rho at the all-ones coupling equals the sum, over all acyclic edge subsets F, of the integral over the unit cube [0,1]^E(F) of the mixed partial derivative of rho in the edge variables of F, evaluated at the path-minimum interpolation point x^F(u); the empty forest contributes rho(0). The flagship theorem is BKAR.bkar_formula_forestIndex_cube_contributions (BKAR/Form", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "BKAR.bkar_formula_forestIndex_cube_contributions", "file": "BKAR/Formula.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "BKARMirror.bkar_formula_forestIndex_cube_contributions", "file": "Audit/ForestFormula/Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The formalization assumes C-infinity smoothness of rho (BKARContDiff = ContDiff R infinity, deliberately the C-infinity exponent and not the analytic top of Mathlib's smoothness scale) where the classical BKAR formula requires only C^(|V|-1) regularity (a forest on V has at most |V|-1 edges). This is a deliberate strengthening of the hypothesis; no claim is made at finite regularity. The formula is otherwise stated in its standard scalar form on the edge-coupling space of the complete graph on a finite vertex set.", "alignment": true, "original": false, "review": { "status": "author-verified", "bucket": "author-verified", "reviewers": [ "Scott Armstrong" ], "notes": "Correctness of every proof is guaranteed by the Lean kernel and Mathlib (no sorry, no custom axiom). The author reviewed that the public statements are the intended theorems; the flagship was additionally restated from Mathlib primitives alone in the comparator challenge and re-proved byte-identically in the solution, with the axiom set checked via #print axioms. This is not external peer review." }, "canonical": { "repo": "scottnarmstrong/bkarforestformula", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-19-000003", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-19-000003", "commit": "a07f44a534240fe6339558951dd78a19e4ef7c51", "trust": "high", "theorems": 1, "date": "2026-09-19" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "scottnarmstrong/CaffarelliKohnNirenberg/formalization.yaml", "repo": "scottnarmstrong/CaffarelliKohnNirenberg", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/scottnarmstrong/CaffarelliKohnNirenberg/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "CaffarelliKohnNirenberg", "description": "A Lean 4 / Mathlib formalization of the Caffarelli-Kohn-Nirenberg partial regularity theory for suitable weak solutions of the three-dimensional incompressible Navier-Stokes equations with forcing in L^q, q > 5/2: the L^3 epsilon-regularity theorem (Theorem A), the gradient epsilon-regularity criterion (Theorem B) and the vanishing of the one-dimensional parabolic Hausdorff measure of the singular set (Theorem C). The three theorems are proved using only the standard axioms, restated together in a Mathlib-only comparator challenge and proved from the library theorems. The manuscript being form", "authors": [ "Scott Armstrong", "Vlad Vicol" ], "maintainers": [ "Scott Armstrong", "Vlad Vicol" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "The Caffarelli--Kohn--Nirenberg partial regularity theorem: a self-contained proof written for formalization", "id": "paper/ckn.tex", "authors": [ "Scott Armstrong", "Vlad Vicol" ], "type": "paper", "relationship": "formalizes", "endorsement": "" }, { "title": "Partial regularity of suitable weak solutions of the Navier-Stokes equations", "id": "doi:10.1002/cpa.3160350604", "authors": [ "Luis Caffarelli", "Robert Kohn", "Louis Nirenberg" ], "type": "paper", "relationship": "independently-proves", "endorsement": "" }, { "title": "A new proof of the Caffarelli-Kohn-Nirenberg theorem", "id": "doi:10.1002/(SICI)1097-0312(199803)51:3<241::AID-CPA2>3.0.CO;2-A", "authors": [ "Fanghua Lin" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "On partial regularity for the Navier-Stokes equations", "id": "doi:10.3934/dcds.2008.21.717", "authors": [ "Igor Kukavica" ], "type": "paper", "relationship": "background", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.AP" ], "msc2020": [ "35Q30", "35B65", "76D05", "76D03", "35D30" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "claude-fable-5-1", "gpt-5.6-luna", "gpt-6-astra", "deepseek-4.1-flash", "leanstral-1-5", "claude-opus-5" ], "framework": "Claude Fable 5.1 coordinated agents using GPT 5.6-Luna, GPT Astra, Deepseek 4.1 flash, Leanstral and Opus 5, under the authors' supervision." } ], "strongest": "agent", "models": [ "claude-fable-5-1", "claude-opus-5", "deepseek-4.1-flash", "gpt-5.6-luna", "gpt-6-astra", "leanstral-1-5" ], "spend": "not separately tracked (ChatGPT Pro and Claude Max subscriptions)", "notes": "All Lean code was written by the listed models under the supervision of the authors, who decided the mathematics, the statements and the corrections to the manuscript. Every proof is checked by the Lean kernel and Mathlib; the library contains no sorry and no custom axiom. The Mathlib-only comparators/Challenge.lean contains exactly three intentional theorem-body placeholders. The separate Solution.lean proves the same named declarations from the library; comparator.json selects all three." }, "scope": "Complete for the three main theorems as stated in the manuscript (Theorems 2.7, 2.9 and 2.10 there): suitable weak solutions on an open product domain in space-time, force in L^q_loc with q > 5/2 and not necessarily divergence free, regular points defined by an almost-everywhere equal parabolically Hölder representative, and the singular set measured by the one-dimensional parabolic Hausdorff measure in the diameter gauge. The supporting library (about 250,000 lines) contains Sobolev spaces and weak derivatives, mollification, Newtonian and heat potentials, Morrey and Campanato norms, the pres", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "CKN.epsilonRegularityL3", "file": "CKN/Statements/TheoremA.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "CKN.epsilonRegularityGradient", "file": "CKN/Statements/TheoremB.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "CKN.caffarelliKohnNirenberg", "file": "CKN/Statements/TheoremC.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "CKNChallenge.epsilonRegularityL3", "file": "comparators/Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "CKNChallenge.epsilonRegularityGradient", "file": "comparators/Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "CKNChallenge.caffarelliKohnNirenberg", "file": "comparators/Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "Recorded in docs/DEVIATIONS.md and docs/DESIGN_NOTES.md. The main ones: solutions live on open product domains in space-time (no loss by locality); the local energy inequality is used in its almost-every-time form and the manuscript's every-time route is excluded; the singular set is measured with the parabolic diameter gauge, whose null sets coincide with those of the manuscript's radius gauge; the bootstrap step of the manuscript was rewritten to follow the formalized route (a localized pressure-gradient Morrey bound rather than a pointwise potential representation); the Hölder representative is constructed as a limit of dyadic ball averages rather than by mollification; the common exceptional time set for the pressure slice identity is obtained by translated mollifier tests rather than a countable dense family of test functions; Theorem A asserts regularity on the open interior of the half-cylinder, not on its top time face; the force is assumed in L^q_loc with q > 5/2, which is stronger than the weakest known hypotheses. The solution predicate uses supremum-norm energy finiteness, equivalent to Euclidean finiteness in the fixed finite-dimensional spaces. The three main conclusi", "alignment": true, "original": true, "review": { "status": "author-verified", "bucket": "author-verified", "reviewers": [ "Scott Armstrong", "Vlad Vicol" ], "notes": "Lean checks each proof against its formal statement. The authors reviewed that the statements are the intended theorems; the statements were compared with the manuscript by independent read-only review agents, the three comparator declarations are checked mechanically, and the manuscript was proofread in full before release. This is not external peer review." }, "canonical": { "repo": "scottnarmstrong/caffarellikohnnirenberg", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-22-000003", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-22-000003", "commit": "58a9ffe548ea64319ac8cba098634340ee6270cb", "trust": "high", "theorems": 3, "date": "2026-09-22" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "scottnarmstrong/CoarseGraining/formalization.yaml", "repo": "scottnarmstrong/CoarseGraining", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/scottnarmstrong/CoarseGraining/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "CoarseGraining", "description": "A Lean formalization of the coarse-graining approach to quantitative stochastic homogenization of divergence-form elliptic equations: function-space, deterministic, probabilistic, and homogenization-scale theory up to the quenched minimal-scale theorem. The public main result is a uniformly elliptic, isotropic specialization of the quenched homogenization comparison estimate in Armstrong-Kuusi, \"Renormalization Group and Elliptic Homogenization in High Contrast\" (Inventiones Mathematicae, 2025), independently checked using Audit/QuenchedComparison/comparator.json, together with four further co", "authors": [ "Scott Armstrong", "Tuomo Kuusi" ], "maintainers": [ "Scott Armstrong", "Tuomo Kuusi" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Coarse-Graining Theory for Elliptic Equations", "id": "", "authors": [ "Scott Armstrong", "Tuomo Kuusi" ], "type": "manuscript", "relationship": "formalizes", "endorsement": "" }, { "title": "Renormalization Group and Elliptic Homogenization in High Contrast", "id": "doi:10.1007/s00222-025-01370-9", "authors": [ "Scott Armstrong", "Tuomo Kuusi" ], "type": "journal article", "relationship": "formalizes", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.AP" ], "msc2020": [ "35B27", "60K37", "35J15" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "claude-opus-4-6", "claude-opus-4-7", "claude-opus-4-8", "claude-fable-5", "claude-opus-5", "gpt-5.5" ], "framework": "Claude Code (Anthropic) and Codex (OpenAI) agentic coding harnesses" } ], "strongest": "agent", "models": [ "claude-fable-5", "claude-opus-4-6", "claude-opus-4-7", "claude-opus-4-8", "claude-opus-5", "gpt-5.5" ], "spend": "≈ $1000 (ChatGPT Pro and Claude Max subscriptions)", "notes": "All Lean code in this repository was written by the listed models under the close supervision of the authors. Every proof is checked by the Lean kernel and Mathlib; the verified library contains no sorry and no custom axiom. The Mathlib-only comparator challenge files (Audit/*/Challenge.lean) each contain exactly one intentional statement-level sorry (the theorem body), which is filled by the corresponding Audit/*/Solution.lean and confirmed by leanprover/comparator (\"Lean default kernel accepts the solution\" / \"Your solution is okay!\")." }, "scope": "A Lean 4 / Mathlib formalization of the coarse-graining approach to quantitative stochastic homogenization of divergence-form elliptic equations: function-space, deterministic, probabilistic, and homogenization-scale theory up to the quenched minimal-scale theorem (Chapter 5). The supporting library also includes finite-exponent cube Calderón-Zygmund, Sobolev, fractional-Sobolev, and local coarse-graining theory. Every theorem stated in the manuscript is formalized; narrower or extra-hypothesis cases are flagged in manuscript footnotes. The two main results are exposed publicly in Homogenizati", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "Homogenization.Book.MainResults.annealedConvergence_uniformEllipticity", "file": "Homogenization/Book/MainResults.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "Homogenization.Book.MainResults.homogenizationComparison_uniformEllipticity", "file": "Homogenization/Book/MainResults.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Homogenization.StatementAudit.homogenizationComparison_uniformEllipticity", "file": "Audit/QuenchedComparison/Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "Where a formalized statement is less general than the manuscript or carries an additional hypothesis, this is flagged in a manuscript footnote. Substantive cases: a W^{2,q} estimate formalized only for q=2; the cube-wise Besov pairing only for p=2; a Caccioppoli iteration on the normalized interval [1/3,1]; deterministic domains taken bounded-open-convex rather than Lipschitz; integer moment exponent xi>2d; one Chapter-5 statement with extra hypothesis alpha= 1 (the paper", "alignment": true, "original": false, "review": { "status": "author-verified", "bucket": "author-verified", "reviewers": [ "Scott Armstrong", "Tuomo Kuusi", "Amélie Loher" ], "notes": "The Lean kernel checks the formal proofs (no unresolved sorry or custom axiom). The authors reviewed the theorem statements and their interpretation; separate agent reviews checked the statements and their dependencies. 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A uniform Kolmogorov displacement-moment bound yields a unique measurable kernel from each starting point to continuous-path space, with the prescribed finite-dimensional distributions. The library also develops strong Markov properties, generators and resolvents, stopping and killing, and Brownian motion as an example. The existence and uniqueness theorem is independently checked using Audit/ContinuousMarkovProcess/comparator.json.", "authors": [ "Scott Armstrong" ], "maintainers": [ "Scott Armstrong" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Markov Processes: Characterization and Convergence", "id": "doi:10.1002/9780470316658", "authors": [ "Stewart N. Ethier", "Thomas G. 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This Lean development proves an exact finite rational upper envelope for the number of subgroups K with H ≤ K ≤ G and the strict bound c(p) n^((log_p n)/4), where c(p) is an Euler-product times a Gaussian theta series. It formalizes the bandwise moment inequality that drives the proof, a universal base-two bound with an exact decimal coefficient, the exact optimal answer for prime-power indices, and elementary-abelian asymptotics showing that both the constant and the exponent 1/4 are optimal", "authors": [ "Sébastien Palcoux" ], "maintainers": [ "Sébastien Palcoux" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Sharp bounds for intervals in finite subgroup lattices", "id": "", "authors": [ "Sébastien Palcoux", "Pablo Spiga" ], "type": "paper", "relationship": "formalizes", "endorsement": "participated" }, { "title": "A good upper-bound for the cardinal of an interval of finite groups", "id": "https://mathoverflow.net/questions/240351/a-good-upper-bound-for-the-cardinal-of-an-interval-of-finite-groups", "authors": [ "Sébastien Palcoux" ], "type": "web discussion", "relationship": "background", "endorsement": "participated" }, { "title": "Maximal subgroups in finite and profinite groups", "id": "doi:10.1090/S0002-9947-96-01665-0", "authors": [ "A. 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Spiga" ], "type": "paper", "relationship": "background", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.GR", "math.CO" ], "msc2020": [ "20D30", "20E15", "05A16" ] }, "automation": { "methods": [ { "method": "autonomous", "models": [ "GPT-5.6 Sol" ], "framework": "ChatGPT" }, { "method": "agent", "models": [], "framework": "ChatGPT" }, { "method": "manual", "models": [], "framework": "Lean command-line tools" } ], "strongest": "autonomous", "models": [ "GPT-5.6 Sol" ], "spend": "subscription-based usage; not separately tracked", "notes": "Sébastien Palcoux is responsible for repository maintenance. Sébastien Palcoux and Pablo Spiga take responsibility for the current manuscript, which incorporates Spiga's two rounds of comments and acknowledges assistance from GPT-5.6 Sol and GPT-6 Astra. The original generation history above is retained as a historical record. No new independent human audit of the informal-to-formal alignment is claimed." }, "scope": "The substantive proof development contains formal statements and proofs corresponding to the registered source draft, as detailed in CERTIFICATION.md. The current submission manuscript has shorter exposition; some auxiliary results remain only in the formal development and the historical crosswalk. The source numbering in alignment.statements below refers to the registered draft, not the current submission. The Palomar Comparator surface deliberately selects nine headline declarations: the bandwise moment estimate; the exact finite and strict sharp bounds; the universal decimal bound; the deci", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The registered draft's extremal notation F(p^a) is a supremum over finite groups of arbitrary underlying types. The formalization replaces that universe-sized set by the logically equivalent and stronger-to-use pair of statements that the Gaussian total bounds every finite inclusion of index p^a and that one explicit elementary-abelian inclusion attains it. This is documented in CERTIFICATION.md. For five Palomar-facing declarations, `Nat.card` replaces `Fintype.card`; the wrappers are proved from the original declarations using the standard equality between these cardinalities on a finite type. This is an elaboration-level adapter, not a mathematical weakening. No other theorem-statement divergence is known from the project's own crosswalk, but that crosswalk has not yet been independently audited by a human. The Palomar surface selects nine headline results rather than every supporting declaration in the full proof development.", "alignment": true, "original": false, "review": { "status": "unchecked", "bucket": "unchecked", "reviewers": [], "notes": "The current manuscript incorporates two rounds of comments by Pablo Spiga. The unchecked status and empty reviewers list here concern independent human review of the formalization and its informal-to-formal alignment, which has not been established by this update. The user previously ran the pinned Lean build and the repository's automated no-proof-hole and axiom checks successfully. The repository workflow additionally runs Comparator and NanoDa; its result is attached to each exact Git commit. These mechanical checks do not establish novelty or bibliographic accuracy." }, "canonical": { "repo": "sebastienpalcoux/sharp-intermediate-subgroup-bounds", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-03-000005", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-03-000005", "commit": "522d6dfcb5d25d47455375e3df0f8a561b7b780b", "trust": "high", "theorems": 9, "date": "2026-09-03" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "sebastienpalcoux/sqrt6-kissing-bound/formalization.yaml", "repo": "sebastienpalcoux/sqrt6-kissing-bound", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/sebastienpalcoux/sqrt6-kissing-bound/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "The optimal universal exponential base for Euclidean kissing numbers", "description": "For every positive integer n, the Euclidean kissing number tau_n is an attained maximum and satisfies tau_n <= (sqrt(6))^n. This Lean formalization proves the bound by packing explicit bodies of revolution in the unit ball, including the geometric separation, volume integrals, and every dimension case. An explicit six-point planar configuration proves tau_2 = 6. Consequently sqrt(6) is the least nonnegative (and least positive) universal exponential base without a prefactor; the admissible bases are exactly alpha >= sqrt(6), and the supremum of tau_n^(1/n) over positive dimensions equals sqrt(", "authors": [ "Sébastien Palcoux" ], "maintainers": [ "Sébastien Palcoux" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "A Universal Square-Root-of-Six Bound for Kissing Numbers", "id": "", "authors": [ "Sébastien Palcoux" ], "type": "paper", "relationship": "formalizes", "endorsement": "participated" }, { "title": "Upper bound of the kissing number in n dimensions", "id": "https://mathoverflow.net/questions/282644/", "authors": [ "Sébastien Palcoux" ], "type": "web discussion", "relationship": "background", "endorsement": "participated" }, { "title": "Answer to Upper bound of the kissing number in n dimensions", "id": "https://mathoverflow.net/a/282685/", "authors": [ "Henry Cohn" ], "type": "web discussion", "relationship": "background", "endorsement": "not-contacted" } ], "related": [], "classification": { "arxiv": [ "math.MG", "math.CO" ], "msc2020": [ "52C17", "52C35" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "GPT-6 Astra" ], "framework": "ChatGPT" } ], "strongest": "agent", "models": [ "GPT-6 Astra" ], "spend": "", "notes": "The mathematical proof is substantive AI-generated work, not human-written mathematics merely typeset by an assistant. Mechanical checks and AI inspection are distinguished from human peer review. Cost, complete prompt logs, and exact total wall time were not recorded." }, "scope": "The full project contains the Euclidean-volume proof and its geometric, analytic, numerical, and optimality steps. Comparator selects the eight principal claims listed in comparator.json, not a separate comparison of every intermediate lemma. Four targets are existing substantive theorems. Four proved adapters expose genuine maximum cardinalities without the implementation-specific bounded-search definition. The Challenge definitions do not build in any claimed upper bound. The proof code is checked transitively, but Lean does not certify the prose or independently audit informal-to-formal ali", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The manuscript defines tau_n as the greatest realizable cardinality. The implementation constructs that number by a bounded search and separately proves its IsGreatest property. The Palomar interface instead states existence of an attained maximum and formulates the number-level consequences for any function of genuine maxima. Solution proves these adapters from the implementation, so there is no extra geometric assumption, cutoff, or weakening of the quantifiers. Universal bases are stated as nonnegative; the theorem characterizes them as alpha >= sqrt(6), hence also gives the manuscript formulation for positive bases. The revised manuscript shortens the exposition of cone separation, bicone containment, and the high-dimensional volume estimates; the Lean sources retain the original intermediate lemmas. The separate spherical-cap proof is not part of the certificate.", "alignment": false, "original": false, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [], "notes": "Palomar mechanically verified and automatically reviewed repository commit cc87870ad897f0fe15e3cdffd3d7acd15f0c1ba9 and registered it as PALOMAR-2026-09-05-000002 version 1 on 5 September 2026. The automated editorial review identified no problems. Its record is https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-05-000002&version=1. This later two-proof manuscript and documentation revision is not a new Palomar registration. 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Monthly 113 (2006), no. 10, 937-938; doi:10.2307/27642094", "authors": [ "Filip Saidak" ], "type": "article", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "Another proof of the infinite primes theorem", "id": "Amer. Math. Monthly 72 (1965), no. 3, 305", "authors": [ "Marvin C. Wunderlich" ], "type": "article", "relationship": "adapts", "endorsement": "not-contacted" }, { "title": "Identity (18 + 17*sqrt 2)^3 + (18 - 17*sqrt 2)^3 = 42^3, giving the irrationality of sqrt 2 from Fermat's Last Theorem for exponent 3", "id": "https://math.stackexchange.com/a/4438248", "authors": [ "Frazer Jarvis", "Tom Meekin" ], "type": "web post", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "Euclid's proof of the infinitude of primes", "id": "Elements, Book IX, Proposition 20", "authors": [ "Euclid" ], "type": "book", "relationship": "formalizes", "endorsement": "n/a" }, { "title": "Goldbach's proof of the infinitude of primes via the pairwise coprimality of the Fermat numbers", "id": "", "authors": [ "Christian Goldbach" ], "type": "folklore", "relationship": "formalizes", "endorsement": "n/a" }, { "title": "Euler's proof of the infinitude of primes from the divergence of the harmonic series and the Euler product", "id": "", "authors": [ "Leonhard Euler" ], "type": "folklore", "relationship": "formalizes", "endorsement": "n/a" }, { "title": "Infinitude of primes from the irrationality of pi^2 and the Euler product for zeta(2)", "id": "https://math.stackexchange.com/a/50012/350772", "authors": [], "type": "web post", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "Euclid-style proofs that there are infinitely many primes congruent to 1 and to 3 mod 4", "id": "", "authors": [], "type": "folklore", "relationship": "formalizes", "endorsement": "n/a" }, { "title": "Proofs of Fermat's little theorem (Wikipedia)", "id": "https://en.wikipedia.org/wiki/Proofs_of_Fermat%27s_little_theorem", "authors": [], "type": "web post", "relationship": "formalizes", "endorsement": "n/a" }, { "title": "Classical proofs of the irrationality of sqrt 2: infinite descent, and the 2-adic valuation argument", "id": "", "authors": [], "type": "folklore", "relationship": "formalizes", "endorsement": "n/a" }, { "title": "Classical proofs of Fermat's theorem on sums of two squares via quadratic reciprocity and via Wilson's theorem", "id": "", "authors": [], "type": "folklore", "relationship": "formalizes", "endorsement": "n/a" }, { "title": "Classical solutions of the Basel problem: Parseval's identity for the Fourier series of x, and Cauchy's cotangent squeeze", "id": "", "authors": [ "Augustin-Louis Cauchy" ], "type": "folklore", "relationship": "formalizes", "endorsement": "n/a" }, { "title": "Pascal's rule and the hockey-stick identity for binomial coefficients", "id": "", "authors": [], "type": "folklore", "relationship": "formalizes", "endorsement": "n/a" } ], "related": [ { "id": "https://github.com/leanprover-community/mathlib4", "relationship": "builds-on", "note": "Mathlib is this project's library dependency, and it is also a prior formalization of several of the same statements. The overlapping declarations listed above are not used here: each theorem family declares its own `Prop`-valued statement in a `*/Defs.lean` file, and every proof in this project proves that `Prop` afresh rather than deriving it from the mathlib counterpart. Some of those proofs ma" } ], "classification": { "arxiv": [ "math.NT", "math.CO", "math.HO", "cs.LO" ], "msc2020": [ "11A41", "11A07", "11E25", "11J72", "11M06", "05A19", "05B45", "52C20", "68V20" ] }, "automation": { "methods": [ { "method": "manual", "models": [], "framework": "" }, { "method": "agent", "models": [ "claude-opus-5", "claude-opus-4-8", "claude-opus-4-7", "claude-fable-5" ], "framework": "Claude Code" }, { "method": "agent", "models": [], "framework": "OpenAI Codex" }, { "method": "agent", "models": [], "framework": "GitHub Copilot coding agent" }, { "method": "agent", "models": [], "framework": "Aristotle (Harmonic)" } ], "strongest": "agent", "models": [ "claude-fable-5", "claude-opus-4-7", "claude-opus-4-8", "claude-opus-5" ], "spend": "subscription-based usage", "notes": "The repository's CONTRIBUTING.md requires that AI-assisted contributions be disclosed in the pull request and checked line by line before submission, and that every theorem statement be verified to say exactly what is intended. 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Where Wagon says \"place the rectangle in standard position\", the formalization instead bases the integrand, lattice, or grid at the tiled rectangle's own corner, which avoids translating the tiling; and several of his per-proof local arguments are factored into shared lemmas (`IsTiling.sum_fgArea`, `Grid`, `CornerCount`, `Cells`, `Walks`) reused across proofs. - Wagon's proof 11 (minimal cut-set, Seymour) is not formalized; it needs planar duality and a planar min-cut theorem, which mathlib does not have. - Proof 14 (Sperner's lemma, Schmerl) does not prove geometric Sperner in general; mathlib has no such lemma either (leanprover-community/mathlib4#25231 tracks it, and mathlib's ", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Seewoo Lee" ], "notes": "Reviewed by the project author rather than by an independent reviewer. 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See SOURCE_PROVENANCE.md and EXTRACTION_MANIFEST.json for exact content hashes and roles." } ], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11B83", "37P99" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Cursor Agent", "Codex" ], "framework": "Codex/Cursor with author-supplied mathematical research and Palomar harnesses" } ], "strongest": "agent", "models": [ "Codex", "Cursor Agent" ], "spend": "subscription-based; not separately tracked", "notes": "AI tools assisted extraction and packaging. Idris Ali Shaik directed the work and accepts responsibility for all claims." }, "scope": "Eighteen selected statements. The original eleven mathematical statements are preserved under public renaming. Seven companions expose the actual geometric reference law, full-group fan, exact profile, actual residual and complete local mixing recipe. Profiles cover exact allocation/hinge identity, each real-moment lower bound and existential paid-level density for canonical and admissible targets. Entropy covers positive-index increments, a finite monotone limit with explicit tail, and zero normalized growth and positive-index infimum. 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Their depth-eleven formulas match production closed forms; the original fractional statements use an existential p satisfying W/2 < p <= W, witnessed by the actual residual. The individual-root statement omits seed-pool membership and score details. The seven new companions expose the actual law and exact mixing recipe and identify them by structural equivalence proofs. Profile density specializes the fixed-level paper theorem to one existential positive paid residual before all clocks, for convergence to one or each admissible target. Entropy uses the full compact-envelope tail, which implies the printed power-tail bound; the zero-rate infimum uses positive normalized indices. The Renyi-order right limit and complementary-capacity branch are not selected. Deeper tables, full optimizers, coefficient-size comparisons and Appendix C amplification remain outside this release.", "alignment": false, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "Author-directed local review of eighteen selected statements. Challenge/Solution compilation, eighteen standard-axiom reports, semantic lint and style checks passed locally. The private Linux build and fresh axiom-report workflow runs against 52f2559e0475d80d3f1fdefa4e7d6f17f796b38b. This publication-metadata update changes no Lean source, dependency, Comparator configuration or workflow. Comparator, NanoDa and official Palomar verification remain pending; no whole-paper formalization or registry acceptance is claimed." }, "canonical": { "repo": "shaikidris/collatzconvergencepositivedensity", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-23-000002", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-23-000002", "commit": "f0fda887ee60dc8ac4ef0d864935148b03734726", "trust": "high", "theorems": 18, "date": "2026-09-23" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "shaikidris/CollatzEndpointTransport-Palomar/formalization.yaml", "repo": "shaikidris/CollatzEndpointTransport-Palomar", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/shaikidris/CollatzEndpointTransport-Palomar/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Refined Arbitrary-Target Transport Through Collatz Endpoints", "description": "A Lean 4 formalization of a quantitative stability theorem for the one-block endpoint-pullback operator of the shortcut Collatz map. If an arbitrary target set has at most C N^(1-D) exceptions among the positive integers up to N, then, for every central Renyi order 1/2 < theta < 1, every admissible fixed central window, and every positive transport coefficient below the explicit asymptotic slope, its pullback through one floor(log_2 n)-step endpoint is (K(1+C^(theta/(1+theta)))D^(-theta/(2(1+theta))), transport*D)-dense whenever D is sufficiently small. The cutoff and K depend only on the fixe", "authors": [], "maintainers": [], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "Quantitative Endpoint Transport for the Collatz Map with a Natural-Density Descent Application", "id": "doi:10.5281/zenodo.22544006", "authors": [ "Idris Ali Shaik" ], "type": "paper", "relationship": "formalizes", "endorsement": "participated" } ], "related": [ { "id": "doi:10.5281/zenodo.21797535", "relationship": "builds-on", "note": "Earlier frozen Lean 4.15.0 release of the complete CET article artifact. This selected revision starts from registered Palomar version 2 and adds only the exact refined operator dependency cone." }, { "id": "https://github.com/shaikidris/FirstPassageLinearTransport", "relationship": "independent", "note": "Independent Collatz first-passage formalization with stronger almost-all landing conclusions. It does not import CET and does not state the arbitrary-target endpoint-pullback theorem selected here." } ], "classification": { "arxiv": [ "math.NT", "math.PR" ], "msc2020": [ "11B83", "37P99", "60F10" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "OpenAI Codex" ], "framework": "Codex with author-supplied mathematical research and verification protocols" } ], "strongest": "agent", "models": [ "OpenAI Codex" ], "spend": "subscription-based; not separately tracked", "notes": "OpenAI Codex materially assisted the Lean development. Idris Ali Shaik directed the work, reviewed the outputs, and accepts responsibility for every claim." }, "scope": "This repository is a substantive formalization, not a wrapper. The immutable commit named by a Palomar submission is the authoritative substantive formalization snapshot for that intake. The compared declaration proves one uniform quantitative one-block endpoint-transport theorem for arbitrary target sets. It includes the shortcut map, exact floor(log_2 n) endpoint, complement-count definition of (C,D)-density, central-window constraints, explicit asymptotic transport slope, uniform small-D cutoff, sublinear target-constant power, and refined inverse-density power. It does not prove pointwise ", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "CollatzEndpointTransportPalomar.quantitative_arbitrary_target_endpoint_transport", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "None in the selected theorem. Challenge and Solution expose the manuscript's refined arbitrary-target endpoint transport with the same map, endpoint, density notion, parameter hypotheses, transport exponent, and prefactor powers. No eventual-descent or first-passage theorem is selected.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "The author and Codex checked the selected statement against the manuscript, rebuilt the exact theorem closure on Lean 4.33.0, and audited admitted obligations and project-specific axioms. 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The Palomar package adds the literal paper-strength Corollary 1.2(1) wrapper and ports the resulting 83 modules to Lean 4.33.0." } ], "classification": { "arxiv": [ "math.NT", "math.LO" ], "msc2020": [ "11B83", "37P99", "03B35" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "OpenAI Codex (GPT-5.6)" ], "framework": "Codex with author-supplied mathematical research and release harnesses" } ], "strongest": "agent", "models": [ "OpenAI Codex (GPT-5.6)" ], "spend": "subscription-based; not separately tracked", "notes": "OpenAI Codex contributed to the proof validation and Lean development. Idris Ali Shaik directed the work through explicit objectives, constraints, protocols, and audits, and accepts responsibility for all claims. Other generative-AI tools were used only for exposition or minor review and did not contribute to the formalized proof argument." }, "scope": "Challenge.lean advertises four declarations. The moving-endpoint theorem accepts a uniformly bounded shell-dependent exponent profile whose explicit entropy buffer tends to infinity and gives both a natural-density-one same-witness set and a quantitative dyadic-shell exceptional ratio. The fixed-exponent theorem gives, for every A above the explicit first-passage entropy threshold, c above 2/log(4/3), beta > 0, and every 0 < gamma < kappa_*(A - A_FP), separate positive landing and exceptional constants and the eventual O(X (log X)^(-gamma)) exceptional count. 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The same-witness stretched-logarithmic declaration is intentionally a qualitative projection: it keeps every 0 < delta < 1, natural density one, a same-witness landing and orbit ceiling, but exposes the formally proved fixed 6953/1000 clock. The separate quantitative declaration retains every strict 0 < sigma < 1 - delta and the explicit prefactor 5, but counts the unclocked landing predicate. It does not expose the manuscript's endpoint sigma = 1 - delta or its joint clock/ceiling predicate. The literal critical log-log/triple-log specializations are not public declarations in this package. All four are exposed through definitionally equivalent Challenge/Solution namespaces. The frozen proof cone and exact-rate wrapper were ported to Lean 4.33.0 using import-path, parser, elaboration, and tactic-compatibility edits only; their mathematical hypotheses and conclusions were not strengthened during the port.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "The author and Codex audited the four statements against the manuscript and frozen Lean exports, built the complete transitive proof cones, and checked for admitted obligations and project-specific axioms. The registered version-1 fixed theorem previously passed Palomar's independent Comparator/Landrun/NanoDa workflow. This expanded version must receive a fresh Comparator replay before registration. 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The results concern bowtie components—two triangles sharing exactly one vertex—in the subgraph induced by even-degree vertices. They include removal preserving specified endpoints, a sharp full-incidence star expansion, composition under explicit boundary hypotheses, and corollaries for graphs with two bowtie components. 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Full model/cost history not reconstructed; no claim of line-by-line human review." }, "scope": "Six selected structural declarations; extracted release verification in progress.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "Gallai.Palomar.theoremA", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": null, "comparator": true, "literature": [] }, { "declaration": "Gallai.Palomar.expansion", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": null, "comparator": true, "literature": [] }, { "declaration": "Gallai.Palomar.expansionSharp", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": null, "comparator": true, "literature": [] }, { "declaration": "Gallai.Palomar.theoremC", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": null, "comparator": true, "literature": [] }, { "declaration": "Gallai.Palomar.corollaryA1", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": null, "comparator": true, "literature": [] }, { "declaration": "Gallai.Palomar.corollaryA2", "file": "Solution.lean", "description": "", "axioms": [], "sorry_count": null, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "Indexed oriented nonempty simple paths represent edge partitions. Isolated vertices contribute no path. A1 allows distinct witnesses for the two choices of centre. No intentional weakening of selected statements.", "alignment": false, "original": false, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [], "notes": "Source native formal gates passed; extracted Comparator, independent-kernel and current Palomar replay are pending. Not human peer reviewed or registered." }, "canonical": { "repo": "shaikidris/gallai-bowtie-removal", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-14-000005", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-14-000005", "commit": "2f19c6ff50ec0a7cb0d979f33870378299d78fea", "trust": "high", "theorems": 6, "date": "2026-09-14" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "shaikidris/prime-cover-power-band-spectra-palomar/formalization.yaml", "repo": "shaikidris/prime-cover-power-band-spectra-palomar", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/shaikidris/prime-cover-power-band-spectra-palomar/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Density-one prescribed-rank spectra of prime-cover graphs", "description": "For finite prime-cover graphs, the ordered adjacency eigenvalues at prescribed arithmetic ranks obey density-one squared-error estimates in every fixed power range 0 < theta < 1/2. Relative to the prime-counting main term and explicit first-exit correction, the error is bounded by a constant times a/log(X) + log(X)^D sqrt(X/(a log(X))), with quantitative terminal exception counts and zero limiting failure density. This gives a positive power improvement over the sqrt(X)/log(X) block bound. Separate sharp every-terminal-centre statements on 21/61 < theta < 1/2 at error O(a/log(X)) remain open o", "authors": [ "Idris Ali Shaik" ], "maintainers": [ "Idris Ali Shaik" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Prescribed Eigenvalues in Power Bands of Finite Prime-Cover Graphs", "id": "https://papers.ssrn.com/abstract=7441718", "authors": [ "Idris Ali Shaik" ], "type": "paper", "relationship": "formalizes", "endorsement": "participated" }, { "title": "Ordered comparison and density-one power-band estimates", "id": "", "authors": [ "Idris Ali Shaik" ], "type": "other", "relationship": "formalizes", "endorsement": "participated" } ], "related": [ { "id": "https://github.com/shaikidris/prime-star-spectra-formalization/tree/59176e4ce57e7d6578cafd1de0029fed6fc13ccf", "relationship": "builds-on", "note": "Pinned Paper I graph, star-forest, first-exit, and analytic infrastructure." } ], "classification": { "arxiv": [ "math.CO", "math.NT" ], "msc2020": [ "05C50", "11N05" ] }, "automation": { "methods": [ { "method": "agent", "models": [], "framework": "OpenAI Codex" } ], "strongest": "agent", "models": [], "spend": "", "notes": "AI agents assisted with proof implementation, porting, documentation and verification. This records the workflow without asserting an exact model inventory for earlier runs or a separate human line-by-line proof review." }, "scope": "Phase 1 D31 entry: Theorem 1.3 and Corollary 1.4 are the two selected almost-all results proved, with two sharp terminal-band obligations open on the preserved development branch. This D31 submission tree contains no Solution admissions. The unchanged proof source passed local and private protected checks before this SSRN-reference metadata update. Exact-source verification, public-access and handoff receipts are maintained on the separate readiness audit branch; only receipts matching the intended intake commit establish its final readiness.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "PrimeCoverPowerBand.palomar_almostAll_powerBand", "file": "PrimeCoverPowerBandSolution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "PrimeCoverPowerBand.palomar_almostAll_subBlock", "file": "PrimeCoverPowerBandSolution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "", "alignment": false, "original": false, "review": { "status": "phase 1 metadata aligned with ssrn submission; exact verification receipts tracked separately", "bucket": "other", "reviewers": [], "notes": "R1--R8 passed the local D31 checklist. Both public almost-all roots have standard-axiom proofs. Terminal and global conclusions use the same error parameters and retain the initial segment. The sharp theorem and its BHP input were not strengthened or removed. Internal Corollaries 10.2 and 10.3 have their separately stated ranges, and 10.3 retains both operator terms. All error comparisons concern squared eigenvalues; strict power gains are for each fixed interior theta, not uniform up to an endpoint. The formal D31 proof uses a sufficient O(a/log(X)) molecule estimate in place of the manuscript's stronger intermediate o(1) estimate, preserving the selected final statements. Pinned Paper I already covers a uniform logarithmic centre window; the current result concerns power bands. A pre-rep" }, "canonical": { "repo": "shaikidris/prime-cover-power-band-spectra-palomar", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-10-000004", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-10-000004", "commit": "72288e1f32ba491a873e63647023b70d195f7139", "trust": "high", "theorems": 2, "date": "2026-09-10" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "shaikidris/prime-star-spectra-formalization/formalization.yaml", "repo": "shaikidris/prime-star-spectra-formalization", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/shaikidris/prime-star-spectra-formalization/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Uniform finite-prime-sum spectra and dynamics of finite prime-cover graphs", "description": "A Lean 4 formalization of the ordered adjacency spectrum of finite prime-cover graphs. Its root-centred specialization proves that the square of the largest ordered eigenvalue equals pi(X) plus four times the finite prime sum of 1/(p-1), up to o(1). More generally, for every fixed finite set S of deleted primes and epsilon > 0, the same law with the explicit centre correction c(a) holds uniformly over allowed centres a <= (log X)^(1/2-epsilon). The same window admits a uniform centre--shell evolution theorem at the explicit first-exit frequency, with a finite localization-plus-phase error boun", "authors": [ "Idris Ali Shaik" ], "maintainers": [ "Idris Ali Shaik" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Prime-Star Localization and Second-Order Spectra of Finite Divisibility Cover Graphs", "id": "https://doi.org/10.5281/zenodo.22163492", "authors": [ "Idris Ali Shaik" ], "type": "paper", "relationship": "formalizes", "endorsement": "participated" }, { "title": "On the eigenstructure of sparse matrices related to the prime number theorem", "id": "https://doi.org/10.1016/j.laa.2019.09.022", "authors": [ "J. 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The author also supplied review text generated by several other language-model systems; those reads were advisory and are not represented as independent human review." }, "scope": "The compared declarations are the undeleted root-centred headline, the full uniform finite-prime-sum form of manuscript Theorem 9.6, and the finite explicit-frequency dynamics bound underlying Corollary 9.9, for natural cutoffs X and finite deletion sets S. The submitted proof development also contains the finite star decomposition, perturbation, localization, first-exit, quantitative PNT, and prime-Mertens layers used in the proof. 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The Lean eigenvalue statement uses natural cutoffs X, the finite graph's exact arithmetic rank, and an epsilon-delta formulation of the manuscript's uniform o(1). The dynamics declaration fixes the first-exit correction by literal finite prime sums, identifies the evolution as exp(-itA), displays the localization and Feshbach constants, normalizes the centre and shell states, and proves positivity of both frequencies and their sum. It states the finite explicit bound from which the manuscript's time-window asymptotic follows. The inherited substantive source is https://github.com/shaikidris/prime-star-spectra-formalization at commit e40d375396f2965757600d7e0351ccdb69423c9b. This submitted snapshot is a toolchain compatibility port with a definitionally synchronized Palomar Challenge/Solution adapter. The current statement surface adds the root specialization and a project-local explicit-frequency dynamics module to that inherited proof cone, and selects all three results in one Comparator entry.", "alignment": true, "original": false, "review": { "status": "self-assessed and machine-checked; independent human expert review pending", "bucket": "self-assessed", "reviewers": [ "Idris Ali Shaik" ], "notes": "The source-to-Lean quantifiers and constants were audited repeatedly, the complete package was rebuilt on the pinned toolchain, and the selected public theorem reports only propext, Classical.choice, and Quot.sound. Language-model reviews do not count as independent human validation." }, "canonical": { "repo": "shaikidris/prime-star-spectra-formalization", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-29-000017", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-29-000017", "commit": "fe3fe9fa41f5c69d9e1e16abc6da144511f778b0", "trust": "high", "theorems": 3, "date": "2026-08-31" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "ShengtongZhang-alt/BN/formalization.yaml", "repo": "ShengtongZhang-alt/BN", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/ShengtongZhang-alt/BN/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "A proof of the Bollobás–Nikiforov conjecture in Lean", "description": "A complete Lean 4 / Mathlib formalization of the Bollobás–Nikiforov conjecture (2007) and of the matrix theorems that prove it. Let G be a finite simple graph with adjacency eigenvalues lambda_1 >= lambda_2 >= ..., m edges and clique number omega(G). The conjecture states that every noncomplete graph on at least two vertices satisfies lambda_1^2 + lambda_2^2 <= 2 (1 - 1/omega(G)) m. The formalization proves it (BN.lambda1_sq_add_lambda2_sq_le) as a consequence of a weighted spectral inequality: for every symmetric entrywise nonnegative matrix B with zero diagonal supported on the edges of G, t", "authors": [ "Gabriel Coutinho", "Yinchen Liu", "Thomás Jung Spier", "Quanyu Tang", "Shengtong Zhang" ], "maintainers": [ "Shengtong Zhang" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "The Bollobás–Nikiforov inequality for nonnegative edge weights", "id": "docs/sol.tex (this repository, at the submitted commit)", "authors": [ "Gabriel Coutinho", "Yinchen Liu", "Thomás Jung Spier", "Quanyu Tang", "Shengtong Zhang" ], "type": "paper", "relationship": "formalizes", "endorsement": "participated" }, { "title": "Cliques and the spectral radius", "id": "doi:10.1016/j.jctb.2006.12.002", "authors": [ "Béla Bollobás", "Vladimir Nikiforov" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Conic programming to understand sums of squares of eigenvalues of graphs", "id": "arXiv:2411.08184", "authors": [ "Gabriel Coutinho", "Thomás Jung Spier", "Shengtong Zhang" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Maxima for graphs and a new proof of a theorem of Turán", "id": "doi:10.4153/CJM-1965-053-6", "authors": [ "Theodore S. Motzkin", "Ernst G. Straus" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Some inequalities for the largest eigenvalue of a graph", "id": "doi:10.1017/S0963548301004928", "authors": [ "Vladimir Nikiforov" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Proof of a conjectured lower bound on the chromatic number of a graph", "id": "doi:10.1016/j.laa.2015.08.007", "authors": [ "Tsuyoshi Ando", "Minghua Lin" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Eigenvalues and triangles in graphs", "id": "doi:10.1017/S0963548320000462", "authors": [ "Huiqiu Lin", "Bo Ning", "Baoyindureng Wu" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "On the first two eigenvalues of regular graphs", "id": "doi:10.1016/j.laa.2024.01.002", "authors": [ "Shengtong Zhang" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Bollobás–Nikiforov conjecture for graphs with not so many triangles", "id": "arXiv:2407.19341", "authors": [ "Hitesh Kumar", "Shivaramakrishna Pragada" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Bollobás–Nikiforov conjecture holds asymptotically almost surely", "id": "arXiv:2501.07137", "authors": [ "Chunmeng Liu", "Changjiang Bu" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "A note on the Bollobás–Nikiforov conjecture", "id": "doi:10.1016/j.laa.2025.01.037", "authors": [ "Jiasheng Zeng", "Xiao-Dong Zhang" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "The Bollobás–Nikiforov conjecture for complete multipartite graphs and dense K_4-free graphs", "id": "arXiv:2603.26379", "authors": [ "Piero Giacomelli" ], "type": "paper", "relationship": "background", "endorsement": "" } ], "related": [ { "id": "https://github.com/ShengtongZhang-alt/SqOmega/tree/e988319dceceb94ae956890a3547af0062465e3e", "relationship": "builds-on", "note": "Lean 4 / Mathlib formalization (Liu, Tang, Zhang; public since July 2026) of the square-energy strengthening of Turán's theorem sqrt(s^+(G)), sqrt(s^-(G)) <= (1 - 1/omega(G)) n. It contains a doubly-nonnegative Motzkin–Straus inequality (SqOmega/DNN/MotzkinStraus.lean, dnnMotzkinStraus_*) proved via the Caro–Wei random partition, with vertex-count normalization. Code reuse: BN/Basic/Graph.lean (tu" } ], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05C50", "05C35", "15B48", "15A18" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Grok 4.6 (xAI)" ], "framework": "Cursor" }, { "method": "agent", "models": [ "Claude Fable 5.1 (Anthropic)" ], "framework": "Cursor" }, { "method": "other", "models": [ "GPT 6 Astra (OpenAI)" ], "framework": "" } ], "strongest": "agent", "models": [ "Claude Fable 5.1 (Anthropic)", "GPT 6 Astra (OpenAI)", "Grok 4.6 (xAI)" ], "spend": "not tracked", "notes": "Human-directed, agent-executed workflow. Shengtong Zhang is the responsible maintainer of this repository. Grok 4.6 agents produced essentially all of the roughly 15,700 lines of Lean code (phase 1 above); a Claude Fable 5.1 agent produced the submission packaging (phase 2); GPT 6 Astra generated the manuscript (phase 0). Correctness of proofs rests on the Lean kernel; the statements in Challenge.lean are the human-audited surface." }, "scope": "Fully formalized: (1) the Bollobás–Nikiforov inequality lambda_1(G)^2 + lambda_2(G)^2 <= 2 (1 - 1/omega(G)) |E(G)| for every finite noncomplete simple graph on at least two vertices (BN.lambda1_sq_add_lambda2_sq_le); (2) the weighted spectral inequality F(B) <= (1 - 1/omega(G)) ||B||_F^2 for symmetric entrywise nonnegative B with zero diagonal supported on E(G) (BN.weighted); (3) complete positivity of M(X) for the Gram matrix X of planar vectors in a closed half-plane through the origin (BN.matrix_theorem); (4) the rank-two Gram inequality sum_{i,j} (A_G)_ij (X_ij)_+^2 <= (1 - 1/omega(G)) ||X", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "BN.lambda1_sq_add_lambda2_sq_le", "file": "BN/Main.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "BN.weighted", "file": "BN/Spectral/Weighted.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "BN.matrix_theorem", "file": "BN/M/Main.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "BN.gram_le", "file": "BN/Spectral/Gram.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "BN.chiVec3_eq_cliqueNum", "file": "BN/Spectral/Conic.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "None known in the headline statements. Presentation and typeclass differences: (a) eigenvalues are Mathlib's Matrix.IsHermitian.eigenvalues₀ of the real adjacency matrix, antitone on Fin n and counted with algebraic multiplicity, so lambda1 and lambda2 are the values at indices 0 and 1; (b) the paper's F(B) (sum of squares of the two largest positive eigenvalues, missing terms replaced by 0) is BN.F, defined as (max lambda_1 0)^2 + (max lambda_2 0)^2 with the second term omitted when n = 1 and F = 0 when n = 0; (c) BN.weighted carries a [Nontrivial n] hypothesis inherited from its ambient section (for a one-vertex graph the only admissible B is 0, so nothing is lost), and BN.matrix_theorem and BN.M carry a [LinearOrder n] hypothesis that fixes the meaning of i < j in the sum defining M, exactly as the paper's indexing 1, ..., n does; (d) the sum in BN.gram_le runs over all ordered pairs (i, j), counting each edge twice, as in the paper's display (eq:gram); (e) chi''_{vec,3} is defined as sSup of the objective = ||X||_F^2 over the feasible set of Coutinho–Spier–Zhang's program (8) (BN.ChiVec3Feasible: X positive semidefinite, rank X <= 2, = 1, X_ij >=", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Shengtong Zhang" ], "notes": "The five compared statements and the definitions they use (adjacencyEigenvalues₀, lambda1, lambda2, F, turanFactor, inner, posPart, e, IsCompletelyPositive, laplacianCoeff, M, gram, offEdgeMatrix, ChiVec3Feasible, chiVec3) were audited by the human maintainer against the source note (see FORMALIZATION.md for the correspondence). Every proof is machine-checked by Lean; `#print axioms` for each headline theorem reports exactly propext, Classical.choice and Quot.sound, and the pinned Comparator toolchain (with the NanoDa kernel) accepts Solution against Challenge. No independent third-party review of the formalization has been performed." }, "canonical": { "repo": "shengtongzhang-alt/bn", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-07-000002", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-07-000002", "commit": "4625b2daaa48a5d1781315e78d933cc78ea6e0fb", "trust": "high", "theorems": 5, "date": "2026-09-07" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "ShengtongZhang-alt/Nikodym/formalization.yaml", "repo": "ShengtongZhang-alt/Nikodym", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/ShengtongZhang-alt/Nikodym/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "The sharp finite-field Nikodym exponent", "description": "A complete Lean 4 / Mathlib formalization of the sharp power saving in the finite-field Nikodym problem. A set N in F_q^d is a Nikodym set if through every point x there is a line all of whose points other than x lie in N. Lower bound: for every finite field F_q, every d >= 2 and every Nikodym set N in F_q^d, |N| >= q^d - (8 d^2 + 1) q^(d - 2^(1-d)); the proof goes through a carrier theorem for integral projective varieties containing many affine lines with private points, proved by finite-grid Hermite interpolation, Hilbert-function estimates (normalized monotonicity and a degree upper bound ", "authors": [ "Ting-Wei Chao", "Zach Hunter", "Cosmin Pohoata", "Hung-Hsun Hans Yu", "Shengtong Zhang" ], "maintainers": [ "Shengtong Zhang" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Nikodym sets: a power saving from finite-grid interpolation", "id": "docs/Nikodym_sharp_power_saving.tex (this repository, at the submitted commit)", "authors": [ "Ting-Wei Chao", "Zach Hunter", "Cosmin Pohoata", "Hung-Hsun Hans Yu", "Shengtong Zhang" ], "type": "paper", "relationship": "formalizes", "endorsement": "participated" }, { "title": "A prime-uniform construction of small finite-field Nikodym sets", "id": "docs/nikodym_construction.tex (this repository, at the submitted commit)", "authors": [ "Ting-Wei Chao", "Zach Hunter", "Cosmin Pohoata", "Hung-Hsun Hans Yu", "Shengtong Zhang" ], "type": "paper", "relationship": "formalizes", "endorsement": "participated" }, { "title": "Finite field Kakeya and Nikodym sets in three dimensions", "id": "doi:10.1137/17M1146099", "authors": [ "Ben Lund", "Shubhangi Saraf", "Charles Wolf" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Finite field Nikodym problem for spread line sets", "id": "arXiv:2601.20851", "authors": [ "Ting-Wei Chao", "Hung-Hsun Hans Yu" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "New Nikodym set constructions over finite fields", "id": "arXiv:2511.07721", "authors": [ "Terence Tao" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Large point-line matchings and small Nikodym sets", "id": "arXiv:2601.19879", "authors": [ "Zach Hunter", "Cosmin Pohoata", "Jacques Verstraete", "Shengtong Zhang" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "New affine-invariant codes from lifting", "id": "doi:10.1145/2422436.2422494", "authors": [ "Alan Guo", "Swastik Kopparty", "Madhu Sudan" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Some inequalities in functional analysis, combinatorics, and probability theory", "id": "doi:10.37236/330", "authors": [ "Chunrong Feng", "Liangpan Li", "Jian Shen" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "On the size of Kakeya sets in finite fields", "id": "doi:10.1090/S0894-0347-08-00607-3", "authors": [ "Zeev Dvir" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Sharp density bounds on the finite field Kakeya problem", "id": "doi:10.19086/da.30707", "authors": [ "Boris Bukh", "Ting-Wei Chao" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Nikodym set", "id": "https://en.wikipedia.org/wiki/Nikodym_set", "authors": [], "type": "web discussion", "relationship": "background", "endorsement": "" }, { "title": "Formal Conjectures (Google DeepMind): finite-field Kakeya conjecture", "id": "https://github.com/google-deepmind/formal-conjectures", "authors": [], "type": "other", "relationship": "background", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.CO", "math.AG" ], "msc2020": [ "05B25", "11T06", "13D40", "14G15" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Claude Fable 5.1 (Anthropic)", "Grok 4.6 (xAI)" ], "framework": "Cursor" } ], "strongest": "agent", "models": [ "Claude Fable 5.1 (Anthropic)", "Grok 4.6 (xAI)" ], "spend": "not tracked", "notes": "Human-directed, agent-executed workflow. The human author (a coauthor of the source notes) fixed the Lean statements in Nikodym/Definition.lean and Nikodym/Main.lean, wrote or approved the blueprints, and reviewed the final statements; AI agents produced essentially all of the ~15,000 lines of proof code. Correctness of proofs rests on the Lean kernel; the statements in Challenge.lean are the human-audited surface. Separately from the formalization, GPT 5.6 and GPT 6 Astra were used in the ideation of the mathematics recorded in the source notes (see README.md)." }, "scope": "Fully formalized: (1) the lower bound |N| >= q^d - (8 d^2 + 1) q^(d - 2^(1-d)) for every finite field F_q (any characteristic, q any prime power), every d >= 2 and every Nikodym set N in F_q^d, in both a denominator-free natural-number form (Nikodym.card_compl_pow_mul_card_le) and the displayed real form with real exponents (Nikodym.card_ge_pow_sub); (2) the upper bound: for every d >= 2 and epsilon > 0 there is q_0 such that every finite field of prime cardinality q >= q_0 contains a Nikodym set of size at most q^d - q^(d - 2^(1-d) - epsilon) (Nikodym.exists_isNikodym_card_le). Not formalized", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "Nikodym.card_compl_pow_mul_card_le", "file": "Nikodym/Main.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Nikodym.card_ge_pow_sub", "file": "Nikodym/Main.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Nikodym.exists_isNikodym_card_le", "file": "Nikodym/Main.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "None known in the headline statements. Presentation differences: the notes state the lower bound for the complement, |F_q^d \\ N| <= (8 d^2 + 1) q^(d - 2^(1-d)); the Lean statements give the equivalent bound on |N| and, separately, the equivalent natural-number form obtained by raising to the power 2^(d-1). Real exponents use Mathlib's Real.rpow. The construction is stated only for prime q (as in the note's prime-uniform corollary); the lower bound holds for every finite field with no restriction on the characteristic (as in the note). The set produced by the construction is only asserted to exist; its explicit description is internal to the proof.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Shengtong Zhang" ], "notes": "The three compared statements and the definition Nikodym.IsNikodym were audited by the human author against the source notes (see FORMALIZATION.md for the correspondence). Every proof is machine-checked by Lean; `#print axioms` for each headline theorem reports exactly propext, Classical.choice and Quot.sound. No independent third-party review of the formalization has been performed." }, "canonical": { "repo": "shengtongzhang-alt/nikodym", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-06-000002", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-06-000002", "commit": "0cd3550000630f4653fac67db43c2d264a0ca426", "trust": "high", "theorems": 3, "date": "2026-09-06" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "shhyang/n4code_lean/formalization.yaml", "repo": "shhyang/n4code_lean", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/shhyang/n4code_lean/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "N4Code: optimal finite-length block codes of size four for binary symmetric channels", "description": "A Lean 4 and mathlib formalization of the Dong-Yang classification of optimal (n,4) binary block codes for binary symmetric channels. The registered results are Theorems 1-5 of the paper: existence of an optimal linear or Class-I code, the explicit linear optimal codes by residue class modulo 3, the universal strict-improvement reductions outside the linear, Class-I, and Class-II classes, the Class-I condition reduction to linear codes, and the finite classification for blocklengths 2 <= n <= 8. The Lean statements use the strict-improvement contrapositive forms.", "authors": [ "Shenghao Yang", "Yanyan Dong" ], "maintainers": [ "Shenghao Yang" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "On Optimal Finite-Length Block Codes of Size Four for Binary Symmetric Channels", "id": "10.1109/TIT.2024.3504823", "authors": [ "Yanyan Dong", "Shenghao Yang" ], "type": "article", "relationship": "formalizes", "endorsement": "endorsed" } ], "related": [], "classification": { "arxiv": [ "cs.IT", "math.IT" ], "msc2020": [ "94B05" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Codex" ], "framework": "OpenAI Codex" } ], "strongest": "agent", "models": [ "Codex" ], "spend": "not tracked", "notes": "The proof scripts were developed with substantial AI assistance. Authors wrote the theorem statements, definitions, and overall architecture, and reviewed the machine-checked proofs." }, "scope": "The Comparator targets are the self-contained statements of Theorems 1-5 in Challenge.lean. Theorem 1 states that for every n >= 2 and every crossover probability there is an optimal code that is linear or Class-I. Theorem 2 gives the explicit linear optimal codes in the residue classes n = 3k-1, 3k, and 3k+1, plus n = 3. Theorem 3 states that any code not equivalent to a linear, Class-I, or Class-II code is universally strictly dominated. Theorem 4 states the corresponding reduction to linear codes under the paper's Class-I condition. Theorem 5 gives the finite classification for 2 <= n <= 8,", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "Known supporting-result mismatch: the paper's Theorem 16 equality clause for n = 3 presumes a genuine code with pairwise distinct codewords, but Lean's Code type permits duplicate rows. The Lean development states class1_one with a DistinctRows hypothesis and handles duplicate-row Class-I codes separately. This discrepancy is confined to that supporting reduction and does not alter the registered Theorem 1-5 statements or their proofs, which are formalized in the enlarged Code type and do not rely on the paper's unqualified n = 3 equality clause.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "The Lean kernel checks every proof. A separate axiom audit reports only propext, Classical.choice, and Quot.sound for the headline theorems." }, "canonical": { "repo": "shhyang/n4code_lean", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-05-000003", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-05-000003", "commit": "1a501146e430d46c355aa937689f88f24e5ea822", "trust": "high", "theorems": 8, "date": "2026-09-05" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "sloisel/MPE-Formal/formalization.yaml", "repo": "sloisel/MPE-Formal", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/sloisel/MPE-Formal/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Local convergence of MPE and RRE for nonlinear fixed-point iterations", "description": "Machine-checked proofs of three probabilistic convergence theorems for restarted Minimal Polynomial Extrapolation (MPE) applied to nonlinear fixed-point iterations. Dithered restarts keep the error below the doubly exponential schedule delta^(theta^m), for any theta in (1,2), from every starting point -- the breakdown set included -- with an explicit failure-probability exponent over the dither; in the root sense the convergence order is at least theta. Under simple spectrum, the error follows a quadratically shrinking dither schedule (root order at least 2) with failure probability linear in ", "authors": [ "Sébastien Loisel" ], "maintainers": [ "Sébastien Loisel" ], "license": "MIT", "role": "", "substantive": "", "sources": [ { "title": "On the local convergence of minimal polynomial and reduced rank extrapolation for nonlinear fixed-point iterations", "id": "Submitted manuscript, July 2026; no public identifier yet", "authors": [ "Sébastien Loisel" ], "type": "article", "relationship": "formalizes", "endorsement": "participated" }, { "title": "A polynomial extrapolation method for finding limits and antilimits of vector sequences", "id": "SIAM J. Numer. Anal. 13 (1976) 734-752", "authors": [ "S. Cabay", "L. W. Jackson" ], "type": "article", "relationship": "background", "endorsement": "" }, { "title": "Extrapolation methods for vector sequences", "id": "SIAM Rev. 29 (1987) 199-233", "authors": [ "D. A. Smith", "W. F. Ford", "A. Sidi" ], "type": "article", "relationship": "background", "endorsement": "" }, { "title": "Some results about vector extrapolation methods and related fixed-point iterations", "id": "J. Comput. Appl. Math. 36 (1991) 385-398", "authors": [ "K. Jbilou", "H. Sadok" ], "type": "article", "relationship": "background", "endorsement": "" }, { "title": "Review of two vector extrapolation methods of polynomial type with applications to large-scale problems", "id": "J. Comput. Sci. 3 (2012) 92-101", "authors": [ "A. Sidi" ], "type": "article", "relationship": "background", "endorsement": "" }, { "title": "Minimal polynomial and reduced rank extrapolation methods are related", "id": "Adv. Comput. Math. 43 (2017) 151-170", "authors": [ "A. Sidi" ], "type": "article", "relationship": "background", "endorsement": "" }, { "title": "Vector Extrapolation Methods with Applications", "id": "SIAM, Philadelphia, 2017", "authors": [ "A. Sidi" ], "type": "book", "relationship": "background", "endorsement": "" }, { "title": "A convergence study for reduced rank extrapolation on nonlinear systems", "id": "Numer. Algorithms 84 (2020) 957-982", "authors": [ "A. Sidi" ], "type": "article", "relationship": "background", "endorsement": "" }, { "title": "Acceleration methods for fixed-point iterations", "id": "Acta Numer. 34 (2025) 805-890", "authors": [ "Y. Saad" ], "type": "article", "relationship": "background", "endorsement": "" }, { "title": "Anderson acceleration for fixed-point iterations", "id": "SIAM J. Numer. Anal. 49 (2011) 1715-1735", "authors": [ "H. F. Walker", "P. Ni" ], "type": "article", "relationship": "background", "endorsement": "" }, { "title": "Convergence analysis for Anderson acceleration", "id": "SIAM J. Numer. Anal. 53 (2015) 805-819", "authors": [ "A. Toth", "C. T. Kelley" ], "type": "article", "relationship": "background", "endorsement": "" }, { "title": "A proof that Anderson acceleration improves the convergence rate in linearly converging fixed-point methods (but not in those converging quadratically)", "id": "SIAM J. Numer. Anal. 58 (2020) 788-810", "authors": [ "C. Evans", "S. Pollock", "L. Rebholz", "M. Xiao" ], "type": "article", "relationship": "background", "endorsement": "" }, { "title": "Regularized nonlinear acceleration", "id": "Math. Program. 179 (2020) 47-83", "authors": [ "D. Scieur", "A. d'Aspremont", "F. Bach" ], "type": "article", "relationship": "background", "endorsement": "" }, { "title": "Estimating the largest eigenvalue by the power and Lanczos algorithms with a random start", "id": "SIAM J. Matrix Anal. Appl. 13 (1992) 1094-1122", "authors": [ "J. Kuczynski", "H. Wozniakowski" ], "type": "article", "relationship": "background", "endorsement": "" }, { "title": "On nonconvex optimization for machine learning: gradients, stochasticity, and saddle points", "id": "J. ACM 68 (2021) 1-29", "authors": [ "C. Jin", "P. Netrapalli", "R. Ge", "S. M. Kakade", "M. I. Jordan" ], "type": "article", "relationship": "background", "endorsement": "" }, { "title": "A look-ahead Lanczos algorithm for unsymmetric matrices", "id": "Math. Comp. 44 (1985) 105-124", "authors": [ "B. N. Parlett", "D. R. Taylor", "Z. A. Liu" ], "type": "article", "relationship": "background", "endorsement": "" }, { "title": "Optimality and degeneracy in linear programming", "id": "Econometrica 20 (1952) 160-170", "authors": [ "A. Charnes" ], "type": "article", "relationship": "background", "endorsement": "" }, { "title": "Simulation of simplicity: a technique to cope with degenerate cases in geometric algorithms", "id": "ACM Trans. Graph. 9 (1990) 66-104", "authors": [ "H. Edelsbrunner", "E. P. Mucke" ], "type": "article", "relationship": "background", "endorsement": "" }, { "title": "The dynamics of Rayleigh quotient iteration", "id": "SIAM J. Numer. Anal. 26 (1989) 624-636", "authors": [ "S. Batterson", "J. Smillie" ], "type": "article", "relationship": "background", "endorsement": "" }, { "title": "Rayleigh quotient iteration for nonsymmetric matrices", "id": "Math. Comp. 55 (1990) 169-178", "authors": [ "S. Batterson", "J. Smillie" ], "type": "article", "relationship": "background", "endorsement": "" }, { "title": "On an extremal problem for polynomials in n variables", "id": "Math. USSR Izv. 7 (1973) 345-356", "authors": [ "Yu. A. Brudnyi", "M. I. Ganzburg" ], "type": "article", "relationship": "background", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.NA" ], "msc2020": [ "65B05", "65F10", "65H10", "41A17", "60D05" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "claude-opus-5", "claude-opus-4-x (earlier sessions)" ], "framework": "Claude Code" }, { "method": "manual", "models": [], "framework": "" } ], "strongest": "agent", "models": [ "claude-opus-4-x (earlier sessions)", "claude-opus-5" ], "spend": "subscription-based usage", "notes": "The corresponding disclosure in the manuscript states that Claude Code was used to discuss the problem and to formalize the three theorems." }, "scope": "All three main theorems of the manuscript, for ambient dimension n >= 2 (the first is registered at the general-window Gram convention; see fidelity note (2)). The setting: a fixed-point iteration x_{j+1} = f(x_j) in R^n, with f smooth near a fixed point placed at 0 (C^2 for the two dithered theorems, C^3 for the undithered one) and Borel measurable globally -- a hypothesis satisfied by any continuous iteration, needed because the random process is defined through the global map -- with Jacobian A = Df(0), and A - I invertible. Minimal Polynomial Extrapolation (MPE) accelerates the iteration: ", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "MPE.mpe_dithered", "file": "Formal/Statement.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "MPE.mpe_quadratic", "file": "Formal/Statement.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "MPE.mpe_undithered", "file": "Formal/Statement.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "(1) Noise model: every random variable is uniform on a cube rather than a ball. The dither at cycle m is delta_m times a variable uniform on [-1,1]^n (clamped coordinatewise; the clamp is almost surely the identity under the product measure), and the starting point of Theorem 5.5 is rho times a variable uniform on [-1,1]^n, where the manuscript draws both from balls. This changes constants only; every exponent is as in the manuscript. (2) Window convention in Theorem 4.7: the registered statement (MPE.mpe_dithered) is proved at the standing window k = deg m_A in the Gram form for every k, with failure exponent (2-theta)/(2k); the manuscript's convention switches at the full window k = n to the determinant form, giving the sharper exponent (2-theta)/n there. The sharper full-window variant is also proved in this repository (MPE.mpe_dithered_C2_stmt, Formal/SevenFull.lean) but is not among the registered statements. (3) The ambient dimension is n = M + 2, so the statements cover n >= 2 only; in dimension 1 restarted MPE is Steffensen's method and the phenomenon being controlled does not occur. (4) The restarted map is used in its cleared form S = Ntilde/sigmatilde throughout; Theorem", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Sébastien Loisel" ], "notes": "The author directed the development and audited the three registered statements against the manuscript, hypothesis by hypothesis. 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Solution verified through human audit by the maintainers with 0 errors, 0 sorries, and strictly standard kernel axioms." }, "canonical": { "repo": "sneed-and-feed/lean-theorems-1", "directory": "palomar/colorful_helly" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "sneed-and-feed/lean-theorems-1/palomar/crossing_lemma/formalization.yaml", "repo": "sneed-and-feed/lean-theorems-1", "path": "palomar/crossing_lemma/formalization.yaml", "directory": "palomar/crossing_lemma", "origins": [ "search" ], "url": "https://github.com/sneed-and-feed/lean-theorems-1/blob/HEAD/palomar/crossing_lemma/formalization.yaml", "version": "v0.4", "missing": [], "name": "Székely's Algebraic Expectation Amplification Lemma for Graph Crossings", "description": "A machine-checked Lean 4 formalization of Székely's Algebraic Expectation Amplification Lemma (László Székely, 1997) for the Crossing Lemma (Ajtai, Chvátal, Newborn, Szemerédi 1982; Leighton 1983) and its application to the Szemerédi–Trotter point-line incidence theorem. 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Proves that for any non-zero real polynomial p(X) over ℝ, the number of positive real roots counted with algebraic multiplicity is bounded above by the number of sign variations in its coefficient sequence (ignoring zero coefficients), and their difference is an even non-negative integer. 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Solution verified through maintainer audit with 0 errors, 0 warnings, 0 sorries, and strictly standard kernel axioms. Imports and connects to Mathlib.Algebra.Polynomial.RuleOfSigns for the upper bound, while formalizing the complete sign-variation parity theorem and list-based interface." }, "canonical": { "repo": "sneed-and-feed/lean-theorems-1", "directory": "palomar/descartes_rule_of_signs" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "sneed-and-feed/lean-theorems-1/palomar/dilworth_mirsky/formalization.yaml", "repo": "sneed-and-feed/lean-theorems-1", "path": "palomar/dilworth_mirsky/formalization.yaml", "directory": "palomar/dilworth_mirsky", "origins": [ "search" ], "url": "https://github.com/sneed-and-feed/lean-theorems-1/blob/HEAD/palomar/dilworth_mirsky/formalization.yaml", "version": "v0.4", "missing": [], "name": "Dilworth's Poset Chain Decomposition Theorem and Mirsky's Dual Theorem", "description": "A complete, machine-checked Lean 4 formalization of Dilworth's Theorem (Robert P. 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To the maintainers' knowledge, the theorem was not found in an exact declaration name, docstring, and type signature search of the pinned Mathlib revision (Mathlib v4.34.0-rc1).", "authors": [ "Noah Cohen" ], "maintainers": [ "Noah Cohen" ], "license": "CC0-1.0", "role": "", "substantive": "", "sources": [ { "title": "Intersection theorems for systems of finite sets", "id": "", "authors": [ "P. Erdős", "C. Ko", "R. Rado" ], "type": "paper", "relationship": "formalizes", "endorsement": "" }, { "title": "A simple proof of the Erdös-Chao Ko-Rado theorem", "id": "", "authors": [ "G. O. H. Katona" ], "type": "paper", "relationship": "adapts", "endorsement": "" }, { "title": "Some intersection theorems for systems of finite sets", "id": "", "authors": [ "A. J. W. Hilton", "E. C. 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Solution verified through human audit by the maintainers with 0 errors, 0 sorries, and strictly standard kernel axioms." }, "canonical": { "repo": "sneed-and-feed/lean-theorems-1", "directory": "palomar/erdos_ko_rado" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "sneed-and-feed/lean-theorems-1/palomar/erdos_szekeres_convex/formalization.yaml", "repo": "sneed-and-feed/lean-theorems-1", "path": "palomar/erdos_szekeres_convex/formalization.yaml", "directory": "palomar/erdos_szekeres_convex", "origins": [ "search" ], "url": "https://github.com/sneed-and-feed/lean-theorems-1/blob/HEAD/palomar/erdos_szekeres_convex/formalization.yaml", "version": "v0.4", "missing": [], "name": "The Happy Ending Theorem (Erdős–Szekeres Convex Polygon Theorem, 1935)", "description": "A comprehensive, machine-checked Lean 4 formalization of the Erdős–Szekeres Convex Polygon Theorem (1935). Formalizes orientation determinants, general position, 2D rotation transformations with finite slope avoidance (rotating any finite set to have mutually distinct x-coordinates), x-coordinate sorting, cup-cap sequences with Pascal-type binomial split recurrences, halfspace separation for convex hulls, and rotation invariance of convex polygons. Proves that any finite set of at least choose(2k - 4, k - 2) + 1 points in general position in ℝ² contains the vertices of a strictly convex k-gon,", "authors": [ "Noah Cohen" ], "maintainers": [ "Noah Cohen" ], "license": "CC0-1.0", "role": "", "substantive": "", "sources": [ { "title": "A combinatorial problem in geometry", "id": "", "authors": [ "Paul Erdős", "George Szekeres" ], "type": "paper", "relationship": "formalizes", "endorsement": "" }, { "title": "Problem 1835", "id": "", "authors": [ "Esther Klein" ], "type": "paper", "relationship": "background", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.CO", "math.MG" ], "msc2020": [ "52C10", "05D10", "52A10" ] }, "automation": { "methods": [ { "method": "agent", "models": [], "framework": "" } ], "strongest": "agent", "models": [], "spend": "", "notes": "" }, "scope": "", "sorry_count": null, "sorry_in_definitions": null, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "", "alignment": false, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Noah Cohen" ], "notes": "Challenge contains deliberate proof holes (`:= sorry` for automated benchmark evaluation), while all definitions and theorem statements are completely specified. Solution verified through human audit by the maintainers with 0 errors, 0 sorries, and strictly standard kernel axioms." }, "canonical": { "repo": "sneed-and-feed/lean-theorems-1", "directory": "palomar/erdos_szekeres_convex" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "sneed-and-feed/lean-theorems-1/palomar/erdos_unit_distances/formalization.yaml", "repo": "sneed-and-feed/lean-theorems-1", "path": "palomar/erdos_unit_distances/formalization.yaml", "directory": "palomar/erdos_unit_distances", "origins": [ "search" ], "url": "https://github.com/sneed-and-feed/lean-theorems-1/blob/HEAD/palomar/erdos_unit_distances/formalization.yaml", "version": "v0.4", "missing": [], "name": "Spencer–Szemerédi–Trotter Bound on the Erdős Unit Distance Problem", "description": "A complete machine-checked Lean 4 formalization of the Spencer–Szemerédi–Trotter (1984) upper bound u(n) = O(n^(4/3)) on the Erdős unit distance problem in the Euclidean plane, using László Székely's circle crossing lemma approach (1997). Defines the UnitDistanceSystem reduction carrier interface linking point-circle incidences to topological circular arc crossings in the plane. Proves explicit non-asymptotic bounds u(n) <= 2 n^(4/3) + 4n, the tight linear bound u(n) <= 2 n^(4/3) + 5/2 n, the uniform bound u(n) <= 4 n^(4/3) for n >= 8, the global bound u(n) <= 6 n^(4/3) for all n >= 1, and the", "authors": [ "Noah Cohen" ], "maintainers": [ "Noah Cohen" ], "license": "CC0-1.0", "role": "", "substantive": "", "sources": [ { "title": "Unit distances in the Euclidean plane", "id": "", "authors": [ "Joel Spencer", "Endre Szemerédi", "William T. Trotter" ], "type": "paper", "relationship": "formalizes", "endorsement": "" }, { "title": "On Sets of Distances of n Points", "id": "", "authors": [ "Paul Erdős" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Crossing numbers and hard Erdős problems in discrete geometry", "id": "", "authors": [ "László A. 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Solution verified through human audit by the maintainers with 0 errors, 0 sorries, and strictly standard kernel axioms." }, "canonical": { "repo": "sneed-and-feed/lean-theorems-1", "directory": "palomar/erdos_unit_distances" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "sneed-and-feed/lean-theorems-1/palomar/euler_polyhedron/formalization.yaml", "repo": "sneed-and-feed/lean-theorems-1", "path": "palomar/euler_polyhedron/formalization.yaml", "directory": "palomar/euler_polyhedron", "origins": [ "search" ], "url": "https://github.com/sneed-and-feed/lean-theorems-1/blob/HEAD/palomar/euler_polyhedron/formalization.yaml", "version": "v0.4", "missing": [], "name": "Combinatorial Map Parity, Face-Degree Handshaking Bounds & Combinatorial Genus Invariants", "description": "A machine-checked Lean 4 formalization of Tutte–Edmonds combinatorial map invariants, face-degree handshaking inequalities, and planar edge bounds across two carrier frameworks: (1) Tutte–Edmonds Combinatorial Maps (dart rotation systems on finite sets with edge involution alpha and vertex permutation sigma), proving the universal even parity of the Euler characteristic chi(M) via the permutation signature homomorphism, the face-cycle handshaking inequality k * F <= 2E derived from absence of monogons and face degree >= k, planar edge bounds E <= 3V - 6 and E <= 2V - 4 derived conditionally un", "authors": [ "Noah Cohen" ], "maintainers": [ "Noah Cohen" ], "license": "CC0-1.0", "role": "", "substantive": "", "sources": [ { "title": "Elementa doctrinae solidorum", "id": "", "authors": [ "Leonhard Euler" ], "type": "paper", "relationship": "inspired_by", "endorsement": "" }, { "title": "Recherches sur les polyèdres", "id": "", "authors": [ "Augustin-Louis Cauchy" ], "type": "paper", "relationship": "formalizes", "endorsement": "" }, { "title": "A combinatorial representation for polyhedral surfaces", "id": "", "authors": [ "Jack Edmonds" ], "type": "paper", "relationship": "formalizes", "endorsement": "" }, { "title": "A census of planar maps", "id": "", "authors": [ "W. T. Tutte" ], "type": "paper", "relationship": "adapts", "endorsement": "" }, { "title": "Über das Problem der Nachbargebiete", "id": "", "authors": [ "Lothar Heffter" ], "type": "paper", "relationship": "adapts", "endorsement": "" } ], "related": [ { "id": "https://github.com/leanprover-community/mathlib4/blob/master/Mathlib/Combinatorics/SimpleGraph/Acyclic.lean", "relationship": "builds-on", "note": "The tree theorem `tree_euler_formula` is an arithmetic wrapper of Mathlib's pre-existing `SimpleGraph.IsTree.card_edgeFinset` theorem, reformulating |E| = |V| - 1 into the Euler form |V| - |E| + 1 = 2." } ], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05C10", "52B05", "57M15" ] }, "automation": { "methods": [ { "method": "agent", "models": [], "framework": "" } ], "strongest": "agent", "models": [], "spend": "", "notes": "" }, "scope": "", "sorry_count": null, "sorry_in_definitions": null, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "", "alignment": false, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Noah Cohen" ], "notes": "Challenge contains deliberate proof holes (`:= sorry` for automated benchmark evaluation), while all definitions and theorem statements are completely specified with 100% AST parity. Human review verified mathematical statements, carrier fidelity, absence of custom axioms, and clean sequential Lean 4 builds." }, "canonical": { "repo": "sneed-and-feed/lean-theorems-1", "directory": "palomar/euler_polyhedron" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "sneed-and-feed/lean-theorems-1/palomar/frankl_wilson/formalization.yaml", "repo": "sneed-and-feed/lean-theorems-1", "path": "palomar/frankl_wilson/formalization.yaml", "directory": "palomar/frankl_wilson", "origins": [ "search" ], "url": "https://github.com/sneed-and-feed/lean-theorems-1/blob/HEAD/palomar/frankl_wilson/formalization.yaml", "version": "v0.4", "missing": [], "name": "The Frankl–Wilson Theorem on Modulo-p Intersecting Set Families", "description": "A complete machine-checked Lean 4 formalization of the Frankl–Wilson theorem (1981) for modulo-p intersecting set families. Uses the polynomial linear algebra method over ZMod p to prove both the general non-uniform bound |F| <= sum_{i=0}^s choose(n, i) and the uniform cardinality bound |F| <= choose(n, s). To the maintainers' knowledge, the theorem was not found in an exact declaration name, docstring, and type signature search of the pinned Mathlib revision (Mathlib v4.34.0-rc1).", "authors": [ "Noah Cohen" ], "maintainers": [ "Noah Cohen" ], "license": "CC0-1.0", "role": "", "substantive": "", "sources": [ { "title": "Intersection theorems with geometric consequences", "id": "", "authors": [ "P. Frankl", "R. M. Wilson" ], "type": "paper", "relationship": "formalizes", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05D05", "11B75", "15A03" ] }, "automation": { "methods": [ { "method": "agent", "models": [], "framework": "" } ], "strongest": "agent", "models": [], "spend": "", "notes": "" }, "scope": "", "sorry_count": null, "sorry_in_definitions": null, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "", "alignment": false, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Noah Cohen" ], "notes": "Challenge contains deliberate proof holes (`:= sorry` for automated benchmark evaluation), while all definitions and theorem statements are completely specified. Solution verified through maintainer audit with 0 errors, 0 sorries, and strictly standard kernel axioms." }, "canonical": { "repo": "sneed-and-feed/lean-theorems-1", "directory": "palomar/frankl_wilson" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "sneed-and-feed/lean-theorems-1/palomar/friendship_theorem/formalization.yaml", "repo": "sneed-and-feed/lean-theorems-1", "path": "palomar/friendship_theorem/formalization.yaml", "directory": "palomar/friendship_theorem", "origins": [ "search" ], "url": "https://github.com/sneed-and-feed/lean-theorems-1/blob/HEAD/palomar/friendship_theorem/formalization.yaml", "version": "v0.4", "missing": [], "name": "The Friendship Theorem on Graphs with Unique Common Neighbors", "description": "A complete, machine-checked Lean 4 formalization of the Friendship Theorem (P. Erdős, A. Rényi, and V. T. Sós, 1966; H. S. Wilf, 1971). Proves that any finite simple graph on at least three vertices (|V| ≥ 3) in which every pair of distinct vertices shares exactly one common neighbor must contain a universal vertex (the 'politician'), structuring the graph as a collection of triangles sharing a central vertex (a windmill graph). To the maintainers' knowledge, the theorem was not found in an exact declaration name, docstring, and type signature search of the pinned Mathlib revision (Mathlib v4.", "authors": [ "Noah Cohen" ], "maintainers": [ "Noah Cohen" ], "license": "CC0-1.0", "role": "", "substantive": "", "sources": [ { "title": "On a problem of graph theory", "id": "", "authors": [ "P. Erdős", "A. Rényi", "V. T. Sós" ], "type": "paper", "relationship": "formalizes", "endorsement": "" }, { "title": "The friendship theorem", "id": "", "authors": [ "H. S. Wilf" ], "type": "paper", "relationship": "adapts", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05C50", "05C75", "05E30" ] }, "automation": { "methods": [ { "method": "agent", "models": [], "framework": "" } ], "strongest": "agent", "models": [], "spend": "", "notes": "" }, "scope": "", "sorry_count": null, "sorry_in_definitions": null, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "", "alignment": false, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Noah Cohen" ], "notes": "Challenge contains deliberate statement holes. Solution verified through human audit by the maintainers with 0 errors, 0 sorries, and strictly standard kernel axioms." }, "canonical": { "repo": "sneed-and-feed/lean-theorems-1", "directory": "palomar/friendship_theorem" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "sneed-and-feed/lean-theorems-1/palomar/friendship_windmill/formalization.yaml", "repo": "sneed-and-feed/lean-theorems-1", "path": "palomar/friendship_windmill/formalization.yaml", "directory": "palomar/friendship_windmill", "origins": [ "search" ], "url": "https://github.com/sneed-and-feed/lean-theorems-1/blob/HEAD/palomar/friendship_windmill/formalization.yaml", "version": "v0.4", "missing": [], "name": "The Friendship Windmill Structure Theorem (Erdős–Rényi–Sós 1966)", "description": "A complete, machine-checked Lean 4 formalization of the Friendship Windmill Structure Theorem (Erdős, Rényi, and Sós, 1966). Proves that any finite graph in which every pair of distinct vertices shares exactly one common neighbor possesses a universal vertex w, has an odd number of vertices |V| = 2k + 1, induces a 1-regular perfect matching on V \\ {w}, and is isomorphic to the Dutch windmill graph Wd(k, 2) To the maintainers' knowledge, the theorem was not found in an exact declaration name, docstring, and type signature search of the pinned Mathlib revision (Mathlib v4.34.0-rc1).", "authors": [ "Noah Cohen" ], "maintainers": [ "Noah Cohen" ], "license": "CC0-1.0", "role": "", "substantive": "", "sources": [ { "title": "On a problem of graph theory", "id": "", "authors": [ "Paul Erdős", "Alfréd Rényi", "Vera T. Sós" ], "type": "paper", "relationship": "formalizes", "endorsement": "" }, { "title": "Proofs from THE BOOK", "id": "", "authors": [ "Martin Aigner", "Günter M. Ziegler" ], "type": "book", "relationship": "background", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05C75", "05C50", "05C07" ] }, "automation": { "methods": [ { "method": "agent", "models": [], "framework": "" } ], "strongest": "agent", "models": [], "spend": "", "notes": "" }, "scope": "", "sorry_count": null, "sorry_in_definitions": null, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "", "alignment": false, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Noah Cohen" ], "notes": "Challenge contains deliberate proof holes (`:= sorry` for automated benchmark evaluation), while all definitions and theorem statements are completely specified. Solution verified through human audit by the maintainers with 0 errors, 0 sorries, and strictly standard kernel axioms." }, "canonical": { "repo": "sneed-and-feed/lean-theorems-1", "directory": "palomar/friendship_windmill" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "sneed-and-feed/lean-theorems-1/palomar/graham_pollak/formalization.yaml", "repo": "sneed-and-feed/lean-theorems-1", "path": "palomar/graham_pollak/formalization.yaml", "directory": "palomar/graham_pollak", "origins": [ "search" ], "url": "https://github.com/sneed-and-feed/lean-theorems-1/blob/HEAD/palomar/graham_pollak/formalization.yaml", "version": "v0.4", "missing": [], "name": "The Graham–Pollak Theorem on Bipartite Partitions of Complete Graphs", "description": "A complete, machine-checked Lean 4 formalization of the Graham–Pollak theorem (1971) and its tightness on complete bipartite partitions of complete graphs. Proves that any partition of the edges of the complete graph K_n into m complete bipartite subgraphs requires at least n - 1 bipartite graphs (m >= n - 1), adapting Helge Tverberg's 1982 linear-algebraic quadratic form argument. Also formalizes the canonical star partition To the maintainers' knowledge, the theorem was not found in an exact declaration name, docstring, and type signature search of the pinned Mathlib revision (Mathlib v4.34.", "authors": [ "Noah Cohen" ], "maintainers": [ "Noah Cohen" ], "license": "CC0-1.0", "role": "", "substantive": "", "sources": [ { "title": "On embedding graphs in squashed cubes", "id": "", "authors": [ "R. L. Graham", "H. O. Pollak" ], "type": "paper", "relationship": "formalizes", "endorsement": "" }, { "title": "On the decomposition of K_n into complete bipartite graphs", "id": "", "authors": [ "Helge Tverberg" ], "type": "paper", "relationship": "adapts", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05C50", "05C70", "15A03" ] }, "automation": { "methods": [ { "method": "agent", "models": [], "framework": "" } ], "strongest": "agent", "models": [], "spend": "", "notes": "" }, "scope": "", "sorry_count": null, "sorry_in_definitions": null, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "", "alignment": false, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Noah Cohen" ], "notes": "Challenge contains deliberate statement holes. Solution verified through human audit by the maintainers with 0 errors, 0 sorries, and strictly standard kernel axioms." }, "canonical": { "repo": "sneed-and-feed/lean-theorems-1", "directory": "palomar/graham_pollak" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "sneed-and-feed/lean-theorems-1/palomar/hall_marriage/formalization.yaml", "repo": "sneed-and-feed/lean-theorems-1", "path": "palomar/hall_marriage/formalization.yaml", "directory": "palomar/hall_marriage", "origins": [ "search" ], "url": "https://github.com/sneed-and-feed/lean-theorems-1/blob/HEAD/palomar/hall_marriage/formalization.yaml", "version": "v0.4", "missing": [], "name": "Hall's Marriage Theorem on Systems of Distinct Representatives", "description": "A complete, machine-checked Lean 4 formalization of Hall's Marriage Theorem (Philip Hall 1935, Halmos–Vaughan 1950; Freek Wiedijk #87). 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Vaughan" ], "type": "paper", "relationship": "adapts", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05A05", "05C70", "05D15" ] }, "automation": { "methods": [ { "method": "agent", "models": [], "framework": "" } ], "strongest": "agent", "models": [], "spend": "", "notes": "" }, "scope": "", "sorry_count": null, "sorry_in_definitions": null, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "", "alignment": false, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Noah Cohen" ], "notes": "Challenge contains deliberate statement holes. Solution verified through human audit by the maintainers with 0 errors, 0 sorries, and strictly standard kernel axioms." }, "canonical": { "repo": "sneed-and-feed/lean-theorems-1", "directory": "palomar/hall_marriage" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "sneed-and-feed/lean-theorems-1/palomar/kneser_lovasz/formalization.yaml", "repo": "sneed-and-feed/lean-theorems-1", "path": "palomar/kneser_lovasz/formalization.yaml", "directory": "palomar/kneser_lovasz", "origins": [ "search" ], "url": "https://github.com/sneed-and-feed/lean-theorems-1/blob/HEAD/palomar/kneser_lovasz/formalization.yaml", "version": "v0.4", "missing": [], "name": "Kneser's Graph Coloring Upper Bound", "description": "A complete, machine-checked Lean 4 formalization of Kneser's explicit graph coloring upper bound (Kneser, 1955). Formalizes the Kneser graph KG(n, k) on k-subsets of an n-element set, defines Kneser's combinatorial fiber coloring based on minimal elements, proves proper vertex coloring via fiberwise pigeonhole on intervals, and establishes that chi(KG(n, k)) <= n - 2k + 2. Distinguishes Kneser's constructive coloring upper bound from Lovász's (1978) topological lower bound based on Borsuk-Ulam / homotopy theory. To the maintainers' knowledge, the theorem was not found in an exact declaration n", "authors": [ "Noah Cohen" ], "maintainers": [ "Noah Cohen" ], "license": "CC0-1.0", "role": "", "substantive": "", "sources": [ { "title": "Aufgabe 360", "id": "", "authors": [ "Martin Kneser" ], "type": "paper", "relationship": "formalizes", "endorsement": "" }, { "title": "Kneser's conjecture, chromatic number, and homotopy", "id": "", "authors": [ "László Lovász" ], "type": "paper", "relationship": "background", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05C15", "05D05" ] }, "automation": { "methods": [ { "method": "agent", "models": [], "framework": "" } ], "strongest": "agent", "models": [], "spend": "", "notes": "" }, "scope": "", "sorry_count": null, "sorry_in_definitions": null, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "", "alignment": false, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Noah Cohen" ], "notes": "Challenge contains deliberate proof holes (`:= sorry` for automated benchmark evaluation), while all definitions and theorem statements are completely specified. Solution verified through maintainer audit with 0 errors, 0 sorries, and strictly standard kernel axioms." }, "canonical": { "repo": "sneed-and-feed/lean-theorems-1", "directory": "palomar/kneser_lovasz" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "sneed-and-feed/lean-theorems-1/palomar/konig_matching/formalization.yaml", "repo": "sneed-and-feed/lean-theorems-1", "path": "palomar/konig_matching/formalization.yaml", "directory": "palomar/konig_matching", "origins": [ "search" ], "url": "https://github.com/sneed-and-feed/lean-theorems-1/blob/HEAD/palomar/konig_matching/formalization.yaml", "version": "v0.4", "missing": [], "name": "The Kőnig–Egerváry Theorem on Matchings and Vertex Covers in Bipartite Graphs", "description": "A complete machine-checked Lean 4 formalization of the Kőnig–Egerváry Theorem (1931) and Gallai identities (1959). Proves strong min-max duality matchingNumber(G) = vertexCoverNumber(G) for 2-colorable graphs G via Hall's defect matching theorem, and formalizes Gallai's identity independenceNumber(G) + vertexCoverNumber(G) = |V(G)|. To the maintainers' knowledge, the theorem was not found in an exact declaration name, docstring, and type signature search of the pinned Mathlib revision (Mathlib v4.34.0-rc1).", "authors": [ "Noah Cohen" ], "maintainers": [ "Noah Cohen" ], "license": "CC0-1.0", "role": "", "substantive": "", "sources": [ { "title": "Gráfok és mátrixok", "id": "", "authors": [ "Dénes Kőnig" ], "type": "paper", "relationship": "formalizes", "endorsement": "" }, { "title": "Matrixok kombinatorius tulajdonságairól", "id": "", "authors": [ "Jenő Egerváry" ], "type": "paper", "relationship": "formalizes", "endorsement": "" }, { "title": "Über extreme Punkt- und Kantenmengen", "id": "", "authors": [ "Tibor Gallai" ], "type": "paper", "relationship": "adapts", "endorsement": "" }, { "title": "On Representatives of Subsets", "id": "", "authors": [ "Philip Hall" ], "type": "paper", "relationship": "adapts", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05C70", "05C69", "90C27" ] }, "automation": { "methods": [ { "method": "agent", "models": [], "framework": "" } ], "strongest": "agent", "models": [], "spend": "", "notes": "" }, "scope": "", "sorry_count": null, "sorry_in_definitions": null, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "", "alignment": false, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Noah Cohen" ], "notes": "Challenge contains deliberate proof holes (`:= sorry` for automated benchmark evaluation), while all definitions and theorem statements are completely specified. Solution verified through human audit by the maintainers with 0 errors, 0 sorries, and strictly standard kernel axioms." }, "canonical": { "repo": "sneed-and-feed/lean-theorems-1", "directory": "palomar/konig_matching" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "sneed-and-feed/lean-theorems-1/palomar/ore_dirac_hamiltonian/formalization.yaml", "repo": "sneed-and-feed/lean-theorems-1", "path": "palomar/ore_dirac_hamiltonian/formalization.yaml", "directory": "palomar/ore_dirac_hamiltonian", "origins": [ "search" ], "url": "https://github.com/sneed-and-feed/lean-theorems-1/blob/HEAD/palomar/ore_dirac_hamiltonian/formalization.yaml", "version": "v0.4", "missing": [], "name": "Ore's and Dirac's Theorems on Hamiltonian Cycles", "description": "A complete machine-checked Lean 4 formalization of Ore's theorem (1960) and Dirac's theorem (1952) on the existence of Hamiltonian cycles in simple graphs. 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Solution verified through maintainer audit with 0 errors, 0 sorries, and strictly standard kernel axioms." }, "canonical": { "repo": "sneed-and-feed/lean-theorems-1", "directory": "palomar/ore_dirac_hamiltonian" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "sneed-and-feed/lean-theorems-1/palomar/picks_theorem/formalization.yaml", "repo": "sneed-and-feed/lean-theorems-1", "path": "palomar/picks_theorem/formalization.yaml", "directory": "palomar/picks_theorem", "origins": [ "search" ], "url": "https://github.com/sneed-and-feed/lean-theorems-1/blob/HEAD/palomar/picks_theorem/formalization.yaml", "version": "v0.4", "missing": [], "name": "Pick's Theorem on Lattice Polygons (Wiedijk #92)", "description": "A complete, machine-checked Lean 4 formalization of Pick's Theorem (Georg Alexander Pick, 1899; Freek Wiedijk 100 Theorems #92). Formalizes elementary lattice triangulations of planar polygons, Euler's disk characteristic V - E + F = 1, edge-face incidence double counting 3F = 2E_int + E_bd, boundary gluing valuation additivity, and proves both the real formula Area = i + b/2 - 1 and the discrete formula 2*Area = 2*i + b - 2. To the maintainers' knowledge, the theorem was not found in an exact declaration name, docstring, and type signature search of the pinned Mathlib revision (Mathlib v4.34.", "authors": [ "Noah Cohen" ], "maintainers": [ "Noah Cohen" ], "license": "CC0-1.0", "role": "", "substantive": "", "sources": [ { "title": "Geometrisches zur Zahlenlehre", "id": "", "authors": [ "Georg Alexander Pick" ], "type": "paper", "relationship": "formalizes", "endorsement": "" }, { "title": "Mathematical Snapshots", "id": "", "authors": [ "Hugo Steinhaus" ], "type": "book", "relationship": "background", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.CO", "math.MG" ], "msc2020": [ "52C05", "05C10", "11H06" ] }, "automation": { "methods": [ { "method": "agent", "models": [], "framework": "" } ], "strongest": "agent", "models": [], "spend": "", "notes": "" }, "scope": "", "sorry_count": null, "sorry_in_definitions": null, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "", "alignment": false, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Noah Cohen" ], "notes": "Challenge contains deliberate statement holes. Solution verified through human audit by the maintainers with 0 errors, 0 sorries, and strictly standard kernel axioms." }, "canonical": { "repo": "sneed-and-feed/lean-theorems-1", "directory": "palomar/picks_theorem" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "sneed-and-feed/lean-theorems-1/palomar/radon_helly/formalization.yaml", "repo": "sneed-and-feed/lean-theorems-1", "path": "palomar/radon_helly/formalization.yaml", "directory": "palomar/radon_helly", "origins": [ "search" ], "url": "https://github.com/sneed-and-feed/lean-theorems-1/blob/HEAD/palomar/radon_helly/formalization.yaml", "version": "v0.4", "missing": [], "name": "Radon's Lemma and Helly's Theorem on Convex Sets", "description": "A complete, machine-checked Lean 4 formalization of Radon's Lemma (Johann Radon, 1921) and Helly's Theorem on intersecting families of convex sets (Eduard Helly, 1923; Freek Wiedijk #99) from first principles. Proves that any d + 2 points in ℝ^d admit a Radon partition into two disjoint subsets whose convex hulls intersect, and uses Radon's lemma by induction on the family size to prove Helly's theorem for finite families of convex sets under the general <= d + 1 subfamily intersection hypothesis (covering small families |ι| <= d + 1 as well as large families). To the maintainers' knowledge, t", "authors": [ "Noah Cohen" ], "maintainers": [ "Noah Cohen" ], "license": "CC0-1.0", "role": "", "substantive": "", "sources": [ { "title": "Mengen konvexer Körper, die einen gemeinsamen Punkt enthalten", "id": "", "authors": [ "Johann Radon" ], "type": "paper", "relationship": "formalizes", "endorsement": "" }, { "title": "Über Mengen konvexer Körper mit gemeinschaftlichen Punkten", "id": "", "authors": [ "Eduard Helly" ], "type": "paper", "relationship": "formalizes", "endorsement": "" }, { "title": "Formalizing 100 Theorems", "id": "", "authors": [ "Freek Wiedijk" ], "type": "webpage", "relationship": "background", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.MG", "math.CO" ], "msc2020": [ "52A35", "52A20", "52C10" ] }, "automation": { "methods": [ { "method": "agent", "models": [], "framework": "" } ], "strongest": "agent", "models": [], "spend": "", "notes": "" }, "scope": "", "sorry_count": null, "sorry_in_definitions": null, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "", "alignment": false, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Noah Cohen" ], "notes": "Challenge contains deliberate proof holes (`:= sorry` for automated benchmark evaluation), while all definitions and theorem statements are completely specified. Solution verified through maintainer audit with 0 errors, 0 sorries, and strictly standard kernel axioms. The theorem formalization uses the general <= d + 1 subfamily intersection hypothesis to cleanly encompass both boundary small families (|ι| <= d + 1) and classical large families (|ι| > d + 1)." }, "canonical": { "repo": "sneed-and-feed/lean-theorems-1", "directory": "palomar/radon_helly" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "sneed-and-feed/lean-theorems-1/palomar/schurs_theorem/formalization.yaml", "repo": "sneed-and-feed/lean-theorems-1", "path": "palomar/schurs_theorem/formalization.yaml", "directory": "palomar/schurs_theorem", "origins": [ "search" ], "url": "https://github.com/sneed-and-feed/lean-theorems-1/blob/HEAD/palomar/schurs_theorem/formalization.yaml", "version": "v0.4", "missing": [], "name": "Schur's Theorem in Additive Combinatorics and Group Theory", "description": "A machine-checked Lean 4 formalization of Schur's Theorem on sum-free partitions (Issai Schur, 1916), the multicolor triangle Ramsey theorem on complete graphs, and their structural generalizations to arbitrary multiplicative groups and additive abelian groups. Proves that for every r >= 1, every symmetric edge-coloring of a complete graph with at least ramseyTriangleBound r vertices contains a monochromatic triangle, every r-coloring of {1, ..., ramseyTriangleBound r} contains a monochromatic solution to x + y = z, and for any finite subset S ⊆ G with |S| >= ramseyTriangleBound r, every r-col", "authors": [ "Noah Cohen" ], "maintainers": [ "Noah Cohen" ], "license": "CC0-1.0", "role": "", "substantive": "", "sources": [ { "title": "Über die Kongruenz x^m + y^m ≡ z^m (mod p)", "id": "", "authors": [ "Issai Schur" ], "type": "paper", "relationship": "formalizes", "endorsement": "" }, { "title": "On a Problem in Formal Logic", "id": "", "authors": [ "Frank P. Ramsey" ], "type": "paper", "relationship": "formalizes", "endorsement": "" }, { "title": "Ramsey Theory", "id": "", "authors": [ "R. L. Graham", "B. L. Rothschild", "J. H. 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Solution verified through maintainer audit with 0 errors, 0 sorries, and strictly standard kernel axioms. Adapts Rado (1933) by formalizing the constant-solution zero-sum case and the Schur instance (1, 1, -1)." }, "canonical": { "repo": "sneed-and-feed/lean-theorems-1", "directory": "palomar/schurs_theorem" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "sneed-and-feed/lean-theorems-1/palomar/sperner_3d/formalization.yaml", "repo": "sneed-and-feed/lean-theorems-1", "path": "palomar/sperner_3d/formalization.yaml", "directory": "palomar/sperner_3d", "origins": [ "search" ], "url": "https://github.com/sneed-and-feed/lean-theorems-1/blob/HEAD/palomar/sperner_3d/formalization.yaml", "version": "v0.4", "missing": [], "name": "3D Sperner's Lemma and Parity Invariant on Simplicial 3-Manifolds", "description": "A complete machine-checked Lean 4 formalization of the three-dimensional Sperner's Lemma and the parity conservation invariant on abstract simplicial 3-complexes with boundary. Proves that for any 4-coloring of vertices, the number of fully colored (panchromatic) tetrahedra is congruent modulo 2 to the number of 0-1-2 triangular boundary facets. 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Solution verified through human audit by the maintainers with 0 errors, 0 sorries, and strictly standard kernel axioms." }, "canonical": { "repo": "sneed-and-feed/lean-theorems-1", "directory": "palomar/sperner_3d" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "sneed-and-feed/lean-theorems-1/palomar/sperner_nd/formalization.yaml", "repo": "sneed-and-feed/lean-theorems-1", "path": "palomar/sperner_nd/formalization.yaml", "directory": "palomar/sperner_nd", "origins": [ "search" ], "url": "https://github.com/sneed-and-feed/lean-theorems-1/blob/HEAD/palomar/sperner_nd/formalization.yaml", "version": "v0.4", "missing": [], "name": "General n-Dimensional Sperner's Lemma and Fixed-Point Combinatorics", "description": "A machine-checked Lean 4 formalization of the general n-dimensional combinatorial Sperner parity theorem on abstract n-dimensional pseudomanifolds with boundary for arbitrary dimension n : ℕ. Proves the local door-counting parity invariant on (n+1)-simplices, the global double-counting identity on door-facet incidences, and the master parity congruence modulo 2 relating panchromatic (n+1)-simplices to boundary doors. The theorem is conditional on an odd number of boundary (n-1)-doors; obtaining the classical geometric Sperner lemma for simplicial subdivisions of geometric simplices with bounda", "authors": [ "Noah Cohen" ], "maintainers": [ "Noah Cohen" ], "license": "CC0-1.0", "role": "", "substantive": "", "sources": [ { "title": "Neuer Beweis für die Invarianz der Dimensionszahl und des Gebietes", "id": "", "authors": [ "Emanuel Sperner" ], "type": "paper", "relationship": "adapts", "endorsement": "" }, { "title": "Simplicial Approximation of Fixed Points", "id": "", "authors": [ "Harold W. 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This module generalizes and supersedes the fixed 2D edge-pseudomanifold and 3D face-pseudomanifold parity formalizations in SpernersLemma.lean and Sperner3D.lean." }, "canonical": { "repo": "sneed-and-feed/lean-theorems-1", "directory": "palomar/sperner_nd" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "sneed-and-feed/lean-theorems-1/palomar/sperners_lemma/formalization.yaml", "repo": "sneed-and-feed/lean-theorems-1", "path": "palomar/sperners_lemma/formalization.yaml", "directory": "palomar/sperners_lemma", "origins": [ "search" ], "url": "https://github.com/sneed-and-feed/lean-theorems-1/blob/HEAD/palomar/sperners_lemma/formalization.yaml", "version": "v0.4", "missing": [], "name": "Sperner's Lemma & Simplicial Fixed-Point Combinatorics in 1D, 2D, and 3D", "description": "A complete machine-checked Lean 4 formalization of Emanuel Sperner's foundational 1928 lemma and combinatorial parity theorems across dimensions 1, 2, and 3 (Wiedijk #57). Formalizes: (1) 1D Sperner Lemma: path color-switching parity theorem and existence of bicolored segments; (2) 2D Sperner Lemma: simplicial door-counting on abstract edge-pseudomanifolds with boundary (every edge in 1 or 2 triangles), deriving the global parity identity |T_pan| = |E_bd^01| (mod 2), oddness transfer, and panchromatic triangle existence; (3) 3D Sperner Lemma: tetrahedral face-pseudomanifold door double-countin", "authors": [ "Noah Cohen" ], "maintainers": [ "Noah Cohen" ], "license": "CC0-1.0", "role": "", "substantive": "", "sources": [ { "title": "Neuer Beweis für die Invarianz der Dimensionszahl und des Gebietes", "id": "", "authors": [ "Emanuel Sperner" ], "type": "paper", "relationship": "formalizes", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.CO", "math.AT" ], "msc2020": [ "05E45", "55M20", "57Q05", "52B05" ] }, "automation": { "methods": [ { "method": "agent", "models": [], "framework": "" } ], "strongest": "agent", "models": [], "spend": "", "notes": "" }, "scope": "", "sorry_count": null, "sorry_in_definitions": null, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "", "alignment": false, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Noah Cohen" ], "notes": "Challenge contains deliberate statement holes across 1D, 2D, and 3D dimensions. Solution verified through maintainer audit with 0 errors, 0 sorries, and strictly standard kernel axioms (propext, Quot.sound, Classical.choice)." }, "canonical": { "repo": "sneed-and-feed/lean-theorems-1", "directory": "palomar/sperners_lemma" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "sneed-and-feed/lean-theorems-1/palomar/sylvester_gallai/formalization.yaml", "repo": "sneed-and-feed/lean-theorems-1", "path": "palomar/sylvester_gallai/formalization.yaml", "directory": "palomar/sylvester_gallai", "origins": [ "search" ], "url": "https://github.com/sneed-and-feed/lean-theorems-1/blob/HEAD/palomar/sylvester_gallai/formalization.yaml", "version": "v0.4", "missing": [], "name": "The Sylvester–Gallai Theorem on Ordinary Lines in the Euclidean Plane", "description": "A complete, machine-checked Lean 4 formalization of the Sylvester–Gallai theorem (J. J. Sylvester 1893, T. Gallai 1944, L. M. Kelly 1948; Freek Wiedijk #98). Proves that every finite non-collinear set of points in the real Euclidean plane contains an ordinary line passing through exactly two points of the set, formalized via Kelly's minimal point-line distance argument. To the maintainers' knowledge, the theorem was not found in an exact declaration name, docstring, and type signature search of the pinned Mathlib revision (Mathlib v4.34.0-rc1).", "authors": [ "Noah Cohen" ], "maintainers": [ "Noah Cohen" ], "license": "CC0-1.0", "role": "", "substantive": "", "sources": [ { "title": "Mathematical Question 11851", "id": "", "authors": [ "J. J. Sylvester" ], "type": "paper", "relationship": "formalizes", "endorsement": "" }, { "title": "Solution to Problem 4065", "id": "", "authors": [ "Tibor Gallai" ], "type": "paper", "relationship": "formalizes", "endorsement": "" }, { "title": "A problem of collinear points", "id": "", "authors": [ "H. S. M. Coxeter", "L. M. 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Proves that characters are simultaneous eigenvectors with DFT eigenvalues (circulant_mulVec_character), establishes the orthogonality and non-degeneracy of the Fourier matrix (fourierMatrix_conjTranspose_mul), and derives the exact product formulas for circulant determinants (det_circulant) and characteristic polynomials (charpoly_circulant). To the maintainers' k", "authors": [ "Noah Cohen" ], "maintainers": [ "Noah Cohen" ], "license": "CC0-1.0", "role": "", "substantive": "", "sources": [ { "title": "Circulant Matrices", "id": "", "authors": [ "Philip J. Davis" ], "type": "book", "relationship": "formalizes", "endorsement": "" }, { "title": "Matrix Analysis", "id": "", "authors": [ "Roger A. Horn", "Charles R. 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Solution verified with 0 errors, 0 sorries, and standard kernel axioms." }, "canonical": { "repo": "sneed-and-feed/lean-theorems-2", "directory": "palomar/circulant_spectral_theory" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "sneed-and-feed/lean-theorems-2/palomar/cyclic_shift/formalization.yaml", "repo": "sneed-and-feed/lean-theorems-2", "path": "palomar/cyclic_shift/formalization.yaml", "directory": "palomar/cyclic_shift", "origins": [ "search" ], "url": "https://github.com/sneed-and-feed/lean-theorems-2/blob/HEAD/palomar/cyclic_shift/formalization.yaml", "version": "v0.4", "missing": [], "name": "Characteristic Polynomial of Weighted Cyclic Shift Matrices", "description": "A complete Lean 4 formalization proving the characteristic polynomial of a general weighted cyclic shift matrix over an arbitrary commutative ring R. Proves that charpoly(shiftMatrix n W) = X^n for nilpotent subdiagonal shift matrices, det(upperBidiagonal n W) = \\prod (-C(W i)), and establishes that for any L-dimensional cyclic matrix with weights W, charpoly(cyclicWeightMatrix W) = X^L - C(\\prod W). To the maintainers' knowledge, the theorem was not found in an exact declaration name, docstring, and type signature search of the pinned Mathlib revision (Mathlib v4.34.0-rc1).", "authors": [ "Noah Cohen" ], "maintainers": [ "Noah Cohen" ], "license": "CC0-1.0", "role": "", "substantive": "", "sources": [ { "title": "Matrix Analysis", "id": "", "authors": [ "Roger A. Horn", "Charles R. Johnson" ], "type": "book", "relationship": "background", "endorsement": "" }, { "title": "Circulant Matrices", "id": "", "authors": [ "Philip J. 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Solution verified with 0 errors, 0 sorries, and strictly standard kernel axioms." }, "canonical": { "repo": "sneed-and-feed/lean-theorems-2", "directory": "palomar/cyclic_shift" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "sneed-and-feed/lean-theorems-2/palomar/fishers_inequality/formalization.yaml", "repo": "sneed-and-feed/lean-theorems-2", "path": "palomar/fishers_inequality/formalization.yaml", "directory": "palomar/fishers_inequality", "origins": [ "search" ], "url": "https://github.com/sneed-and-feed/lean-theorems-2/blob/HEAD/palomar/fishers_inequality/formalization.yaml", "version": "v0.4", "missing": [], "name": "Fisher's Inequality for Balanced Incomplete Block Designs", "description": "A complete Lean 4 formalization of Fisher's Inequality in combinatorial design theory (Fisher 1940, Bose 1949). Proves that for any 2-(v, k, \\lambda) balanced incomplete block design with 1 < k < v and \\lambda > 0, the number of blocks b is at least the number of points v (b >= v). The proof establishes the exact Gramian product identity N N^T = (r - \\lambda) I + \\lambda J, computes the determinant det(N N^T) = (r - \\lambda)^{v-1}(r + (v-1)\\lambda), proves its strict positivity, and applies linear algebraic rank bounds. To the maintainers' knowledge, the theorem was not found in an exact decla", "authors": [ "Noah Cohen" ], "maintainers": [ "Noah Cohen" ], "license": "CC0-1.0", "role": "", "substantive": "", "sources": [ { "title": "An examination of the different possible solutions of a problem in incomplete blocks", "id": "", "authors": [ "Ronald A. Fisher" ], "type": "paper", "relationship": "formalizes", "endorsement": "" }, { "title": "A note on Fisher's inequality for balanced incomplete block designs", "id": "", "authors": [ "R. C. 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Solution verified with 0 errors, 0 sorries, and strictly standard kernel axioms." }, "canonical": { "repo": "sneed-and-feed/lean-theorems-2", "directory": "palomar/fishers_inequality" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "sneed-and-feed/lean-theorems-2/palomar/gessel_viennot/formalization.yaml", "repo": "sneed-and-feed/lean-theorems-2", "path": "palomar/gessel_viennot/formalization.yaml", "directory": "palomar/gessel_viennot", "origins": [ "search" ], "url": "https://github.com/sneed-and-feed/lean-theorems-2/blob/HEAD/palomar/gessel_viennot/formalization.yaml", "version": "v0.4", "missing": [], "name": "The Lindström–Gessel–Viennot Lemma (LGV Lemma)", "description": "A complete Lean 4 formalization of the Lindström–Gessel–Viennot (LGV) Lemma (Lindström 1973, Gessel & Viennot 1985) connecting matrix determinants to sums of non-intersecting path systems in directed graphs. Proves the Leibniz determinant expansion over permutation path products, the sign-reversing involution cancellation of intersecting path systems, the full LGV determinant identity, and its specialization to planar DAGs. To the maintainers' knowledge, the theorem was not found in an exact declaration name, docstring, and type signature search of the pinned Mathlib revision (Mathlib v4.34.0-", "authors": [ "Noah Cohen" ], "maintainers": [ "Noah Cohen" ], "license": "CC0-1.0", "role": "", "substantive": "", "sources": [ { "title": "On the vector representations of induced matroids", "id": "", "authors": [ "Bernt Lindström" ], "type": "paper", "relationship": "formalizes", "endorsement": "" }, { "title": "Binomial determinants, paths, and hook length formulae", "id": "", "authors": [ "Ira Gessel", "Gérard Viennot" ], "type": "paper", "relationship": "formalizes", "endorsement": "" }, { "title": "Enumerative Combinatorics, Volume 2", "id": "", "authors": [ "Richard P. 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Solution verified with 0 errors, 0 sorries, and strictly standard kernel axioms." }, "canonical": { "repo": "sneed-and-feed/lean-theorems-2", "directory": "palomar/gessel_viennot" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "sneed-and-feed/lean-theorems-2/palomar/hoffman_singleton/formalization.yaml", "repo": "sneed-and-feed/lean-theorems-2", "path": "palomar/hoffman_singleton/formalization.yaml", "directory": "palomar/hoffman_singleton", "origins": [ "search" ], "url": "https://github.com/sneed-and-feed/lean-theorems-2/blob/HEAD/palomar/hoffman_singleton/formalization.yaml", "version": "v0.4", "missing": [], "name": "The Hoffman–Singleton Moore Graph Classification Theorem", "description": "A complete machine-checked Lean 4 formalization of the Hoffman–Singleton Theorem (1960) on authentic SimpleGraph carriers. Establishes the metric-to-strongly-regular bridge proving that any d-regular graph G of diameter 2 and girth 5 is strongly regular with parameters (1 + d², d, 0, 1) and satisfies the adjacency matrix equation A² + A - (d - 1)I = J. Deduces the spectral decomposition, trace nullity, and multiplicities on EuclideanSpace ℝ V, and proves the complete degree classification theorem: any regular graph of degree d ≥ 2 with diameter 2 and girth 5 must have d ∈ {2, 3, 7, 57}. To the", "authors": [ "Noah Cohen" ], "maintainers": [ "Noah Cohen" ], "license": "CC0-1.0", "role": "", "substantive": "", "sources": [ { "title": "On Moore graphs with diameters 2 and 3", "id": "", "authors": [ "Alan J. Hoffman", "Robert R. 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Solution verified through human audit by the maintainers with 0 errors, 0 sorries, and strictly standard kernel axioms." }, "canonical": { "repo": "sneed-and-feed/lean-theorems-2", "directory": "palomar/hoffman_singleton" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "sneed-and-feed/lean-theorems-2/palomar/ihara_bass/formalization.yaml", "repo": "sneed-and-feed/lean-theorems-2", "path": "palomar/ihara_bass/formalization.yaml", "directory": "palomar/ihara_bass", "origins": [ "search" ], "url": "https://github.com/sneed-and-feed/lean-theorems-2/blob/HEAD/palomar/ihara_bass/formalization.yaml", "version": "v0.4", "missing": [], "name": "The Ihara–Bass Determinantal Formula for Graphs", "description": "A complete Lean 4 formalization of the Ihara–Bass Determinantal Formula (Bass 1992, Ihara 1966) connecting the Hashimoto edge-adjacency operator determinant with the vertex Laplacian-type determinant via block matrix Schur complement factorizations. Proves the exact matrix block factorizations M N = K, K L = KL, and deduces the Ihara–Bass polynomial identity. To the maintainers' knowledge, the theorem was not found in an exact declaration name, docstring, and type signature search of the pinned Mathlib revision (Mathlib v4.34.0-rc1).", "authors": [ "Noah Cohen" ], "maintainers": [ "Noah Cohen" ], "license": "CC0-1.0", "role": "", "substantive": "", "sources": [ { "title": "The Ihara-Selberg zeta function of a tree lattice", "id": "", "authors": [ "Hyman Bass" ], "type": "paper", "relationship": "formalizes", "endorsement": "" }, { "title": "On discrete subgroups of the two by two projective linear group over p-adic fields", "id": "", "authors": [ "Yasutaka Ihara" ], "type": "paper", "relationship": "background", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05C50", "11M36", "15A15" ] }, "automation": { "methods": [ { "method": "agent", "models": [], "framework": "" } ], "strongest": "agent", "models": [], "spend": "", "notes": "" }, "scope": "", "sorry_count": null, "sorry_in_definitions": null, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "", "alignment": false, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Noah Cohen" ], "notes": "Challenge contains deliberate benchmark holes (`:= sorry`), while all definitions and theorem statements are fully specified. Solution verified with 0 errors, 0 sorries, and strictly standard kernel axioms." }, "canonical": { "repo": "sneed-and-feed/lean-theorems-2", "directory": "palomar/ihara_bass" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "sneed-and-feed/lean-theorems-2/palomar/ihara_zeta/formalization.yaml", "repo": "sneed-and-feed/lean-theorems-2", "path": "palomar/ihara_zeta/formalization.yaml", "directory": "palomar/ihara_zeta", "origins": [ "search" ], "url": "https://github.com/sneed-and-feed/lean-theorems-2/blob/HEAD/palomar/ihara_zeta/formalization.yaml", "version": "v0.4", "missing": [], "name": "The Ihara Zeta Function and Hashimoto Edge Adjacency Matrix", "description": "A complete Lean 4 formalization of the Hashimoto edge adjacency operator, dart projection matrices (source, target, involution), and the algebraic foundation of the Ihara zeta function for finite graphs. Proves exact matrix product identities connecting directed dart operators to vertex adjacency matrices and degree diagonals. 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Solution verified with 0 errors, 0 sorries, and strictly standard kernel axioms." }, "canonical": { "repo": "sneed-and-feed/lean-theorems-2", "directory": "palomar/ihara_zeta" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "sneed-and-feed/lean-theorems-2/palomar/jungs_theorem/formalization.yaml", "repo": "sneed-and-feed/lean-theorems-2", "path": "palomar/jungs_theorem/formalization.yaml", "directory": "palomar/jungs_theorem", "origins": [ "search" ], "url": "https://github.com/sneed-and-feed/lean-theorems-2/blob/HEAD/palomar/jungs_theorem/formalization.yaml", "version": "v0.4", "missing": [], "name": "Jung's Theorem on Circumscribed Euclidean Spheres and Chebyshev Radii", "description": "A machine-checked Lean 4 formalization of Jung's Theorem (1901) on the circumradius / Chebyshev radius of bounded sets in d-dimensional Euclidean space. Formulates the enclosing ball predicate, circumradius infimum, and Helly intersection reduction for closed balls via Mathlib's compact Helly theorem. Proves the finite simplex base case via dual quadratic form maximization on the standard simplex and Lagrange's Euclidean distance identity, establishing the circumradius bound R(S) <= sqrt(d / (2(d+1))) * diam(S) with zero custom axioms, and evaluates concrete specializations in dimensions 1, 2,", "authors": [ "Noah Cohen" ], "maintainers": [ "Noah Cohen" ], "license": "CC0-1.0", "role": "", "substantive": "", "sources": [ { "title": "Über die kleinste Kugel, die eine räumliche Figur einschliesst", "id": "", "authors": [ "Heinrich Jung" ], "type": "paper", "relationship": "formalizes", "endorsement": "" }, { "title": "Helly's theorem and its relatives", "id": "", "authors": [ "Ludwig Danzer", "Branko Grünbaum", "Victor Klee" ], "type": "paper", "relationship": "adapts", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.MG" ], "msc2020": [ "52A40", "52A35", "51M04" ] }, "automation": { "methods": [ { "method": "agent", "models": [], "framework": "" } ], "strongest": "agent", "models": [], "spend": "", "notes": "" }, "scope": "", "sorry_count": null, "sorry_in_definitions": null, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "", "alignment": false, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Noah Cohen" ], "notes": "Challenge contains deliberate proof holes (`:= sorry` for automated benchmark evaluation), while all definitions and theorem statements are completely specified. Solution verified through human audit by the maintainers with 0 errors, 0 sorries, and strictly standard kernel axioms." }, "canonical": { "repo": "sneed-and-feed/lean-theorems-2", "directory": "palomar/jungs_theorem" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "sneed-and-feed/lean-theorems-2/palomar/konig_matching/formalization.yaml", "repo": "sneed-and-feed/lean-theorems-2", "path": "palomar/konig_matching/formalization.yaml", "directory": "palomar/konig_matching", "origins": [ "search" ], "url": "https://github.com/sneed-and-feed/lean-theorems-2/blob/HEAD/palomar/konig_matching/formalization.yaml", "version": "v0.4", "missing": [], "name": "The Kőnig–Egerváry Theorem on Matchings and Vertex Covers in Bipartite Graphs", "description": "A complete machine-checked Lean 4 formalization of the Kőnig–Egerváry Theorem (1931) and Gallai identities (1959). Proves strong min-max duality matchingNumber(G) = vertexCoverNumber(G) for 2-colorable graphs G via Hall's defect matching theorem, and formalizes Gallai's identity independenceNumber(G) + vertexCoverNumber(G) = |V(G)|. To the maintainers' knowledge, the theorem was not found in an exact declaration name, docstring, and type signature search of the pinned Mathlib revision (Mathlib v4.34.0-rc1).", "authors": [ "Noah Cohen" ], "maintainers": [ "Noah Cohen" ], "license": "CC0-1.0", "role": "", "substantive": "", "sources": [ { "title": "Gráfok és mátrixok", "id": "", "authors": [ "Dénes Kőnig" ], "type": "paper", "relationship": "formalizes", "endorsement": "" }, { "title": "Matrixok kombinatorius tulajdonságairól", "id": "", "authors": [ "Jenő Egerváry" ], "type": "paper", "relationship": "formalizes", "endorsement": "" }, { "title": "Über extreme Punkt- und Kantenmengen", "id": "", "authors": [ "Tibor Gallai" ], "type": "paper", "relationship": "adapts", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05C70", "05C69", "90C27" ] }, "automation": { "methods": [ { "method": "agent", "models": [], "framework": "" } ], "strongest": "agent", "models": [], "spend": "", "notes": "" }, "scope": "", "sorry_count": null, "sorry_in_definitions": null, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "", "alignment": false, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Noah Cohen" ], "notes": "Challenge contains deliberate proof holes (`:= sorry` for automated benchmark evaluation), while all definitions and theorem statements are completely specified. Solution verified through human audit by the maintainers with 0 errors, 0 sorries, and strictly standard kernel axioms." }, "canonical": { "repo": "sneed-and-feed/lean-theorems-2", "directory": "palomar/konig_matching" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "sneed-and-feed/lean-theorems-2/palomar/macmahons_master_theorem/formalization.yaml", "repo": "sneed-and-feed/lean-theorems-2", "path": "palomar/macmahons_master_theorem/formalization.yaml", "directory": "palomar/macmahons_master_theorem", "origins": [ "search" ], "url": "https://github.com/sneed-and-feed/lean-theorems-2/blob/HEAD/palomar/macmahons_master_theorem/formalization.yaml", "version": "v0.4", "missing": [], "name": "MacMahon's Master Theorem for Products of Linear Forms", "description": "A complete machine-checked Lean 4 formalization of MacMahon's Master Theorem (1915). Relates the coefficients of multi-index powers of linear forms [X^s] prod_i (sum_j A_ij X_j)^(s_i) to the formal power series inverse determinant [X^s] det(I - X A)^(-1) over arbitrary commutative rings. Proves the full n-dimensional identity using multivariate polynomials, matrix adjugate identities, and formal power series inversion without custom axioms. To the maintainers' knowledge, the theorem was not found in an exact declaration name, docstring, and type signature search of the pinned Mathlib revision ", "authors": [ "Noah Cohen" ], "maintainers": [ "Noah Cohen" ], "license": "CC0-1.0", "role": "", "substantive": "", "sources": [ { "title": "Combinatory Analysis, Volume 1", "id": "", "authors": [ "Percy Alexander MacMahon" ], "type": "book", "relationship": "formalizes", "endorsement": "" }, { "title": "A short proof of MacMahon's 'Master Theorem'", "id": "", "authors": [ "I. J. 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Solution verified through human audit by the maintainers with 0 errors, 0 sorries, and strictly standard kernel axioms." }, "canonical": { "repo": "sneed-and-feed/lean-theorems-2", "directory": "palomar/macmahons_master_theorem" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "sneed-and-feed/lean-theorems-2/palomar/matrix_tree_theorem/formalization.yaml", "repo": "sneed-and-feed/lean-theorems-2", "path": "palomar/matrix_tree_theorem/formalization.yaml", "directory": "palomar/matrix_tree_theorem", "origins": [ "search" ], "url": "https://github.com/sneed-and-feed/lean-theorems-2/blob/HEAD/palomar/matrix_tree_theorem/formalization.yaml", "version": "v0.4", "missing": [], "name": "Algebraic Foundations of the Matrix-Tree Theorem and Graph Laplacians", "description": "A complete Lean 4 formalization of the algebraic and spectral foundations of Kirchhoff's Matrix-Tree Theorem and spectral graph theory. 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Establishes min-max duality for internally vertex-disjoint paths vs vertex separators, and edge-disjoint paths vs edge cuts in finite simple graphs. Proves weak duality and characterizes k-vertex-connected graphs via Menger path systems. To the maintainers' knowledge, the theorem was not found in an exact declaration name, docstring, and type signature search of the pinned Mathlib revision (Mathlib v4.34.0-rc1).", "authors": [ "Noah Cohen" ], "maintainers": [ "Noah Cohen" ], "license": "CC0-1.0", "role": "", "substantive": "", "sources": [ { "title": "Zur allgemeinen Kurventheorie", "id": "", "authors": [ "Karl Menger" ], "type": "paper", "relationship": "formalizes", "endorsement": "" }, { "title": "Congruent graphs and the connectivity of graphs", "id": "", "authors": [ "Hassler Whitney" ], "type": "paper", "relationship": "formalizes", "endorsement": "" }, { "title": "Short proof of Menger's theorem", "id": "", "authors": [ "G. A. 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Solution verified with 0 errors, 0 sorries, and strictly standard kernel axioms." }, "canonical": { "repo": "sneed-and-feed/lean-theorems-2", "directory": "palomar/prefix_sparsity" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "sneed-and-feed/lean-theorems-2/palomar/ramanujan_tau/formalization.yaml", "repo": "sneed-and-feed/lean-theorems-2", "path": "palomar/ramanujan_tau/formalization.yaml", "directory": "palomar/ramanujan_tau", "origins": [ "search" ], "url": "https://github.com/sneed-and-feed/lean-theorems-2/blob/HEAD/palomar/ramanujan_tau/formalization.yaml", "version": "v0.4", "missing": [], "name": "Ramanujan Tau Function and Congruence Modulo 691", "description": "Lean 4 formalization of the Ramanujan tau function \\tau(n) and the exact Bernoulli number computation B_{12} = -691/2730. 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Solution verified with 0 errors, 0 sorries, and strictly standard kernel axioms. The formalization documents that the congruence theorem establishes the linear algebraic congruence modulo 691 reduction assuming the modular forms span of M_{12} and existence of E_{12} as explicit premises." }, "canonical": { "repo": "sneed-and-feed/lean-theorems-2", "directory": "palomar/ramanujan_tau" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "sneed-and-feed/lean-theorems-2/palomar/rsk_bijection/formalization.yaml", "repo": "sneed-and-feed/lean-theorems-2", "path": "palomar/rsk_bijection/formalization.yaml", "directory": "palomar/rsk_bijection", "origins": [ "search" ], "url": "https://github.com/sneed-and-feed/lean-theorems-2/blob/HEAD/palomar/rsk_bijection/formalization.yaml", "version": "v0.4", "missing": [], "name": "The Robinson–Schensted–Knuth (RSK) Bijection and Tableaux Combinatorics", "description": "A machine-checked Lean 4 formalization of the Robinson–Schensted–Knuth (RSK) Correspondence (Robinson 1938, Schensted 1961, Knuth 1970). Implements Schensted row-insertion bumping, Standard Young Tableaux (SYT), insertion/recording tableau generation, and formalizes Schensted's LIS theorem, Greene's LDS theorem, the Frobenius sum-of-squares identity sum (f^lambda)^2 = n!, and the Involution Fixed Points Theorem without custom axioms. To the maintainers' knowledge, the theorem was not found in an exact declaration name, docstring, and type signature search of the pinned Mathlib revision (Mathli", "authors": [ "Noah Cohen" ], "maintainers": [ "Noah Cohen" ], "license": "CC0-1.0", "role": "", "substantive": "", "sources": [ { "title": "On the representations of the symmetric group", "id": "", "authors": [ "Gilbert de Beauregard Robinson" ], "type": "paper", "relationship": "formalizes", "endorsement": "" }, { "title": "Longest increasing and decreasing subsequences", "id": "", "authors": [ "Craige Schensted" ], "type": "paper", "relationship": "formalizes", "endorsement": "" }, { "title": "Permutations, matrices, and generalized Young tableaux", "id": "", "authors": [ "Donald E. Knuth" ], "type": "paper", "relationship": "adapts", "endorsement": "" }, { "title": "An extension of Schensted's theorem", "id": "", "authors": [ "Curtis Greene" ], "type": "paper", "relationship": "adapts", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.CO", "math.RT" ], "msc2020": [ "05E10", "05A19", "20C30" ] }, "automation": { "methods": [ { "method": "agent", "models": [], "framework": "" } ], "strongest": "agent", "models": [], "spend": "", "notes": "" }, "scope": "", "sorry_count": null, "sorry_in_definitions": null, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "", "alignment": false, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Noah Cohen" ], "notes": "Challenge contains deliberate proof holes (`:= sorry` for automated benchmark evaluation), while all definitions and theorem statements are completely specified. Solution verified through human audit by the maintainers with 0 errors, 0 sorries, and strictly standard kernel axioms." }, "canonical": { "repo": "sneed-and-feed/lean-theorems-2", "directory": "palomar/rsk_bijection" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "sneed-and-feed/lean-theorems-2/palomar/sperner_antichain/formalization.yaml", "repo": "sneed-and-feed/lean-theorems-2", "path": "palomar/sperner_antichain/formalization.yaml", "directory": "palomar/sperner_antichain", "origins": [ "search" ], "url": "https://github.com/sneed-and-feed/lean-theorems-2/blob/HEAD/palomar/sperner_antichain/formalization.yaml", "version": "v0.4", "missing": [], "name": "Sperner's Theorem on Antichains and the LYM Inequality", "description": "A complete Lean 4 formalization of Sperner's Theorem on Antichains (1928), the Lubell–Yamamoto–Meshalkin (LYM) inequality (1966), and the full extremal equality and stability characterization in the Boolean lattice. Proves that the maximum cardinality of an antichain of subsets of an n-element set is Nat.choose n (n / 2) and characterizes all maximal antichains as middle-level slices. Mathlib contains the upper bound inequality via Mathlib.Combinatorics.SetFamily.LYM; this formalization provides a self-contained combinatorial API and formalizes the complete equality characterization for even a", "authors": [ "Noah Cohen" ], "maintainers": [ "Noah Cohen" ], "license": "CC0-1.0", "role": "", "substantive": "", "sources": [ { "title": "Ein Satz über Untermengen einer endlichen Menge", "id": "", "authors": [ "Emanuel Sperner" ], "type": "paper", "relationship": "formalizes", "endorsement": "" }, { "title": "A short proof of Sperner's lemma", "id": "", "authors": [ "David Lubell" ], "type": "paper", "relationship": "adapts", "endorsement": "" }, { "title": "Logarithmic order of free distributive lattice", "id": "", "authors": [ "Koichi Yamamoto" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Generalization of Sperner's theorem on the number of subsets of a finite set", "id": "", "authors": [ "L. D. Meshalkin" ], "type": "paper", "relationship": "background", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05D05", "06A07" ] }, "automation": { "methods": [ { "method": "agent", "models": [], "framework": "" } ], "strongest": "agent", "models": [], "spend": "", "notes": "" }, "scope": "", "sorry_count": null, "sorry_in_definitions": null, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "", "alignment": false, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Noah Cohen" ], "notes": "Challenge contains deliberate benchmark holes (`:= sorry`), while all definitions and theorem statements are fully specified. Solution verified with 0 errors, 0 sorries, and strictly standard kernel axioms." }, "canonical": { "repo": "sneed-and-feed/lean-theorems-2", "directory": "palomar/sperner_antichain" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "sneed-and-feed/lean-theorems-2/palomar/stanley_sl2/formalization.yaml", "repo": "sneed-and-feed/lean-theorems-2", "path": "palomar/stanley_sl2/formalization.yaml", "directory": "palomar/stanley_sl2", "origins": [ "search" ], "url": "https://github.com/sneed-and-feed/lean-theorems-2/blob/HEAD/palomar/stanley_sl2/formalization.yaml", "version": "v0.4", "missing": [], "name": "Stanley's sl2 Representation Proof of the Strong Sperner Property for L(m, n)", "description": "A machine-checked Lean 4 formalization of Richard Stanley's sl2 representation theorem (1980) on the partition poset L(m, n). 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Proctor" ], "type": "paper", "relationship": "adapts", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.CO", "math.RT" ], "msc2020": [ "05E10", "06A07", "17B10" ] }, "automation": { "methods": [ { "method": "agent", "models": [], "framework": "" } ], "strongest": "agent", "models": [], "spend": "", "notes": "" }, "scope": "", "sorry_count": null, "sorry_in_definitions": null, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "", "alignment": false, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Noah Cohen" ], "notes": "Challenge contains deliberate proof holes (`:= sorry` for automated benchmark evaluation), while all definitions and theorem statements are completely specified. 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Proves that any K_{r+1}-free simple graph on n vertices has at most the number of edges in the complete r-partite Turán graph T(n, r), and that equality holds if and only if G is isomorphic to T(n, r). Mathlib contains the Turán number upper bound; this formalization establishes the exact polynomial equivalence between discrete and continuous edge bounds, Mantel's theorem, and complete multipartite uniquenes", "authors": [ "Noah Cohen" ], "maintainers": [ "Noah Cohen" ], "license": "CC0-1.0", "role": "", "substantive": "", "sources": [ { "title": "Eine Extremalaufgabe aus der Graphentheorie", "id": "", "authors": [ "Pál Turán" ], "type": "paper", "relationship": "formalizes", "endorsement": "" }, { "title": "Vraagstuk XXVIII", "id": "", "authors": [ "W. Mantel" ], "type": "paper", "relationship": "formalizes", "endorsement": "" }, { "title": "A method for solving extremal problems in graph theory", "id": "", "authors": [ "Miklós Simonovits" ], "type": "paper", "relationship": "background", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05C35", "05C75" ] }, "automation": { "methods": [ { "method": "agent", "models": [], "framework": "" } ], "strongest": "agent", "models": [], "spend": "", "notes": "" }, "scope": "", "sorry_count": null, "sorry_in_definitions": null, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "", "alignment": false, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Noah Cohen" ], "notes": "Challenge contains deliberate benchmark holes (`:= sorry`), while all definitions and theorem statements are fully specified. Solution verified with 0 errors, 0 sorries, and strictly standard kernel axioms." }, "canonical": { "repo": "sneed-and-feed/lean-theorems-2", "directory": "palomar/turans_theorem" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "sneed-and-feed/lean-theorems-2/palomar/van_der_waerden/formalization.yaml", "repo": "sneed-and-feed/lean-theorems-2", "path": "palomar/van_der_waerden/formalization.yaml", "directory": "palomar/van_der_waerden", "origins": [ "search" ], "url": "https://github.com/sneed-and-feed/lean-theorems-2/blob/HEAD/palomar/van_der_waerden/formalization.yaml", "version": "v0.4", "missing": [], "name": "Van der Waerden's Theorem on Arithmetic Progressions", "description": "A complete Lean 4 formalization of the one-dimensional Van der Waerden Theorem on Arithmetic Progressions (1927), establishing both the finite version (for every r ≥ 1 colors and progression length k ≥ 1, there exists a finite bound W such that every r-coloring of Fin W contains a monochromatic arithmetic progression of length k) and the infinite version (every r-coloring of ℕ contains an arbitrary length-k monochromatic arithmetic progression). The finite version is derived via combinatorial lines and the Hales–Jewett theorem, and the infinite version via monochromatic homothetic copies.", "authors": [ "Noah Cohen" ], "maintainers": [ "Noah Cohen" ], "license": "CC0-1.0", "role": "", "substantive": "", "sources": [ { "title": "Beweis einer Baudetschen Vermutung", "id": "", "authors": [ "B. L. van der Waerden" ], "type": "paper", "relationship": "formalizes", "endorsement": "" }, { "title": "Ramsey Theory", "id": "", "authors": [ "R. L. Graham", "B. L. Rothschild", "J. H. 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Solution verified with 0 errors, 0 sorries, and strictly standard kernel axioms." }, "canonical": { "repo": "sneed-and-feed/lean-theorems-2", "directory": "palomar/van_der_waerden" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "sneed-and-feed/lean-theorems-2/palomar/vizings_theorem/formalization.yaml", "repo": "sneed-and-feed/lean-theorems-2", "path": "palomar/vizings_theorem/formalization.yaml", "directory": "palomar/vizings_theorem", "origins": [ "search" ], "url": "https://github.com/sneed-and-feed/lean-theorems-2/blob/HEAD/palomar/vizings_theorem/formalization.yaml", "version": "v0.4", "missing": [], "name": "Vizing's Theorem on Edge Colorings & König's Line Coloring Theorem", "description": "A complete Lean 4 formalization of Vizing's Theorem (1964) and König's Line Coloring Theorem (1916) in graph theory. 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Vizing" ], "type": "paper", "relationship": "formalizes", "endorsement": "" }, { "title": "Über Graphen und ihre Anwendung auf Determinantentheorie und Mengenlehre", "id": "", "authors": [ "Dénes König" ], "type": "paper", "relationship": "formalizes", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05C15" ] }, "automation": { "methods": [ { "method": "agent", "models": [], "framework": "" } ], "strongest": "agent", "models": [], "spend": "", "notes": "" }, "scope": "", "sorry_count": null, "sorry_in_definitions": null, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "", "alignment": false, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Noah Cohen" ], "notes": "Challenge contains deliberate benchmark holes (`:= sorry`), while all definitions and theorem statements are fully specified. Solution verified with 0 errors, 0 sorries, and strictly standard kernel axioms." }, "canonical": { "repo": "sneed-and-feed/lean-theorems-2", "directory": "palomar/vizings_theorem" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "sneed-and-feed/lean-theorems-3/formalization.yaml", "repo": "sneed-and-feed/lean-theorems-3", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/sneed-and-feed/lean-theorems-3/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "The 7D Keller Cube-Tiling Bridge and Corrádi–Szabó Disproofs", "description": "A machine-checked Lean 4 formalization of the geometric-to-combinatorial bridge connecting Ott-Heinrich Keller's 1930 cube-tiling conjecture to maximum cliques in the Corrádi–Szabó and Mackey Keller graphs G_{d, s}. Establishes the Szabó 2Z^d-periodic cube packing framework in R^d, the structural partition into 2^d coordinate blocks, the universal clique upper bound \\omega(G_{d, s}) \\le 2^d, and the Corrádi–Szabó / Mackey bridge theorem that \\omega(G_{d, s}) = 2^d implies the failure of Keller's conjecture in dimension d. 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Solution verified with 0 errors, 0 sorries, and strictly standard kernel axioms." }, "canonical": { "repo": "sneed-and-feed/lean-theorems-3", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "sneed-and-feed/lean-theorems-3/palomar/alon_boppana/formalization.yaml", "repo": "sneed-and-feed/lean-theorems-3", "path": "palomar/alon_boppana/formalization.yaml", "directory": "palomar/alon_boppana", "origins": [ "search" ], "url": "https://github.com/sneed-and-feed/lean-theorems-3/blob/HEAD/palomar/alon_boppana/formalization.yaml", "version": "v0.4", "missing": [], "name": "The Alon–Boppana Spectral Bound and Ramanujan Expanders", "description": "A machine-checked Lean 4 formalization of the foundations of the Alon–Boppana spectral lower bound (Alon 1986, Nilli 1991) and Ramanujan expander graphs. 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To the maintainers' knowledge, the theorem was not found in an exact declaration name, docstring, and type signature search of the pinned Mathlib revision (Mathlib ", "authors": [ "Noah Cohen" ], "maintainers": [ "Noah Cohen" ], "license": "CC0-1.0", "role": "", "substantive": "", "sources": [ { "title": "Eigenvalues and expanders", "id": "", "authors": [ "Noga Alon" ], "type": "paper", "relationship": "formalizes", "endorsement": "" }, { "title": "On the second eigenvalue of a graph", "id": "", "authors": [ "Alon Nilli" ], "type": "paper", "relationship": "adapts", "endorsement": "" }, { "title": "Ramanujan graphs", "id": "", "authors": [ "Alexander Lubotzky", "Ralph Phillips", "Peter Sarnak" ], "type": "paper", "relationship": "background", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05C50", "05C12", "15A18" ] }, "automation": { "methods": [ { "method": "agent", "models": [], "framework": "" } ], "strongest": "agent", "models": [], "spend": "", "notes": "" }, "scope": "", "sorry_count": null, "sorry_in_definitions": null, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "", "alignment": false, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Noah Cohen" ], "notes": "Challenge contains deliberate proof holes (`:= sorry` for automated benchmark evaluation), while all definitions and theorem statements are completely specified. Solution verified through human audit by the maintainers with 0 errors, 0 sorries, and strictly standard kernel axioms." }, "canonical": { "repo": "sneed-and-feed/lean-theorems-3", "directory": "palomar/alon_boppana" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "sneed-and-feed/lean-theorems-3/palomar/brieskorn_manifolds/formalization.yaml", "repo": "sneed-and-feed/lean-theorems-3", "path": "palomar/brieskorn_manifolds/formalization.yaml", "directory": "palomar/brieskorn_manifolds", "origins": [ "search" ], "url": "https://github.com/sneed-and-feed/lean-theorems-3/blob/HEAD/palomar/brieskorn_manifolds/formalization.yaml", "version": "v0.4", "missing": [], "name": "Brieskorn Manifolds, Topological Spheres, and the 28 Exotic 7-Spheres", "description": "A complete, machine-checked Lean 4 formalization of Egbert Brieskorn's 1966 theory of singularity links Sigma(a_1, ..., a_n), the Brieskorn–Hirzebruch sphere criterion, the 28 Milnor–Kervaire exotic 7-spheres generated by the family Sigma(2, 2, 2, 3, 6k-1) in C^5 surjecting onto Theta_7 = Z/28Z, and the Milnor fiber signature formula for the Casson invariants of Brieskorn homology 3-spheres with certified evaluations for Sigma(2,3,5), Sigma(2,3,7), Sigma(2,3,11), and Sigma(2,5,7). 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Solution verified through maintainer audit with 0 errors, 0 sorries, and strictly standard kernel axioms." }, "canonical": { "repo": "sneed-and-feed/lean-theorems-3", "directory": "palomar/brieskorn_manifolds" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "sneed-and-feed/lean-theorems-3/palomar/brieskorn_su2_character_variety/formalization.yaml", "repo": "sneed-and-feed/lean-theorems-3", "path": "palomar/brieskorn_su2_character_variety/formalization.yaml", "directory": "palomar/brieskorn_su2_character_variety", "origins": [ "search" ], "url": "https://github.com/sneed-and-feed/lean-theorems-3/blob/HEAD/palomar/brieskorn_su2_character_variety/formalization.yaml", "version": "v0.4", "missing": [], "name": "Irreducible SU(2) Character Varieties of Brieskorn Homology Spheres and Casson Invariants", "description": "A complete, machine-checked Lean 4 formalization of the irreducible SU(2) character varieties of Brieskorn homology 3-spheres Sigma(p, q, r), connecting Seifert presentations to Diophantine spherical angle inequalities, certified representation counts (#R^* = 2 for Sigma(2,3,5) and Sigma(2,3,7); #R^* = 4 for Sigma(2,3,11) and Sigma(2,5,7)), exact equality with the gauge-theoretic Casson invariant lambda_SU(2) = (1/2) #R^* = (1/8) |sigma(p,q,r)|, and the Fricke–Vogt trace variety. To the maintainers' knowledge, the theorem was verified absent from the core Mathlib library (checked against Mathl", "authors": [ "Noah Cohen" ], "maintainers": [ "Noah Cohen" ], "license": "CC0-1.0", "role": "", "substantive": "", "sources": [ { "title": "Representations of knot groups in SU(2)", "id": "", "authors": [ "Ronald Fintushel", "Ronald J. Stern" ], "type": "paper", "relationship": "formalizes", "endorsement": "" }, { "title": "Instanton homology, surgery, and knots", "id": "", "authors": [ "Ronald Fintushel", "Ronald J. Stern" ], "type": "paper", "relationship": "adapts", "endorsement": "" }, { "title": "An invariant for decorated 3-manifolds", "id": "", "authors": [ "Andrew J. 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Solution verified through maintainer audit with 0 errors, 0 sorries, and strictly standard kernel axioms." }, "canonical": { "repo": "sneed-and-feed/lean-theorems-3", "directory": "palomar/brieskorn_su2_character_variety" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "sneed-and-feed/lean-theorems-3/palomar/busemann_petty_zhang_audit/formalization.yaml", "repo": "sneed-and-feed/lean-theorems-3", "path": "palomar/busemann_petty_zhang_audit/formalization.yaml", "directory": "palomar/busemann_petty_zhang_audit", "origins": [ "search" ], "url": "https://github.com/sneed-and-feed/lean-theorems-3/blob/HEAD/palomar/busemann_petty_zhang_audit/formalization.yaml", "version": "v0.4", "missing": [], "name": "Gaoyong Zhang's 1994/1999 Busemann–Petty R^4 Resolution & Audit", "description": "Complete machine-checked Lean 4 formalization of Gaoyong Zhang's 1999 Annals affirmative resolution of the Busemann–Petty problem in dimension 4, Erwin Lutwak's 1988 intersection body comparison theorem, Richard J. 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Formalizes the Calabi–Yau self-duality sum condition sum alpha_i = 2, indicial polynomial I_0(theta) = theta^4, Frobenius cusp monodromy T_0 in Sp_4(Z), nilpotence of index 2 (Type II degeneration N^2 = 0), CY3 Maximally Unipotent Monodromy (Type III, N^4 = 0), infinitesimal and finite symplectic Lie algebra invariance (Griffiths transversality), and classical/multi-instanton BPS Yukawa expansions. To the maintainers' knowledge, ", "authors": [ "Noah Cohen" ], "maintainers": [ "Noah Cohen" ], "license": "CC0-1.0", "role": "", "substantive": "", "sources": [ { "title": "A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory", "id": "", "authors": [ "Philip Candelas", "Xenia C. de la Ossa", "Paul S. Green", "Linda Parkes" ], "type": "paper", "relationship": "formalizes", "endorsement": "" }, { "title": "Mirror symmetry and algebraic geometry", "id": "", "authors": [ "David A. 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Solution verified through maintainer audit with 0 errors, 0 sorries, and strictly standard kernel axioms." }, "canonical": { "repo": "sneed-and-feed/lean-theorems-3", "directory": "palomar/picard_fuchs_mirror_monodromy" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "sneed-and-feed/lean-theorems-3/palomar/universal_monodromy_weight_filtration/formalization.yaml", "repo": "sneed-and-feed/lean-theorems-3", "path": "palomar/universal_monodromy_weight_filtration/formalization.yaml", "directory": "palomar/universal_monodromy_weight_filtration", "origins": [ "search" ], "url": "https://github.com/sneed-and-feed/lean-theorems-3/blob/HEAD/palomar/universal_monodromy_weight_filtration/formalization.yaml", "version": "v0.4", "missing": [], "name": "Deligne–Schmid Mixed Hodge Weight Filtration and Symplectic Monodromy", "description": "A complete, machine-checked Lean 4 formalization of Pierre Deligne's canonical weight filtration formula W_l(N, k) = sum_{j=0}^k (ker(N^{j+1}) cap im(N^{j - l + k})) for nilpotent monodromy operators N with N^{k+1} = 0 over general commutative rings R. Proves universal filtration monotonicity, top extremal saturation W_{2k} = V, fundamental shift inclusion N(W_l) subseteq W_{l-2}, explicit 2-step weight filtrations on Z^4 for Type II unipotent abelian surface monodromy (N^2 = 0), explicit 4-step weight filtrations for Type III CY3 MUM nilpotent operators (N^4 = 0), and Hodge–Riemann symplectic", "authors": [ "Noah Cohen" ], "maintainers": [ "Noah Cohen" ], "license": "CC0-1.0", "role": "", "substantive": "", "sources": [ { "title": "Théorie de Hodge: II", "id": "", "authors": [ "Pierre Deligne" ], "type": "paper", "relationship": "formalizes", "endorsement": "" }, { "title": "Variation of Hodge structure: the singularities of the period mapping", "id": "", "authors": [ "Wilfried Schmid" ], "type": "paper", "relationship": "adapts", "endorsement": "" }, { "title": "Periods of integrals on algebraic manifolds, I", "id": "", "authors": [ "Phillip A. 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Solution verified through maintainer audit with 0 errors, 0 sorries, and strictly standard kernel axioms." }, "canonical": { "repo": "sneed-and-feed/lean-theorems-3", "directory": "palomar/universal_monodromy_weight_filtration" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "stalex444/arithmetic-of-time/formalization.yaml", "repo": "stalex444/arithmetic-of-time", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/stalex444/arithmetic-of-time/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "The Arithmetic of Time: chirality of the Padovan substitution", "description": "A kernel-checked chirality package for the Padovan substitution (0 ↦ 1, 1 ↦ 2, 2 ↦ 01 on Fin 3; its incidence matrix has characteristic polynomial x³ − x − 1, an orientation fact that is not used or compared). Three compared theorems, all at the iterate-factor level — they quantify over factors of the concrete words S^[n] [x], never over an abstract subshift. (1) The successor invariant: in every positive iterate from any seed, the letter 0 (\"a\") is immediately followed by 1 (\"b\"). (2) Chirality: the word 20 (\"ca\") is a factor of an iterate while its reversal 02 (\"ac\") is a factor of no iterat", "authors": [ "Stephanie Alexander" ], "maintainers": [ "Stephanie Alexander" ], "license": "MIT", "role": "", "substantive": "", "sources": [ { "title": "Algebraic Combinatorics on Words", "id": "Cambridge University Press, 2002. ISBN 978-0521812207", "authors": [ "M. Lothaire" ], "type": "book", "relationship": "background", "endorsement": "n/a" }, { "title": "Dynamical Invariants from Asymptotic Composants", "id": "arXiv:2602.18501", "authors": [ "Franz Gähler" ], "type": "article", "relationship": "adapts", "endorsement": "not-contacted" } ], "related": [], "classification": { "arxiv": [ "math.CO", "math.DS" ], "msc2020": [ "68R15", "37B10" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Anthropic Claude (Opus/Sonnet-class, 2026)" ], "framework": "Claude Code" }, { "method": "manual", "models": [], "framework": "n/a" } ], "strongest": "agent", "models": [ "Anthropic Claude (Opus/Sonnet-class, 2026)" ], "spend": "subscription-based", "notes": "Human-directed, machine-verified workflow: the author fixes statements and reviews all mathematics; AI assistance drafts and repairs proofs; the Lean kernel is the acceptance criterion, enforced in CI with a per-declaration axiom audit." }, "scope": "Formalized: the Padovan chirality package on the Fin 3 alphabet — the a-forces-b successor invariant (every positive iterate, every seed), the ca/ac chirality (ca a factor of an iterate; ac a factor of no iterate from any seed), and palindrome-freeness for factors of length ≥ 4 (every seed, every iterate, including the zeroth). All compared statements are at the iterate-factor level: they quantify over factors of the words S^[n] [x], not over an abstract subshift or tiling space. Not formalized: tiling-space-level or subshift-level statements of the reflection asymmetry; primitivity of the sub", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "None known. The compared statements are at the iterate-factor level (they quantify over factors of the concrete words S^[n] [x], not over an abstract subshift or tiling space), and every prose claim in this entry is stated at exactly that level.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Stephanie Alexander" ], "notes": "Statements audited by the author against the informal claims; the settling module carries an independent second formalization in the same repository (SettlingReplica) as a replication check; CI reruns the full build and the per-declaration axiom audit on every push." }, "canonical": { "repo": "stalex444/arithmetic-of-time", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-19-000006", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-19-000006", "commit": "c48eb12817dd1d7ff3dc4d92306c7a4f249e6d7d", "trust": "high", "theorems": 3, "date": "2026-08-19" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "stalex444/Busch-theorem/formalization.yaml", "repo": "stalex444/Busch-theorem", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/stalex444/Busch-theorem/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Busch's theorem: Born-rule uniqueness on quantum effects (finite dimension)", "description": "Kernel-checked proof of Busch's theorem in finite dimension: every generalized probability measure on the effects of a complex matrix algebra — nonnegative on effects, additive whenever a sum of effects is again an effect, normalized at the identity — is represented by a unique density matrix via the trace pairing. Existence and uniqueness are both compared (busch_representation). The result is the uniqueness of the Born rule in the POVM reading; it is valid in every finite dimension (beyond finiteness and decidable equality of the index type, the compared statement assumes only [Nonempty n]) ", "authors": [ "Stephanie Alexander" ], "maintainers": [ "Stephanie Alexander" ], "license": "MIT", "role": "", "substantive": "", "sources": [ { "title": "Quantum states and generalized observables: a simple proof of Gleason's theorem", "id": "Physical Review Letters 91, 120403 (2003)", "authors": [ "Paul Busch" ], "type": "article", "relationship": "formalizes", "endorsement": "n/a" }, { "title": "Measures on the closed subspaces of a Hilbert space", "id": "Journal of Mathematics and Mechanics 6 (1957), 885–893", "authors": [ "Andrew M. Gleason" ], "type": "article", "relationship": "background", "endorsement": "n/a" } ], "related": [ { "id": "https://github.com/leanprover-community/mathlib4 (Matrix.PosSemidef, the Hermitian spectral theorem and eigenvector unitary, Unitary.conjStarAlgAut, the trace API)", "relationship": "builds-on", "note": "The compared statements use only Mathlib vocabulary (Matrix.PosSemidef, Matrix.trace, smul, the real-to-complex coercion) — zero custom definitions on the compared surface. The proof builds on Mathlib's Hermitian spectral theorem (eigenvalues, the eigenvector unitary) and star-algebra conjugation. The frame-function development itself — effect frame functions, the no-continuity homogeneity bootstr" }, { "id": "https://github.com/leanprover-community/physlib (QuantumInfo POVM development)", "relationship": "other", "note": "Physlib defines POVMs (positive semidefinite matrices summing to the identity) and measurement channels but contains no Busch/Gleason-type representation theorem for effect measures (checked at HEAD 4a4de62, 2026-08-24) — adjacent infrastructure, not this theorem." }, { "id": "https://github.com/stalex444/pdt-lean (parent development; pinned: commit d66ffbe813e48636aa636ffcd80070b81cbf10da)", "relationship": "other", "note": "The parent project this entry was developed in. This repository is the complete EXTRACTED proof development, not a wrapper: the full PdtBusch.lean module (the entire proof), the compared surface, and the pinned toolchain build here self-contained, importing nothing from the parent. At the pinned revision the parent's PdtBusch.lean, Challenge.lean, and Solution.lean are verbatim identical to this r" } ], "classification": { "arxiv": [ "quant-ph", "math-ph" ], "msc2020": [ "81P16", "81P15", "46N50" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Anthropic Claude (Opus/Fable-class, 2026)" ], "framework": "Claude Code" }, { "method": "manual", "models": [], "framework": "n/a" } ], "strongest": "agent", "models": [ "Anthropic Claude (Opus/Fable-class, 2026)" ], "spend": "subscription-based", "notes": "Human-directed, machine-verified workflow: the author fixes targets and reviews statements; AI assistance drafts and repairs proofs; the Lean kernel is the acceptance criterion." }, "scope": "Formalized, 2 compared theorems, finite dimension (matrices over the complex numbers; the representation theorem additionally assumes a nonempty index type). busch_representation: existence AND uniqueness — every generalized probability measure on effects satisfies f a = trace (rho * a) for exactly one density matrix rho (positive semidefinite, trace one). homogeneity_automatic: the no-continuity step compared separately — for a normalized measure, nonnegativity plus additivity force f (t • a) = t * f a for every effect a and every scalar t in [0,1]. No physical claim is part of the compared s", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The source states the theorem for a complex Hilbert space of any dimension, with additivity over countable collections of effects (sigma-additivity). This entry is the finite-dimensional case (matrices over the complex numbers), and additivity is weakened to binary additivity — a weaker hypothesis, so the compared theorem implies the paper's finite-dimensional instance. The infinite-dimensional sigma-additive statement is not formalized. The paper's upper bound f <= 1 on effects is derived in the development, not assumed.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Stephanie Alexander" ], "notes": "Statements audited by the author against the source paper's hypotheses; the kernel and the axiom audit are the acceptance criteria. An in-house adversarial mock-referee pass, automated (AI-assisted: Anthropic Claude, in Claude Code, under the author's direction; 2026-08-24) was applied to this surface before submission, and its blocking findings were fixed (2026-08-25)." }, "canonical": { "repo": "stalex444/busch-theorem", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-31-000014", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-31-000014", "commit": "b6801977b666320c38b7027bc130a49583578902", "trust": "high", "theorems": 2, "date": "2026-08-31" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "stalex444/chiral-padovan-histories/formalization.yaml", "repo": "stalex444/chiral-padovan-histories", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/stalex444/chiral-padovan-histories/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Sharp arithmetic separation in chiral Padovan histories", "description": "Sixteen compared statements establish quantitative recurrence for the actual infinite Padovan prefix path and compare cubic and quartic arithmetic record windows. For 0=1, and Padovan substitution stages attain equality at arbitrarily large clocks. The full cubic record window is relatively dense wit", "authors": [ "Stephanie Alexander" ], "maintainers": [ "Stephanie Alexander" ], "license": "MIT", "role": "", "substantive": "", "sources": [ { "title": "Quantitative return bounds for the chiral Padovan prefix path and its arithmetic window comparison", "id": "", "authors": [ "Stephanie Alexander" ], "type": "original-proof", "relationship": "other", "endorsement": "" }, { "title": "Dynamical Invariants from Asymptotic Composants", "id": "arXiv:2602.18501v1", "authors": [ "Franz Gähler" ], "type": "preprint", "relationship": "background", "endorsement": "" }, { "title": "Diffraction of a model set with complex windows", "id": "arXiv:1904.08285v2", "authors": [ "Michael Baake", "Uwe Grimm" ], "type": "article", "relationship": "background", "endorsement": "" }, { "title": "Self-similar Delone sets and Pisot numbers", "id": "arXiv:2608.11867v1", "authors": [ "Christoph Bandt", "Yves Meyer" ], "type": "preprint", "relationship": "background", "endorsement": "" } ], "related": [ { "id": "https://github.com/stalex444/arithmetic-of-time/tree/c48eb12817dd1d7ff3dc4d92306c7a4f249e6d7d", "relationship": "builds-on", "note": "Earlier Padovan factor-language chirality development. This entry uses the author's continuing combinatorial proof development, with the new infinite stream and geometric transport built upon it. Source and packaged fingerprints are recorded locally; no byte-identity claim about this earlier revision or build-time dependency on it is required." }, { "id": "https://github.com/stalex444/golden-family-trace-forms/tree/9528df064df58318213b541048e05ff4c0d9cb67", "relationship": "builds-on", "note": "Earlier trace Gram and signature formalizations of the same mathematical objects. This entry includes trace infrastructure from the author's continuing development, not necessarily byte-identical files from that earlier revision. Observer and history modules extend the construction. The full earlier entry is not re-submitted here." } ], "classification": { "arxiv": [ "math.DS", "math.NT" ], "msc2020": [ "37B10", "11K16", "52C23", "11E04" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "OpenAI GPT-family; exact model revision not recorded" ], "framework": "Codex" }, { "method": "other", "models": [], "framework": "Earlier formalizations listed above" } ], "strongest": "agent", "models": [ "OpenAI GPT-family; exact model revision not recorded" ], "spend": "not tracked", "notes": "Stephanie Alexander supplied research direction and publication decisions. Coding agents wrote proofs, assembled the comparison surface and prose, and performed internal audits. Kernel checking verifies formal proofs; internal agent review is not independent human peer review." }, "scope": "The sixteen statements are listed in comparison order. Return bounds concern the actual infinite Padovan word, transverse distance to the origin, 0 1 descends from the golden ratio (n = 2) through the plastic number (n = 3) toward 1. The family's structural boundary falls at n = 3; this entry proves the morphic half unconditionally at general n and holds the Pisot half as the measure⟺Pisot equivalence, the unconditional n = 4 non-Pisot statement, and the Siegel-conditional strict excess for n >= 4, from the family's own arithmetic, with no per-degree case check above the boundary. Eleven compared statements carry th", "authors": [ "Stephanie Alexander" ], "maintainers": [ "Stephanie Alexander" ], "license": "MIT", "role": "", "substantive": "", "sources": [ { "title": "On the irreducibility of certain trinomials", "id": "Mathematica Scandinavica 4 (1956), 287-302", "authors": [ "Ernst S. Selmer" ], "type": "article", "relationship": "background", "endorsement": "n/a" }, { "title": "Morphic numbers", "id": "Nieuw Archief voor Wiskunde (5) 2 (2001), 56-58", "authors": [ "Jan Aarts", "Robbert Fokkink", "Godfried Kruijtzer" ], "type": "article", "relationship": "formalizes", "endorsement": "n/a" }, { "title": "Algebraic integers whose conjugates lie in the unit circle", "id": "Duke Mathematical Journal 11 (1944), 597-602", "authors": [ "Carl Ludwig Siegel" ], "type": "article", "relationship": "background", "endorsement": "n/a" }, { "title": "On the product of the conjugates outside the unit circle of an algebraic integer", "id": "Bulletin of the London Mathematical Society 3 (1971), 169-175", "authors": [ "Christopher J. Smyth" ], "type": "article", "relationship": "background", "endorsement": "n/a" }, { "title": "Étude de certaines fonctions méromorphes bornées sur le cercle unité. Application à un ensemble fermé d'entiers algébriques", "id": "Annales scientifiques de l'École normale supérieure (3) 72 (1955), no. 1, 69-92", "authors": [ "J. Dufresnoy", "Ch. Pisot" ], "type": "article", "relationship": "background", "endorsement": "n/a" }, { "title": "A problem of Boyd concerning geometric means of polynomials", "id": "Journal of Number Theory 16 (1983), 356-362", "authors": [ "Wayne M. Lawton" ], "type": "article", "relationship": "background", "endorsement": "n/a" }, { "title": "On measures of polynomials in several variables", "id": "Bulletin of the Australian Mathematical Society 23 (1981), 49-63", "authors": [ "Christopher J. Smyth" ], "type": "article", "relationship": "background", "endorsement": "n/a" }, { "title": "On the Conjecture of Lehmer, limit Mahler measure of trinomials and asymptotic expansions", "id": "Uniform Distribution Theory 11 (2016), no. 1, 79-139", "authors": [ "Jean-Louis Verger-Gaugry" ], "type": "article", "relationship": "background", "endorsement": "n/a" }, { "title": "Locating trinomial zeros", "id": "Involve, a Journal of Mathematics 11 (2018), no. 4, 711-720", "authors": [ "Russell Howell", "David Kyle" ], "type": "article", "relationship": "background", "endorsement": "n/a" }, { "title": "Boyd's conjecture", "id": "arXiv:1401.1688 [math.NT] (2014), v2, 11 March 2014, 19 pp. (preprint; no journal version found on Crossref as of 2026-09-01)", "authors": [ "Dragan Stankov" ], "type": "preprint", "relationship": "background", "endorsement": "n/a" }, { "title": "Locating Unimodular Roots", "id": "The College Mathematics Journal 45 (2014), no. 3, 162-168", "authors": [ "Michael Brilleslyper", "Beth Schaubroeck" ], "type": "article", "relationship": "background", "endorsement": "n/a" } ], "related": [ { "id": "https://github.com/leanprover-community/mathlib4 (Mathlib/RingTheory/Polynomial/Selmer.lean; Mathlib/Analysis/Polynomial/MahlerMeasure.lean; Mathlib/NumberTheory/Real/GoldenRatio.lean; Mathlib/Algebra/Polynomial/UnitTrinomial.lean)", "relationship": "builds-on", "note": "Honest accounting of what is consumed, at the pin fabf563a: Selmer's irreducibility theorem (Polynomial.X_pow_sub_X_sub_one_irreducible_rat, Thomas Browning); the Mahler-measure development (Barroero-Wilson: mahlerMeasure with its monic root-product form, mahlerMeasure_eq_leadingCoeff_mul_prod_roots); Real.goldenRatio for the n = 2 witness inside the proof modules; and the Ljunggren unit-trinomial" }, { "id": "https://github.com/stalex444/pdt-lean (parent development; pinned reference revision: commit d66ffbe813e48636aa636ffcd80070b81cbf10da)", "relationship": "other", "note": "The broader public project this entry belongs to. This repository is the complete, self-contained proof development for the compared statements, not a wrapper: the proof modules PdtMorphic and PdtFamilyBoundaries build here with the pinned toolchain and no imports from the parent. The family-boundary module was developed for this entry; the parent development shares the plastic-number conventions " }, { "id": "Registered entry PALOMAR-2026-08-31-000004 (stalex444/mahler-measure-minima, registered revision ca3df22)", "relationship": "other", "note": "CONTEXT IMPORT, NOT COMPARED CONTENT. The registered sibling certifies the n <= 4 degree-wise minimality instances: x^d - x - 1 attains the minimal Mahler measure above one among monic irreducible integer polynomials at each degree d in {2, 3, 4}, with the exact identities M^2 = M + 1, M^3 = M + 1, M^4 = M^3 + 1. None of those instances is re-compared here; of this entry's eleven compared statemen" }, { "id": "Registered entry PALOMAR-2026-09-01-000005 (stalex444/mahler-minimizer-compositum @ d78ce1610608cf4a90d5580c267110c96d2f9af5)", "relationship": "other", "note": "CONTEXT IMPORT, NOT COMPARED CONTENT. The registered compositum entry carries psi = M(x^4 - x - 1) among its registered statements (ruler_mahler_eq_quartic_measure). This entry's Q_lt_psi places the quartic family root strictly below psi via real-arithmetic brackets — a general-family fact about the n = 4 rung, with psi entering hypothesis-style (psi^4 = psi^3 + 1), not a restatement of the regist" }, { "id": "Registered entry PALOMAR-2026-09-01-000008 (stalex444/only-two-morphic-numbers @ 9d3664f4ad3abc3783a9296ad1207a1ef1fee41d)", "relationship": "other", "note": "CONTEXT IMPORT AND PROOF-MODULE REUSE. That entry's compared surface is the Aarts-Fokkink-Kruijtzer classification itself (the iff and its witnesses). This repository reuses its proof module PdtMorphic as machinery: the family morphic boundary compared here consumes the classification and adds the general-n descent argument. The classification is not re-compared; the boundary corollary at general " } ], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11R06", "11R09", "11C08" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Anthropic Claude (Opus/Fable-class, 2026)" ], "framework": "Claude Code" }, { "method": "manual", "models": [], "framework": "n/a" } ], "strongest": "agent", "models": [ "Anthropic Claude (Opus/Fable-class, 2026)" ], "spend": "subscription-based", "notes": "Human-directed, machine-verified workflow: the author fixes targets and reviews statements; AI assistance drafts and repairs proofs; the Lean kernel is the acceptance criterion." }, "scope": "Formalized, eleven compared theorems, hypotheses inline. (1) family_root_exists_unique — existence and uniqueness of the family root r_n for every n >= 2, with strict decrease folded as a second conjunct. (2) family_morphic_boundary — r_n morphic iff n <= 3, unconditional, morphic property spelled inline. (3) no_unit_circle_root — no modulus-1 complex root, any n. (4) family_le_mahlerMeasure — r_n <= M(x^n - x - 1). (5) family_measure_eq_iff_pisot — measure equality iff all other roots strictly inside the unit circle (the measure⟺Pisot equivalence as a theorem, counted once). (6) siegel_condit", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The compared statements are general-n theorems with no single source text to transcribe; fidelity here is design honesty rather than transcription fidelity, and the design choices are: (a) every constant enters hypothesis-style through its defining equation (the root via 1 < r and r^n = r + 1; the plastic number via 1 < rho and rho^3 = rho + 1, reconciled to the proof module's constructed constant by its uniqueness lemma; psi via 1 < psi and psi^4 = psi^3 + 1) — no custom definitions appear in any compared statement; (b) the morphic property is spelled inline exactly as in the AFK paper, zpow-free (p^l * (p - 1) = 1), identically to the morphic-classification entry's registered spelling; (c) Siegel's theorem appears only as the named hypothesis `siegel`, with \"Pisot\" unfolded in-statement via minpoly — a reader can verify by inspection that exactly one compared statement is conditional and on exactly what; (d) the ladder's strict decrease is folded into family_root_exists_unique as a second conjunct (the proof module carries the halves separately); (e) the measure⟺Pisot equivalence is stated as one compared theorem and the description never counts its two readings as separate bound", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Stephanie Alexander" ], "notes": "Statements audited by the author against the sources and the sibling entries' registered surfaces; the kernel and the axiom audit are the acceptance criteria. An in-house adversarial scoping pass, automated (AI-assisted: Anthropic Claude, in Claude Code, under the author's direction), was applied before the build, 2026-09-01: independent prior-art and duplication probes (which identified the registered n <= 4 instances and mandated the context-import handling used here), compiled Lean feasibility probes against the pinned Mathlib, and an adversarial judge whose five conditions — a compared surface that re-registers no sibling content, the measure/Pisot equivalence stated as the single theorem it is, the registered instances carried as labeled context imports, the limit literature confined " }, "canonical": { "repo": "stalex444/golden-family-boundary", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-01-000012", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-01-000012", "commit": "fc43a62561b4d7846ca25273229a4f964dff7ae5", "trust": "high", "theorems": 11, "date": "2026-09-01" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "stalex444/golden-family-trace-forms/formalization.yaml", "repo": "stalex444/golden-family-trace-forms", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/stalex444/golden-family-trace-forms/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "The trace form of a tensor product of algebras and the signature product rule, with the trace forms of Q[x]/(x^3 - x - 1), Q[x]/(x^4 - x - 1) and their compositum as instances", "description": "GENERAL THEOREMS FIRST. For finite free algebras A, B over a commutative ring R, the trace form of the tensor product A ⊗_R B is the tensor product of the trace forms, and over a linearly ordered field the Sylvester signature of a tensor product of quadratic forms obeys a product rule. Seven compared general statements: (1) the trace of a pure tensor is the product of the traces, Tr_{A⊗B}(a ⊗ b) = Tr_A(a)·Tr_B(b); (2) the trace form on pure tensors is the product of the trace forms; (3) the Gram matrix of a tensor basis bA ⊗ bB (Mathlib's Basis.tensorProduct) is the Kronecker product of the tw", "authors": [ "Stephanie Alexander" ], "maintainers": [ "Stephanie Alexander" ], "license": "MIT", "role": "", "substantive": "", "sources": [ { "title": "Extrait d'une lettre de Mr. Ch. Hermite de Paris à Mr. Borchardt de Berlin sur le nombre des racines d'une équation algébrique comprises entre des limites données", "id": "Journal für die reine und angewandte Mathematik 52 (1856), 39-51", "authors": [ "Charles Hermite" ], "type": "article", "relationship": "independently-proves", "endorsement": "n/a" }, { "title": "The Discriminant Matrices of an Algebraic Number Field", "id": "Journal of the London Mathematical Society s1-43 (1968), no. 1, 152-154", "authors": [ "Olga Taussky" ], "type": "article", "relationship": "independently-proves", "endorsement": "n/a" }, { "title": "A Survey of Trace Forms of Algebraic Number Fields", "id": "Series in Pure Mathematics 2, World Scientific, Singapore, 1984", "authors": [ "Pierre E. Conner", "Robert Perlis" ], "type": "book", "relationship": "background", "endorsement": "n/a" }, { "title": "A demonstration of the theorem that every homogeneous quadratic polynomial is reducible by real orthogonal substitutions to the form of a sum of positive and negative squares", "id": "The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 4 (1852), 138-142", "authors": [ "James Joseph Sylvester" ], "type": "article", "relationship": "background", "endorsement": "n/a" }, { "title": "Integral trace forms associated to cubic extensions", "id": "Algebra & Number Theory 4 (2010), no. 6, 681-699", "authors": [ "Guillermo Mantilla-Soler" ], "type": "article", "relationship": "background", "endorsement": "n/a" }, { "title": "Newform 23.1.b.a", "id": "The L-functions and Modular Forms Database, classical newform 23.1.b.a (weight 1, level 23, character 23.b)", "authors": [ "The LMFDB Collaboration" ], "type": "other", "relationship": "background", "endorsement": "n/a" }, { "title": "Introduction to Quadratic Forms over Fields", "id": "Graduate Studies in Mathematics 67, American Mathematical Society, Providence, RI, 2005; ISBN 0-8218-1095-2", "authors": [ "T. Y. Lam" ], "type": "book", "relationship": "background", "endorsement": "n/a" }, { "title": "The discriminant of compositum of algebraic number fields", "id": "International Journal of Number Theory 15 (2019), no. 2, 353-360", "authors": [ "Sudesh K. Khanduja" ], "type": "article", "relationship": "background", "endorsement": "n/a" }, { "title": "Certifying Rings of Integers in Number Fields", "id": "Proceedings of the 14th ACM SIGPLAN International Conference on Certified Programs and Proofs (CPP 2025), ACM, 50-66", "authors": [ "Anne Baanen", "Alain Chavarri Villarello", "Sander R. Dahmen" ], "type": "article", "relationship": "background", "endorsement": "n/a" }, { "title": "Formally certifying number field invariants", "id": "arXiv:2607.26230 (submitted 2026-07-28)", "authors": [ "Alain Chavarri Villarello", "Sander R. Dahmen" ], "type": "article", "relationship": "background", "endorsement": "n/a" }, { "title": "On the irreducibility of certain trinomials", "id": "Mathematica Scandinavica 4 (1956), 287-302", "authors": [ "Ernst S. Selmer" ], "type": "article", "relationship": "background", "endorsement": "n/a" }, { "title": "Number field 4.2.283.1", "id": "The L-functions and Modular Forms Database, number field 4.2.283.1 (x^4 - x - 1)", "authors": [ "The LMFDB Collaboration" ], "type": "other", "relationship": "background", "endorsement": "n/a" }, { "title": "Number field 3.1.23.1", "id": "The L-functions and Modular Forms Database, number field 3.1.23.1 (reduced defining polynomial x^3 - x^2 + 1)", "authors": [ "The LMFDB Collaboration" ], "type": "other", "relationship": "background", "endorsement": "n/a" } ], "related": [ { "id": "https://github.com/leanprover-community/mathlib4 (Mathlib/RingTheory/Trace/Defs.lean, Mathlib/RingTheory/Trace/Basic.lean, Mathlib/RingTheory/Discriminant.lean, Mathlib/LinearAlgebra/QuadraticForm/Signature.lean, Mathlib/RingTheory/AdjoinRoot.lean, Mathlib/RingTheory/Polynomial/Selmer.lean, Mathlib/", "relationship": "builds-on", "note": "Honest accounting of what is consumed, at the pin fabf563a: the Selmer irreducibility theorem (Polynomial.X_pow_sub_X_sub_one_irreducible_rat, Thomas Browning), in the prose only; Newton's identities exist in Mathlib (Mathlib/RingTheory/MvPolynomial/NewtonIdentities.lean) but are NOT used — every trace is computed from the left-multiplication matrix and the relation r^n = r + 1; the trace-form inf" }, { "id": "https://github.com/alainchmt/RingOfIntegersProject and https://github.com/alainchmt/CertifyingInvariantsNF (tag v1) — the Baanen-Chavarri Villarello-Dahmen certification framework", "relationship": "other", "note": "PRIOR ART, NOT CONSUMED. The Lean 4 + Mathlib framework behind the two disclosed papers: certified rings of integers and field discriminants (CPP 2025), then signatures (r_1, r_2) by real-root counting, unit groups modulo p-th powers, and class groups (2026), for LMFDB entries at large. Nothing from it is imported here; its published examples contain neither of this entry's fields (checked 2026-09" }, { "id": "https://github.com/stalex444/pdt-lean (parent development; pinned reference revision: commit add583f3a7fc465a7fa78e2a236fe728ea9ec850)", "relationship": "other", "note": "The broader public project this entry belongs to, and the provenance of the proof modules. The seven proof modules of this repository — PdtSignature and PdtSignatureRho (the explicit Gram matrices and their rational congruences to diagonal form), PdtIrreducible (irreducibility of the two trinomials over Q, by reduction modulo 2, consumed only by the field statement (16)), PdtTraceLink (the identif" }, { "id": "Registered entry PALOMAR-2026-09-01-000005 (stalex444/mahler-minimizer-compositum @ d78ce1610608cf4a90d5580c267110c96d2f9af5)", "relationship": "other", "note": "CONTEXT, NOT COMPARED CONTENT. The registered sibling treats the compositum of the same two fields as Q(rho·Q) (its degree-12 minimal polynomial, conjugate census, and unit-ness of the product of the two roots). Nothing about trace forms, discriminants, or signatures is registered there, and nothing from it is re-compared here. The compositum statements (18)-(23) of this entry are about the tensor" }, { "id": "Registered entry PALOMAR-2026-09-01-000012 (stalex444/golden-family-boundary @ fc43a62561b4d7846ca25273229a4f964dff7ae5)", "relationship": "other", "note": "CONTEXT, NOT COMPARED CONTENT. The registered sibling holds the family x^n - x - 1 at general n (root ladder, morphic boundary, the n = 4 root not Pisot). It supplies the real-root picture that the Hermite reading of the compared signatures matches (one real root of the cubic; two of the quartic), as context only; no root count appears in any compared statement here." } ], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11E04", "11E12", "15A63", "11R04", "11R16", "11R29" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Anthropic Claude (Fable-class orchestration; Opus-class proof agents, 2026)" ], "framework": "Claude Code" }, { "method": "manual", "models": [], "framework": "n/a" } ], "strongest": "agent", "models": [ "Anthropic Claude (Fable-class orchestration; Opus-class proof agents, 2026)" ], "spend": "subscription-based", "notes": "Human-directed, machine-verified workflow: the author fixes targets and reviews statements; AI assistance drafts and repairs proofs; the Lean kernel is the acceptance criterion." }, "scope": "Formalized, twenty-three compared theorems, seven general and sixteen instances. GENERAL, with hypotheses exactly as stated in the Lean: (1) trace_tmul and (2) traceForm_tmul — R a commutative ring, A and B commutative R-algebras that are free and finite as R-modules; the trace of a ⊗ b is Tr(a)·Tr(b) and the trace form on pure tensors is the product of the trace forms. (3) traceMatrix_tensorProduct — same hypotheses, bases indexed by arbitrary types: the trace matrix of bA.tensorProduct bB is the Kronecker product. (4) discr_tensorProduct — index types finite with decidable equality: Algebra.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The compared statements are instances at two fields of a classical theorem, so fidelity is design honesty; the design choices are: (a) the fields enter as the quotient rings AdjoinRoot of the concrete polynomials X^3 - X - 1 and X^4 - X - 1 over Q, and the power bases enter as the families i -> root^i indexed by Fin 3 and Fin 4 — no custom definition appears in any compared statement; (b) Algebra.traceMatrix and Algebra.discr are applied to those families (Mathlib defines both for an arbitrary finite family, with discr_def : discr = det of the trace matrix), so (10)-(11) are the discriminants OF THE POWER BASES; that these equal the field discriminants is cited from the LMFDB records (index 1) and not formalized; (c) the signature is stated through Mathlib's basis-free sigPos and sigNeg, so (14)-(17) are invariants of the quadratic form (Algebra.traceForm).toQuadraticMap on the field, independent of any basis or diagonalisation; the explicit weights of (12)-(13) are ONE diagonalisation, exhibited for concreteness, and the entry's prose never treats the weights themselves as invariants; (d) the classical (r_1 + r_2, r_2) theorem is neither stated nor used: the signatures are proved ", "alignment": true, "original": true, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "Self-assessed by the author. This is version 2 of the entry. Version 1 (sixteen statements: the two fields and their compositum, no general theorem) was reviewed by the registry on 2026-09-02 and blocked on research interest — \"fixed low-degree computations and direct instances or applications of classical general results\" — and on a Challenge scope docstring that still described an earlier ten-statement surface. Version 2 answers the first with the seven general theorems (1)-(7), new kernel content rather than new prose, and the second by rewriting the Challenge's opening account before anything else. In-house checks applied before this submission: the kernel build of every module and of the comparison surface; the axiom audit of all twenty-three compared theorems and of every theorem of " }, "canonical": { "repo": "stalex444/golden-family-trace-forms", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-02-000014", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-02-000014", "commit": "9528df064df58318213b541048e05ff4c0d9cb67", "trust": "high", "theorems": 23, "date": "2026-09-02" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "stalex444/mahler-measure-minima/formalization.yaml", "repo": "stalex444/mahler-measure-minima", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/stalex444/mahler-measure-minima/blob/HEAD/formalization.yaml", "version": "", "missing": [], "name": "The Mahler degree window: minimal Mahler measures at degrees 2, 3, 4", "description": "For each degree d in {2, 3, 4}, the polynomial x^d - x - 1 attains the minimal Mahler measure among monic irreducible integer polynomials of degree d with Mahler measure above one, and the three minimal values are exactly the golden ratio (M^2 = M + 1), the plastic ratio (M^3 = M + 1), and the root of x^4 - x^3 - 1 above one (M^4 = M^3 + 1); by Siegel's classical theorem (1944, cited as background, not formalized here) the latter two are the smallest and second-smallest Pisot numbers. Classically the degree-4 minimizer is not itself a Pisot polynomial, yet its measure lands back on the Pisot l", "authors": [ "Stephanie Alexander" ], "maintainers": [ "Stephanie Alexander" ], "license": "MIT", "role": "", "substantive": "", "sources": [ { "title": "Quantum Mechanics as the Arithmetic of a Complex Place (v8B)", "id": "", "authors": [], "type": "", "relationship": "formalizes", "endorsement": "" }, { "title": "C. L. Siegel, Algebraic integers whose conjugates lie in the unit circle, Duke Math. J. 11 (1944), 597-602", "id": "", "authors": [], "type": "", "relationship": "background", "endorsement": "" }, { "title": "D. H. Lehmer, Factorization of certain cyclotomic functions, Ann. of Math. (2) 34 (1933), 461-479", "id": "", "authors": [], "type": "", "relationship": "formalizes", "endorsement": "" }, { "title": "C. J. Smyth, On the product of the conjugates outside the unit circle of an algebraic integer, Bull. London Math. Soc. 3 (1971), 169-175", "id": "", "authors": [], "type": "", "relationship": "independently-proves", "endorsement": "" }, { "title": "C. Smyth, The Mahler measure of algebraic numbers: a survey, in Number Theory and Polynomials, LMS Lecture Note Series 352, CUP (2008), 322-349", "id": "", "authors": [], "type": "", "relationship": "background", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11R06", "11R09", "12D10" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Anthropic Claude (Opus/Fable-class, 2026)" ], "framework": "Claude Code" } ], "strongest": "agent", "models": [ "Anthropic Claude (Opus/Fable-class, 2026)" ], "spend": "", "notes": "" }, "scope": "", "sorry_count": null, "sorry_in_definitions": null, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "", "alignment": true, "original": true, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Stephanie Alexander" ], "notes": "Statements audited by the author against the cited sources, and the claim-to-Lean alignment block reviewed by the author before submission; the Lean kernel and the axiom audit are the acceptance criteria. An in-house adversarial mock-referee pass, automated (AI-assisted: Anthropic Claude, in Claude Code, under the author's direction; 2026-08-24), was applied to this surface, and its blocking findings — unhedged Pisot epithets, attainment stated without the minimizers' irreducibility on the compared surface, an over-broad source note — were fixed in the staged revision before submission." }, "canonical": { "repo": "stalex444/mahler-measure-minima", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-31-000004", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-31-000004", "commit": "ca3df22ba91ce60e5199aee4a0c35a6ca8ee559d", "trust": "high", "theorems": 9, "date": "2026-08-31" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "stalex444/mahler-minimizer-compositum/formalization.yaml", "repo": "stalex444/mahler-minimizer-compositum", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/stalex444/mahler-minimizer-compositum/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "The compositum of the two Mahler minimizers: exact measure, conjugate census, house, and minimal product recurrence, with the Bedford-Kim plastic family and Lehmer's polynomial", "description": "Kernel-checked certificates for the multiplicative compositum of the two registered Mahler minimizers — rho, the real root of x^3 = x + 1 (the plastic number: Smyth's nonreciprocal minimum, Siegel's smallest Pisot number) and Q, the real root > 1 of x^4 = x + 1 (the degree-4 entry of the minimal-measure tables) — together with the Bedford-Kim spectral family whose limit rho is. Seventeen compared theorems. The spectral half: (1) for n >= 7 the family chi_n(x) = x^(n+1)(x^3 - x - 1) + (x^3 + x^2 - 1) has a canonical root in [117/100, rho), largest below rho, strictly increasing, tending to rho;", "authors": [ "Stephanie Alexander" ], "maintainers": [ "Stephanie Alexander" ], "license": "MIT", "role": "", "substantive": "", "sources": [ { "title": "Dynamics of rational surface automorphisms: linear fractional recurrences", "id": "Journal of Geometric Analysis 19 (2009), 553-583", "authors": [ "Eric Bedford", "Kyounghee Kim" ], "type": "article", "relationship": "formalizes", "endorsement": "n/a" }, { "title": "On the product of the conjugates outside the unit circle of an algebraic integer", "id": "Bulletin of the London Mathematical Society 3 (1971), 169-175", "authors": [ "Christopher J. Smyth" ], "type": "article", "relationship": "background", "endorsement": "n/a" }, { "title": "Algebraic integers whose conjugates lie in the unit circle", "id": "Duke Mathematical Journal 11 (1944), 597-602", "authors": [ "Carl Ludwig Siegel" ], "type": "article", "relationship": "background", "endorsement": "n/a" }, { "title": "Etude de certaines fonctions meromorphes bornees sur le cercle unite. Application a un ensemble ferme d'entiers algebriques", "id": "Annales scientifiques de l'Ecole Normale Superieure (3) 72 (1955), 69-92", "authors": [ "Jacques Dufresnoy", "Charles Pisot" ], "type": "article", "relationship": "background", "endorsement": "n/a" }, { "title": "Factorization of certain cyclotomic functions", "id": "Annals of Mathematics 34 (1933), 461-479", "authors": [ "Derrick H. Lehmer" ], "type": "article", "relationship": "background", "endorsement": "n/a" }, { "title": "Polynomials with small Mahler measure", "id": "Mathematics of Computation 67 (1998), 1697-1705", "authors": [ "Michael J. Mossinghoff" ], "type": "article", "relationship": "background", "endorsement": "n/a" }, { "title": "Minimal Mahler measures", "id": "Experimental Mathematics 17 (2008), no. 4, 451-458", "authors": [ "Michael J. Mossinghoff", "Georges Rhin", "Qiang Wu" ], "type": "article", "relationship": "background", "endorsement": "n/a" }, { "title": "The Mahler measure of algebraic numbers: a survey", "id": "Number Theory and Polynomials, LMS Lecture Note Series 352, Cambridge University Press, 2008, pp. 322-349", "authors": [ "Christopher J. Smyth" ], "type": "article", "relationship": "background", "endorsement": "n/a" }, { "title": "Pisot and Salem Numbers", "id": "Birkhauser Basel, 1992", "authors": [ "Marie-Jose Bertin", "Annette Decomps-Guilloux", "Marthe Grandet-Hugot", "Martine Pathiaux-Delefosse", "Jean-Pierre Schreiber" ], "type": "book", "relationship": "background", "endorsement": "n/a" }, { "title": "Degrees of sums in a separable field extension", "id": "Proceedings of the American Mathematical Society 25 (1970), 638-641", "authors": [ "I. Martin Isaacs" ], "type": "article", "relationship": "background", "endorsement": "n/a" }, { "title": "A degree problem for two algebraic numbers and their sum", "id": "Publicacions Matematiques 56 (2012), no. 2, 413-448", "authors": [ "Paulius Drungilas", "Arturas Dubickas", "Christopher J. Smyth" ], "type": "article", "relationship": "background", "endorsement": "n/a" }, { "title": "On the degree of compositum of two number fields", "id": "Mathematische Nachrichten 286 (2013), no. 2-3, 171-180", "authors": [ "Paulius Drungilas", "Arturas Dubickas", "Florian Luca" ], "type": "article", "relationship": "background", "endorsement": "n/a" }, { "title": "On degrees of three algebraic numbers with zero sum or unit product", "id": "Colloquium Mathematicum 143 (2016), no. 2, 159-167", "authors": [ "Paulius Drungilas", "Arturas Dubickas" ], "type": "article", "relationship": "background", "endorsement": "n/a" }, { "title": "A degree problem for the compositum of two number fields", "id": "Lithuanian Mathematical Journal 59 (2019), 39-47", "authors": [ "Paulius Drungilas", "Lukas Maciulevicius" ], "type": "article", "relationship": "background", "endorsement": "n/a" }, { "title": "Degree of the product of two algebraic numbers one of which is of prime degree", "id": "Mathematics 11 (2023), article 1485", "authors": [ "Paulius Virbalas" ], "type": "article", "relationship": "background", "endorsement": "n/a" }, { "title": "Compositum of two number fields of prime power degree", "id": "Lithuanian Mathematical Journal 65 (2025), 445-454", "authors": [ "Paulius Virbalas" ], "type": "article", "relationship": "background", "endorsement": "n/a" }, { "title": "The product of a quartic and a sextic number cannot be octic", "id": "Open Mathematics 22 (2024)", "authors": [ "Arturas Dubickas", "Lukas Maciulevicius" ], "type": "article", "relationship": "background", "endorsement": "n/a" }, { "title": "Irreducibles and the composed product for polynomials over a finite field", "id": "Discrete Mathematics 65 (1987), 115-139", "authors": [ "Joel V. Brawley", "Leonard Carlitz" ], "type": "article", "relationship": "background", "endorsement": "n/a" }, { "title": "Factorization and irreducibility of composed products", "id": "Designs, Codes and Cryptography 93 (2025), no. 3, 591-614 (online 4 Dec 2024)", "authors": [ "Lukas Kolsch", "Lucas Krompholz", "Gohar Kyureghyan" ], "type": "article", "relationship": "background", "endorsement": "n/a" }, { "title": "Recurrence Sequences", "id": "Mathematical Surveys and Monographs 104, American Mathematical Society, 2003", "authors": [ "Graham Everest", "Alf van der Poorten", "Igor Shparlinski", "Thomas Ward" ], "type": "book", "relationship": "background", "endorsement": "n/a" }, { "title": "Suites recurrentes lineaires: proprietes algebriques et arithmetiques", "id": "L'Enseignement Mathematique (2) 33 (1987), 67-108", "authors": [ "Luigi Cerlienco", "Maurice Mignotte", "Francesco Piras" ], "type": "article", "relationship": "background", "endorsement": "n/a" }, { "title": "Strong primality tests that are not sufficient", "id": "Mathematics of Computation 39 (1982), 255-300", "authors": [ "William Adams", "Daniel Shanks" ], "type": "article", "relationship": "background", "endorsement": "n/a" }, { "title": "Positivity problems for low-order linear recurrence sequences", "id": "Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2014), 366-379", "authors": [ "Joel Ouaknine", "James Worrell" ], "type": "article", "relationship": "background", "endorsement": "n/a" }, { "title": "Ultimate positivity is decidable for simple linear recurrence sequences", "id": "ICALP 2014, Lecture Notes in Computer Science 8573, 330-341", "authors": [ "Joel Ouaknine", "James Worrell" ], "type": "article", "relationship": "background", "endorsement": "n/a" }, { "title": "OEIS A000931 (Padovan), A001608 (Perrin), A017817, A050443", "id": "The On-Line Encyclopedia of Integer Sequences", "authors": [ "OEIS Foundation" ], "type": "misc", "relationship": "background", "endorsement": "n/a" }, { "title": "Enumerative Combinatorics, volume 1 (2nd ed.), Proposition 4.2.5", "id": "Cambridge Studies in Advanced Mathematics 49, Cambridge University Press", "authors": [ "Richard P. Stanley" ], "type": "book", "relationship": "background", "endorsement": "n/a" }, { "title": "Dynamical degrees of birational transformations of projective surfaces", "id": "Journal of the American Mathematical Society 29 (2016), 415-471", "authors": [ "Jeremy Blanc", "Serge Cantat" ], "type": "article", "relationship": "background", "endorsement": "n/a" }, { "title": "Dynamics on blowups of the projective plane", "id": "Publications Mathematiques de l'IHES 105 (2007), 49-89", "authors": [ "Curtis T. McMullen" ], "type": "article", "relationship": "background", "endorsement": "n/a" }, { "title": "Salem/Pisot Numbers in the Weyl Spectrum", "id": "arXiv:2312.17729 (preprint; v2, 20 Jan 2025)", "authors": [ "Kyounghee Kim" ], "type": "misc", "relationship": "background", "endorsement": "n/a" } ], "related": [ { "id": "https://github.com/leanprover-community/mathlib4 (intermediate value theorem, sSup/compactness, monotone convergence, Filter.Tendsto, complex norm, minpoly, rootSet, Set.ncard, Polynomial.mahlerMeasure, IntermediateField, LinearDisjoint)", "relationship": "builds-on", "note": "The compared statements use only core Mathlib vocabulary — zero custom definitions on the compared surface. Mathlib contains no Salem numbers, no Lehmer polynomial, no Pisot theory, and no Hadamard-product theory for linear recurrences (checked at the pin): the family, the compositum certificates, the census, the exact measure evaluations, and the minimal product recurrence are this entry's contri" }, { "id": "Registered entry PALOMAR-2026-08-31-000004 (stalex444/mahler-measure-minima, registered revision ca3df22)", "relationship": "other", "note": "The registered sibling entry that kernel-certifies the pair's selection: for each degree d in {2, 3, 4}, x^d - x - 1 attains the minimal Mahler measure among monic irreducible integer polynomials of degree d with measure above one, with the minima pinned exactly (M^3 = M + 1 at degree 3; M^4 = M^3 + 1 at degree 4). Compared statement (15) re-proves that entry's degree-4 quantity in-repo as the lef" }, { "id": "https://github.com/stalex444/pdt-lean (parent development; pinned: commit d66ffbe813e48636aa636ffcd80070b81cbf10da)", "relationship": "other", "note": "The broader public project this entry belongs to. This repository is the complete, self-contained proof development for the compared statements, not a wrapper: all six proof modules build here with the pinned toolchain and no imports from the parent. PdtIrreducible, PdtSalemFamily, and PdtRulerMinpoly descend from material in the parent development at the pinned revision; PdtRulerCensus, PdtRulerM" } ], "classification": { "arxiv": [ "math.NT", "math.DS" ], "msc2020": [ "11R06", "11R04", "11B37", "14E07", "37B40" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Anthropic Claude (Opus/Fable-class, 2026)" ], "framework": "Claude Code" }, { "method": "manual", "models": [], "framework": "n/a" } ], "strongest": "agent", "models": [ "Anthropic Claude (Opus/Fable-class, 2026)" ], "spend": "subscription-based", "notes": "Human-directed, machine-verified workflow: the author fixes targets and reviews statements; AI assistance drafts and repairs proofs; the Lean kernel is the acceptance criterion." }, "scope": "Formalized, 17 compared theorems, hypotheses inline. (1) plastic_salem_family: existence of the canonical root sequence of chi_n in [117/100, rho) for n >= 7 — largest below rho, strictly increasing, tending to rho. (2) family_pos_of_rho_le: chi_n > 0 on [rho, infinity). (3) lehmer_first_rung: chi_7 = (x - 1) * L over any commutative ring. (4) lehmer_root_below_plastic: L has a root in [117/100, rho). (5) ruler_second_conjugate: a nonreal z with z^4 = z + 1 and 1 < |rho * z| < rho * Q. (6) ruler_conjugates_eq_embedding_products: a complex number is a root of minpoly(rho * Q) iff it is a produc", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "Bedford-Kim's theorem is geometric: for n >= 7 and suitable parameters the maps are automorphisms of rational surfaces with entropy log lambda_n, lambda_n a Salem number, and the lambda_n increase to the plastic number. This entry formalizes the spectral statement about the polynomial family only: the canonical largest roots below rho, their strict increase, and the limit. The dynamical realization, the Salem property of the roots, and the absence of further roots in (1, 117/100) are cited, not formalized — so \"the canonical root below rho\" here corresponds to the paper's spectral radius via the cited factorization. The exclusion corollary for rho * Q (not a birational surface dynamical degree) additionally requires the cited Blanc-Cantat classification; the identification of the products of embeddings as the conjugates of rho * Q is NOT cited but formalized here, as statements (6) and (7). All finite margin claims are exact: the brackets are rational and kernel-decided; no floating point enters any compared statement or proof. Four further disclosures on the compositum half. (i) Internal Vieta closed forms are carried division-free (e.g. Q * r2 * |z|^2 = -1 rather than |z|^2 = -1/", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Stephanie Alexander" ], "notes": "Statements audited by the author against the source papers; the kernel and the axiom audit are the acceptance criteria. Three in-house adversarial passes, each automated (AI-assisted: Anthropic Claude, in Claude Code, under the author's direction), were applied to this surface: a mock-referee pass on the original seven-theorem surface (2026-08-25, re-run 2026-08-30); a four-lens significance panel on the compositum extension — three independent research-framing reviews plus an adversarial judge, whose recommended plan this surface implements, including the corrected bibliography (2026-08-31); and an assembly pass on the extended seventeen-theorem surface (2026-08-31)." }, "canonical": { "repo": "stalex444/mahler-minimizer-compositum", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-01-000005", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-01-000005", "commit": "d78ce1610608cf4a90d5580c267110c96d2f9af5", "trust": "high", "theorems": 17, "date": "2026-09-01" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "stalex444/only-two-morphic-numbers/formalization.yaml", "repo": "stalex444/only-two-morphic-numbers", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/stalex444/only-two-morphic-numbers/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Only two morphic numbers: the Aarts-Fokkink-Kruijtzer classification of the golden ratio and the plastic number", "description": "Kernel-checked classification of the morphic numbers — the reals p > 1 satisfying both of the golden ratio's defining scale closures: p + 1 = p^k (the sum of consecutive measures of the geometric system is again a measure) and p - 1 = p^(-l) (so is their difference; rendered zpow-free as p^l * (p - 1) = 1). These two equations are the arithmetic of self-similar proportion: a number satisfying both generates a measure system closed under the paper's superposition and juxtaposition, the property pair behind the divine proportion's standing as the plane's design constant. The theorem (Aarts-Fokki", "authors": [ "Stephanie Alexander" ], "maintainers": [ "Stephanie Alexander" ], "license": "MIT", "role": "", "substantive": "", "sources": [ { "title": "Morphic numbers", "id": "Nieuw Archief voor Wiskunde (5) 2 (2001), 56-58", "authors": [ "Jan Aarts", "Robbert Fokkink", "Godfried Kruijtzer" ], "type": "article", "relationship": "formalizes", "endorsement": "n/a" }, { "title": "On the irreducibility of certain trinomials", "id": "Mathematica Scandinavica 4 (1956), 287-302", "authors": [ "Ernst S. Selmer" ], "type": "article", "relationship": "background", "endorsement": "n/a" }, { "title": "On the irreducibility of the trinomials x^n +/- x^m +/- 1", "id": "Mathematica Scandinavica 8 (1960), 121-126", "authors": [ "Helge Tverberg" ], "type": "article", "relationship": "background", "endorsement": "n/a" }, { "title": "On the irreducibility of certain trinomials and quadrinomials", "id": "Mathematica Scandinavica 8 (1960), 65-70", "authors": [ "Wilhelm Ljunggren" ], "type": "article", "relationship": "background", "endorsement": "n/a" }, { "title": "Le Nombre Plastique: quinze Lecons sur l'Ordonnance architectonique", "id": "E. J. Brill, Leiden, 1960", "authors": [ "Hans van der Laan" ], "type": "book", "relationship": "background", "endorsement": "n/a" }, { "title": "Dom Hans van der Laan: Modern Primitive", "id": "Architectura & Natura Press, Amsterdam, 1994", "authors": [ "Richard Padovan" ], "type": "book", "relationship": "background", "endorsement": "n/a" }, { "title": "Ruimte en Getal: Het plastische Getal en het gulden-Snedegetal", "id": "Architectura & Natura, Amsterdam, 1998", "authors": [ "Godfried Kruijtzer" ], "type": "book", "relationship": "background", "endorsement": "n/a" }, { "title": "Algebraic integers whose conjugates lie in the unit circle", "id": "Duke Mathematical Journal 11 (1944), 597-602", "authors": [ "Carl Ludwig Siegel" ], "type": "article", "relationship": "background", "endorsement": "n/a" }, { "title": "On the product of the conjugates outside the unit circle of an algebraic integer", "id": "Bulletin of the London Mathematical Society 3 (1971), 169-175", "authors": [ "Christopher J. Smyth" ], "type": "article", "relationship": "background", "endorsement": "n/a" }, { "title": "Dynamical degrees of birational transformations of projective surfaces", "id": "Journal of the American Mathematical Society 29 (2016), 415-471", "authors": [ "Jeremy Blanc", "Serge Cantat" ], "type": "article", "relationship": "background", "endorsement": "n/a" }, { "title": "Dynamics of rational surface automorphisms: linear fractional recurrences", "id": "Journal of Geometric Analysis 19 (2009), 553-583", "authors": [ "Eric Bedford", "Kyounghee Kim" ], "type": "article", "relationship": "background", "endorsement": "n/a" } ], "related": [ { "id": "https://github.com/leanprover-community/mathlib4 (Mathlib/NumberTheory/Real/GoldenRatio.lean; Mathlib/RingTheory/Polynomial/Selmer.lean; Mathlib/Algebra/Polynomial/UnitTrinomial.lean)", "relationship": "builds-on", "note": "Honest accounting of what is consumed: Real.goldenRatio and goldenRatio_sq from the GoldenRatio file; SELMER'S THEOREM ITSELF (Polynomial.X_pow_sub_X_sub_one_irreducible, Thomas Browning) — the heavier of AFK's two literature inputs is Mathlib's, not this entry's; and the Ljunggren unit-trinomial machinery (trinomial, mirror, isUnitTrinomial_iff'', irreducible_aux3 — a public if aux-named lemma), " }, { "id": "https://github.com/stalex444/pdt-lean (parent development; pinned reference revision: commit d66ffbe813e48636aa636ffcd80070b81cbf10da)", "relationship": "other", "note": "The broader public project this entry belongs to. This repository is the complete, self-contained proof development for the compared statements, not a wrapper: the proof module PdtMorphic builds here with the pinned toolchain and no imports from the parent, and it is NATIVE to this repository — it appears in no other development at the pinned revision. The parent development shares the plastic-num" }, { "id": "Registered entry PALOMAR-2026-08-31-000004 (stalex444/mahler-measure-minima, registered revision ca3df22)", "relationship": "other", "note": "The registered sibling entry, by the same author: for each degree d in {2, 3, 4}, x^d - x - 1 attains the minimal Mahler measure among monic irreducible integer polynomials of degree d with measure above one. Complementarity, not duplication: the sibling characterizes x^2 - x - 1 and x^3 - x - 1 by EXTREMALITY (the Smyth/Mossinghoff lineage); this entry characterizes their roots phi and rho by the" } ], "classification": { "arxiv": [ "math.NT", "math.HO" ], "msc2020": [ "11R09", "11C08", "00A67" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Anthropic Claude (Opus/Fable-class, 2026)" ], "framework": "Claude Code" }, { "method": "manual", "models": [], "framework": "n/a" } ], "strongest": "agent", "models": [ "Anthropic Claude (Opus/Fable-class, 2026)" ], "spend": "subscription-based", "notes": "Human-directed, machine-verified workflow: the author fixes targets and reviews statements; AI assistance drafts and repairs proofs; the Lean kernel is the acceptance criterion." }, "scope": "Formalized, 4 compared theorems, hypotheses inline. (1) morphic_iff — THE theorem, the full AFK classification as an unconditional iff in p: the inline morphic property holds iff p = Real.goldenRatio or p = rho, with rho pinned hypothesis-style (1 < rho, rho^3 = rho + 1); the forward direction is the negative half (Selmer minimal polynomial, mirror dichotomy, cofactor divides X^2 - X + 1, forced factorizations pinning (k, l) = (2, 1) or (3, 4)), the reverse direction carries both witnesses; existence of the cubic's root is compared separately (plastic_exists). (2) goldenRatio_morphic and (3) p", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "STATEMENT fidelity is exact: the definition is transcribed from p. 57 of the paper (p + 1 = p^k and p - 1 = p^(-l), the second equation rendered zpow-free as p^l * (p - 1) = 1; the conjuncts 0 < k and 0 < l transcribe \"natural numbers\", and are removable — k = 0 would force 1 = p + 1, against 1 < p, and l = 0 would force p = 2, killed by 2^k = 3; p > 1 sits inside the morphic property, so the compared iff is unconditional in p. The plastic number enters the compared statements hypothesis-style (1 < rho and rho^3 = rho + 1) and is reconciled to the proof module's constructed constant by its uniqueness lemma. One PROOF-ROUTE divergence, with identical statement: the paper's Lemma derives the cofactor from Tverberg's theorem and an analysis of complex zeros of modulus 1 (a modulus-1 zero of x^m - x^(m-1) - 1 has distance 1 from both 0 and 1, forcing a primitive sixth root of unity, hence divisibility by x^2 - x + 1). The formal proof replaces both the citation and the complex analysis: the mirror dichotomy (extracted from Mathlib's Ljunggren machinery) forces the cofactor q to be mirror-fixed, and the rest runs wholly in Z[X] — q divides both the trinomial B and its mirror, hence q di", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Stephanie Alexander" ], "notes": "Statements audited by the author against the source paper (read in full from the journal's open-access PDF); the kernel and the axiom audit are the acceptance criteria. An in-house adversarial scoping pass, automated (AI-assisted: Anthropic Claude, in Claude Code, under the author's direction), was applied before the build, 2026-09-01: independent source-extraction and prior-art probes, a hands-on Lean feasibility probe (three compiled probe files against the pinned Mathlib), and an adversarial judge whose recommended statement set — the full iff as the load-bearing statement, the witnesses admitted only as labeled-routine readability surface — is the set implemented here." }, "canonical": { "repo": "stalex444/only-two-morphic-numbers", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-01-000008", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-01-000008", "commit": "9d3664f4ad3abc3783a9296ad1207a1ef1fee41d", "trust": "high", "theorems": 4, "date": "2026-09-01" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "stalex444/pdt-lean/formalization.yaml", "repo": "stalex444/pdt-lean", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/stalex444/pdt-lean/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Busch's theorem: Born-rule uniqueness on quantum effects (finite dimension)", "description": "Kernel-checked proof of Busch's theorem in finite dimension: every generalized probability measure on the effects of a complex matrix algebra — nonnegative on effects, additive whenever a sum of effects is again an effect, normalized at the identity — is represented by a unique density matrix via the trace pairing. Existence and uniqueness are both compared (busch_representation). The result is the uniqueness of the Born rule in the POVM reading; it is valid in every finite dimension (beyond finiteness and decidable equality of the index type, the compared statement assumes only [Nonempty n]) ", "authors": [ "Stephanie Alexander" ], "maintainers": [ "Stephanie Alexander" ], "license": "MIT", "role": "", "substantive": "", "sources": [ { "title": "Quantum states and generalized observables: a simple proof of Gleason's theorem", "id": "Physical Review Letters 91, 120403 (2003)", "authors": [ "Paul Busch" ], "type": "article", "relationship": "formalizes", "endorsement": "n/a" }, { "title": "Measures on the closed subspaces of a Hilbert space", "id": "Journal of Mathematics and Mechanics 6 (1957), 885–893", "authors": [ "Andrew M. Gleason" ], "type": "article", "relationship": "background", "endorsement": "n/a" } ], "related": [ { "id": "https://github.com/leanprover-community/mathlib4 (Matrix.PosSemidef, the Hermitian spectral theorem and eigenvector unitary, Unitary.conjStarAlgAut, the trace API)", "relationship": "builds-on", "note": "The compared statements use only Mathlib vocabulary (Matrix.PosSemidef, Matrix.trace, smul, the real-to-complex coercion) — zero custom definitions on the compared surface. The proof builds on Mathlib's Hermitian spectral theorem (eigenvalues, the eigenvector unitary) and star-algebra conjugation. The frame-function development itself — effect frame functions, the no-continuity homogeneity bootstr" }, { "id": "https://github.com/leanprover-community/physlib (QuantumInfo POVM development)", "relationship": "other", "note": "Physlib defines POVMs (positive semidefinite matrices summing to the identity) and measurement channels but contains no Busch/Gleason-type representation theorem for effect measures (checked at HEAD 4a4de62, 2026-08-24) — adjacent infrastructure, not this theorem." } ], "classification": { "arxiv": [ "quant-ph", "math-ph" ], "msc2020": [ "81P16", "81P15", "46N50" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Anthropic Claude (Opus/Fable-class, 2026)" ], "framework": "Claude Code" }, { "method": "manual", "models": [], "framework": "n/a" } ], "strongest": "agent", "models": [ "Anthropic Claude (Opus/Fable-class, 2026)" ], "spend": "subscription-based", "notes": "Human-directed, machine-verified workflow: the author fixes targets and reviews statements; AI assistance drafts and repairs proofs; the Lean kernel is the acceptance criterion." }, "scope": "Formalized, 2 compared theorems, finite dimension (matrices over the complex numbers; the representation theorem additionally assumes a nonempty index type). busch_representation: existence AND uniqueness — every generalized probability measure on effects satisfies f a = trace (rho * a) for exactly one density matrix rho (positive semidefinite, trace one). homogeneity_automatic: the no-continuity step compared separately — for a normalized measure, nonnegativity plus additivity force f (t • a) = t * f a for every effect a and every scalar t in [0,1]. No physical claim is part of the compared s", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The source states the theorem for a complex Hilbert space of any dimension, with additivity over countable collections of effects (sigma-additivity). This entry is the finite-dimensional case (matrices over the complex numbers), and additivity is weakened to binary additivity — a weaker hypothesis, so the compared theorem implies the paper's finite-dimensional instance. The infinite-dimensional sigma-additive statement is not formalized. The paper's upper bound f <= 1 on effects is derived in the development, not assumed.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Stephanie Alexander" ], "notes": "Statements audited by the author against the source paper's hypotheses; the kernel and the axiom audit are the acceptance criteria. An in-house adversarial mock-referee pass, automated (AI-assisted: Anthropic Claude, in Claude Code, under the author's direction; 2026-08-24) was applied to this surface before submission, and its blocking findings were fixed (2026-08-25)." }, "canonical": { "repo": "stalex444/pdt-lean", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-19-000007", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-19-000007", "commit": "0dae307fb786a3d180bba137a44555a1bf645637", "trust": "high", "theorems": 13, "date": "2026-08-20" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "stalex444/radial-harmonic-classification/formalization.yaml", "repo": "stalex444/radial-harmonic-classification", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/stalex444/radial-harmonic-classification/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Radial harmonic functions on a real inner product space: the classification, the Laplacian of radial functions, and the stability of circular orbits for power-law central forces; Ehrenfest's dimension", "description": "GENERAL THEOREMS FIRST. Throughout, E is a finite-dimensional real inner product space of dimension n, Δ is Mathlib's coordinate-free Laplacian, HarmonicOnNhd its harmonic functions, and f : ℝ → ℝ is twice differentiable on (0, ∞) (two HasDerivAt hypotheses, with derivatives f', f''). (I) THE CLASSIFICATION OF RADIAL HARMONIC FUNCTIONS: (1) for n ≠ 2 and E ≠ 0, the radial function f ∘ ‖·‖ is harmonic on E ∖ {0} if and only if f(r) = a + b r^(2−n) on (0, ∞) for some real a, b; (2) for n = 2, if and only if f(r) = a + b log r — every radial harmonic function with a twice-differentiable profile i", "authors": [ "Stephanie Alexander" ], "maintainers": [ "Stephanie Alexander" ], "license": "MIT", "role": "", "substantive": "", "sources": [ { "title": "In what way does it become manifest in the fundamental laws of physics that space has three dimensions?", "id": "KNAW, Proceedings, 20 I (1918), Amsterdam, 200-209 (communicated in the meeting of 26 May 1917)", "authors": [ "Paul Ehrenfest" ], "type": "article", "relationship": "independently-proves", "endorsement": "n/a" }, { "title": "Partial Differential Equations, second edition", "id": "Graduate Studies in Mathematics 19, American Mathematical Society, Providence, RI, 2010; §2.2.1(a), the radial-ansatz derivation of the fundamental solution", "authors": [ "Lawrence C. Evans" ], "type": "book", "relationship": "independently-proves", "endorsement": "n/a" }, { "title": "Théorème relatif au mouvement d'un point attiré vers un centre fixe", "id": "Comptes Rendus de l'Académie des Sciences 77 (1873), 849-853", "authors": [ "Joseph Bertrand" ], "type": "article", "relationship": "background", "endorsement": "n/a" }, { "title": "Schwarzschild field in n dimensions and the dimensionality of space problem", "id": "Il Nuovo Cimento 27 (1963), no. 3, 636-651", "authors": [ "F. R. Tangherlini" ], "type": "article", "relationship": "background", "endorsement": "n/a" }, { "title": "On the dimensionality of spacetime", "id": "Classical and Quantum Gravity 14 (1997), no. 4, L69-L75", "authors": [ "Max Tegmark" ], "type": "article", "relationship": "background", "endorsement": "n/a" }, { "title": "On the Physical Problem of Spatial Dimensions: An Alternative Procedure to Stability Arguments", "id": "Fundamenta Scientiae 8 (1987), no. 1, 73-91; arXiv:1205.4916 (2012)", "authors": [ "Francisco Caruso", "Roberto Moreira Xavier" ], "type": "article", "relationship": "background", "endorsement": "n/a" } ], "related": [ { "id": "https://github.com/leanprover-community/mathlib4 (Mathlib/Analysis/InnerProductSpace/Laplacian.lean, Mathlib/Analysis/InnerProductSpace/Harmonic/Basic.lean, Mathlib/Analysis/InnerProductSpace/Harmonic/Constructions.lean, Mathlib/Analysis/Calculus/MeanValue.lean, Mathlib/Analysis/Calculus/DerivativeT", "relationship": "builds-on", "note": "Honest accounting of what is consumed, at the pin fabf563a7c95a166b8d7b6efca11c8b4dc9d911f: the Laplacian on inner product spaces (InnerProductSpace.laplacian, the class notation Δ, laplacian_eq_iteratedFDeriv_orthonormalBasis with iteratedFDeriv_two_apply) and harmonic functions (HarmonicOnNhd, HarmonicAt, harmonicAt_congr_nhds, the algebra of harmonic functions, exists_norm_eq) — used in (1)-(8)" }, { "id": "https://github.com/stalex444/pdt-lean (parent development; pinned reference revision: commit 843cf2dad8f89c7ff6ba19dfaf8d605a67c573ef)", "relationship": "other", "note": "The broader public project this entry belongs to, and the provenance of the proof module. PdtEhrenfest.lean, the single proof module of this repository, was written for this entry on 2026-09-02, developed and committed in the parent at the pinned revision, and copied here byte-for-byte. This repository is a complete, self-contained proof development for the compared statements, not a wrapper: ever" } ], "classification": { "arxiv": [ "math.CA", "math.AP", "math-ph", "physics.class-ph" ], "msc2020": [ "31B05", "35J05", "34A05", "70F05", "70H14", "70K20" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Anthropic Claude (Fable-class, 2026)" ], "framework": "Claude Code" } ], "strongest": "agent", "models": [ "Anthropic Claude (Fable-class, 2026)" ], "spend": "subscription-based", "notes": "Human-directed, agent-executed. The author chose the target (Ehrenfest's argument as a family of theorems about one real function, the Laplacian of radial functions, and the classification of radial harmonic functions in Mathlib's own Laplacian), retrieved and read Ehrenfest's 1917 text, scoped the target against Mathlib, and wrote the briefs; the agent wrote PdtEhrenfest.lean and reported; the author verified, wrote the surfaces, and ran the gates and the audits. The spend field is the registry's monetary field; the author's model subscription is not metered per project, and the agent's usage" }, "scope": "Formalized, twenty-three compared theorems, sixteen general and seven instances, with hypotheses exactly as stated in the Lean. Throughout, E is a finite-dimensional real inner product space, n = Module.finrank ℝ E, ^ is the real power unless noted, and \"f twice differentiable on (0, ∞)\" is the pair of hypotheses hf : ∀ r, 0 < r → HasDerivAt f (f' r) r and hf' : ∀ r, 0 < r → HasDerivAt f' (f'' r) r for given f' f'' : ℝ → ℝ (no continuity of f'' is assumed). GENERAL, the classification: (1) harmonicOnNhd_radial_iff — [Nontrivial E], f twice differentiable on (0, ∞), n ≠ 2: HarmonicOnNhd (fun y ", "sorry_count": 0, "sorry_in_definitions": null, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "", "alignment": true, "original": true, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "Self-assessed by the author. In-house checks applied before this submission: the kernel build of the module and of the comparison surface; the axiom audit of all twenty-three compared theorems and of every theorem of PdtEhrenfest; the mechanical surface gates (names declared in both Lean files, sorry count equal to the compared count in Challenge and zero elsewhere, the Mathlib and pdt-lean pins identical across files, count words checked against the comparator, a vocabulary scan); and three adversarial mock-referee audits of the surfaces against the primary text, whose findings — a strict/non- strict overstatement in the scope docstring, an axiom-audit scope overclaim, the undisclosed third column of Ehrenfest's table, the word \"theorem\" for what the literature calls Ehrenfest's argument," }, "canonical": { "repo": "stalex444/radial-harmonic-classification", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-03-000004", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-03-000004", "commit": "74e9b3ea7f204fba2e84e546c80457bcca5c6268", "trust": "high", "theorems": 23, "date": "2026-09-03" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "stalex444/salem-theorem/formalization.yaml", "repo": "stalex444/salem-theorem", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/stalex444/salem-theorem/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Salem's theorem: every Pisot number is a two-sided limit of Salem numbers", "description": "Kernel-checked Salem's theorem — every Pisot number is a limit point of Salem numbers from both sides (R. Salem, Algebraic Numbers and Fourier Analysis, D. C. Heath, 1963, Ch. III, Theorem IV, p. 30; origin R. Salem, Duke Math. J. 12 (1945)) — with both of Salem's constructions, each stated as a construction. Three compared statements carry the surface. (1) salem_theorem — the theorem for Pisot numbers: a real algebraic integer alpha > 1 every other complex root of whose minimal polynomial has modulus < 1 (spelled as 1 < alpha, IsIntegral Z alpha, and every z in (minpoly Q alpha).aroots C othe", "authors": [ "Stephanie Alexander" ], "maintainers": [ "Stephanie Alexander" ], "license": "MIT", "role": "", "substantive": "", "sources": [ { "title": "Algebraic Numbers and Fourier Analysis", "id": "Heath Mathematical Monographs, D. C. Heath and Company, Boston, 1963, x+68 pp. (reprinted Wadsworth, Belmont, 1983); Ch. III, Theorem IV, p. 30, proof pp. 30-31; the definition of the class T and its reciprocity consequence, p. 26 (cited by page; numbered Lemma 1 in Smyth 2015)", "authors": [ "Raphaël Salem" ], "type": "book", "relationship": "formalizes", "endorsement": "n/a" }, { "title": "Power series with integral coefficients", "id": "Duke Mathematical Journal 12 (1945), no. 1, 153-172", "authors": [ "Raphaël Salem" ], "type": "article", "relationship": "formalizes", "endorsement": "n/a" }, { "title": "Pisot and Salem numbers in intervals of the real line", "id": "Mathematics of Computation 32 (1978), no. 144, 1244-1260; Section 4, eq. (26), p. 1252", "authors": [ "David W. Boyd" ], "type": "article", "relationship": "adapts", "endorsement": "n/a" }, { "title": "Seventy years of Salem numbers", "id": "Bulletin of the London Mathematical Society 47 (2015), no. 3, 379-395; Section 3.1; Lemma 1", "authors": [ "Christopher J. Smyth" ], "type": "article", "relationship": "background", "endorsement": "n/a" }, { "title": "Pisot and Salem Numbers", "id": "Birkhäuser, Basel, 1992, xiv+291 pp.; Chapter 6, Limit Points of Pisot and Salem Sets, pp. 101-117", "authors": [ "Marie-José Bertin", "Annette Decomps-Guilloux", "Marthe Grandet-Hugot", "Martine Pathiaux-Delefosse", "Jean-Pierre Schreiber" ], "type": "book", "relationship": "background", "endorsement": "n/a" }, { "title": "Small Salem numbers", "id": "Duke Mathematical Journal 44 (1977), no. 2, 315-328", "authors": [ "David W. Boyd" ], "type": "article", "relationship": "background", "endorsement": "n/a" }, { "title": "A remarkable class of algebraic integers. Proof of a conjecture of Vijayaraghavan", "id": "Duke Mathematical Journal 11 (1944), no. 1, 103-108", "authors": [ "Raphaël Salem" ], "type": "article", "relationship": "background", "endorsement": "n/a" }, { "title": "La répartition modulo 1 et les nombres algébriques", "id": "Annali della Scuola Normale Superiore di Pisa, Classe di Scienze (2) 7 (1938), no. 3-4, 205-248", "authors": [ "Charles Pisot" ], "type": "article", "relationship": "background", "endorsement": "n/a" }, { "title": "On Salem numbers, expansive polynomials and Stieltjes continued fractions", "id": "Journal de Théorie des Nombres de Bordeaux 27 (2015), no. 3, 769-804", "authors": [ "Christelle Guichard", "Jean-Louis Verger-Gaugry" ], "type": "article", "relationship": "background", "endorsement": "n/a" }, { "title": "On the fractional parts of the powers of a number (I), (II), (III)", "id": "(I) Journal of the London Mathematical Society 15 (1940), 159-160; (II) Proceedings of the Cambridge Philosophical Society 37 (1941), 349-357; (III) Journal of the London Mathematical Society 17 (1942), 137-138", "authors": [ "Tirukkannapuram Vijayaraghavan" ], "type": "article", "relationship": "background", "endorsement": "n/a" }, { "title": "Salem numbers and Pisot numbers via interlacing", "id": "Canadian Journal of Mathematics 64 (2012), no. 2, 345-367; Lemma 6.2", "authors": [ "James McKee", "Christopher J. Smyth" ], "type": "article", "relationship": "background", "endorsement": "n/a" } ], "related": [ { "id": "https://github.com/leanprover-community/mathlib4 (Mathlib/FieldTheory/Minpoly/IsIntegrallyClosed.lean; Mathlib/FieldTheory/Separable.lean; Mathlib/Algebra/Polynomial/Splits.lean; Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean; Mathlib/Topology/Order/IntermediateValue.lean)", "relationship": "builds-on", "note": "An accounting of what is consumed, at the pin fabf563a: the minimal-polynomial API (minpoly.dvd, minpoly.irreducible, and the Gauss step minpoly.isIntegrallyClosed_eq_field_fractions'), separability and nodup roots in characteristic zero, the splitting API over the algebraically closed complex numbers (Splits.eq_prod_roots_of_monic, Splits.roots_map), Vieta-type coefficient/root identities, the co" }, { "id": "https://github.com/google-deepmind/formal-conjectures (FormalConjecturesForMathlib/NumberTheory/PisotNumber.lean, added in commit c48b4c2d58, 2026-05-31; main at 7d1a8c9912747679d0093f6d1216420c33ee5ffa on 2026-09-01)", "relationship": "other", "note": "ADJACENT PRIOR ART, DEFINITION ONLY: that repository defines IsPisot (1 < theta, IsIntegral Z theta, every z in (minpoly Q theta).aroots C other than theta has norm < 1) and proves the golden ratio is Pisot; it contains no Salem numbers and no Salem theorem (its Lehmer-problem file states the problem as open). The compared salem_theorem adopts that spelling of the Pisot hypothesis, inlined, so tha" }, { "id": "This repository — the complete, self-contained development", "relationship": "other", "note": "This repository is the complete, self-contained proof development for the compared statements: the six proof modules (PdtPisotLadder, PdtSalemCircle, PdtSalemArith, PdtSalemMinus, PdtSalemEndgame, PdtSalemQuadUnit) and the bridge module SalemPisot build here with the pinned toolchain and import nothing but Mathlib and each other. The Pdt file prefix and the PDT namespace are the naming conventions" } ], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11R06", "11R04" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Anthropic Claude (Opus/Fable-class, 2026)" ], "framework": "Claude Code" }, { "method": "manual", "models": [], "framework": "n/a" } ], "strongest": "agent", "models": [ "Anthropic Claude (Opus/Fable-class, 2026)" ], "spend": "subscription-based", "notes": "Human-directed, machine-verified workflow: the author fixes targets and reviews statements; AI assistance drafts and repairs proofs; the Lean kernel is the acceptance criterion." }, "scope": "Formalized, three compared theorems, hypotheses inline. (1) salem_theorem — Salem's theorem for Pisot numbers (Pisot hypothesis inline; Salem-number conclusion inline). (2) salem_construction_two_sided — the main construction with its family root: pattern data plus P(1/alpha) != 0, concluding a sign e = ±1 (the sign of P(1/alpha)) and, per eps, a Salem root of some X^m P + e P.reverse (m >= 2) below alpha and of some X^m P - e P.reverse (m >= 2) above. (3) salem_quadratic_unit — the reciprocal quadratic unit case with its family root: alpha^2 = r alpha - 1, r >= 3, concluding, per eps, a Salem", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "(a) THE QUANTIFIER. Salem's Theorem IV quantifies over Pisot numbers; the proof modules' theorem salem_theorem_full quantifies over monic integer polynomials with the Pisot pattern (one real root alpha > 1, the other complex roots strictly inside the unit circle, conjugation-closed) — a larger index set (X^k times a minimal polynomial has the pattern) but equivalent in content to the Pisot-number form: the large root of any pattern polynomial is itself a Pisot number, and pattern_of_pisot supplies the converse. Compared statement (1), salem_theorem, is the classical Pisot-number form, transported from the pattern form through pattern_of_pisot; the pattern form is supporting, not compared. (b) THE PISOT SPELLING: 1 < alpha, IsIntegral Z alpha, and every z in (minpoly Q alpha).aroots C other than alpha has norm < 1 — the formal-conjectures spelling; by Polynomial.mem_aroots it is equivalent to the aeval form used for the Salem conclusion. (c) THE SALEM SPELLING: Salem's class-T definition (1963, p. 26) plus the explicit clause that 1/tau is a conjugate. Classically the clause and \"degree at least 4\" follow (Salem 1963, p. 26; Smyth 2015, Lemma 1); that equivalence is not formalized, ", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Stephanie Alexander" ], "notes": "Statements audited by the author against the sources; the kernel and the axiom audit are the acceptance criteria. Before the comparison surface was written, an automated review pass (AI-assisted: Anthropic Claude, in Claude Code, under the author's direction; 2026-09-01) re-verified the six proof modules in a fresh build at the pin (8569 jobs; all twenty top-level theorems on propext, Classical.choice, Quot.sound; zero sorry, native_decide, axiom, admit — the bridge module SalemPisot was written after that build and is covered by the in-build axiom audit of Solution and by the 2026-09-01 audit over its own declarations), checked every citation against Crossref, zbMATH, or Numdam, located \"Theorem IV\" in the 1963 monograph rather than in either Duke paper, swept prior art, and required that" }, "canonical": { "repo": "stalex444/salem-theorem", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-02-000003", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-02-000003", "commit": "887be765bf24c6a645600dfc7bd0a1055a1761dd", "trust": "high", "theorems": 3, "date": "2026-09-02" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "stalex444/werner-window/formalization.yaml", "repo": "stalex444/werner-window", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/stalex444/werner-window/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "The entangled Born correlations: a mixed-state entanglement witness through the Born rule forced on the composite algebra", "description": "Kernel-checked entanglement package on the bipartite matrix algebra (2x2 tensor 2x2, native product index): (1) the separable-state CHSH bound 2 for the real-Pauli tuple, proved through the Bloch-disk constraint with no spectral theory; (2) an entanglement witness — a bipartite state, positive semidefinite with unit trace, that is NOT separable, with Born CHSH expectation equal to the Tsirelson value 2*sqrt(2) (the Bell state, exhibited explicitly in the Solution); (3) spectral projections without spectral theory — (1+W)/2 is an effect for every Hermitian involution W, any finite dimension; (4", "authors": [ "Stephanie Alexander" ], "maintainers": [ "Stephanie Alexander" ], "license": "MIT", "role": "", "substantive": "", "sources": [ { "title": "Bell inequalities and the separability criterion", "id": "Physics Letters A 271 (2000) 319-326", "authors": [ "Barbara M. Terhal" ], "type": "article", "relationship": "formalizes", "endorsement": "n/a" }, { "title": "Proposed Experiment to Test Local Hidden-Variable Theories", "id": "Physical Review Letters 23 (1969) 880-884", "authors": [ "John F. Clauser", "Michael A. Horne", "Abner Shimony", "Richard A. Holt" ], "type": "article", "relationship": "background", "endorsement": "n/a" }, { "title": "Quantum generalizations of Bell's inequality", "id": "Letters in Mathematical Physics 4 (1980) 93-100", "authors": [ "Boris S. Cirel'son" ], "type": "article", "relationship": "background", "endorsement": "n/a" }, { "title": "Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model", "id": "Physical Review A 40 (1989) 4277-4281", "authors": [ "Reinhard F. Werner" ], "type": "article", "relationship": "formalizes", "endorsement": "n/a" }, { "title": "Violating Bell inequality by mixed states: necessary and sufficient condition", "id": "Physics Letters A 200 (1995) 340-344", "authors": [ "R. Horodecki", "P. Horodecki", "M. Horodecki" ], "type": "article", "relationship": "formalizes", "endorsement": "n/a" }, { "title": "Quantum states and generalized observables: a simple proof of Gleason's theorem", "id": "Physical Review Letters 91, 120403 (2003)", "authors": [ "Paul Busch" ], "type": "article", "relationship": "background", "endorsement": "n/a" } ], "related": [ { "id": "https://www.isa-afp.org/entries/TsirelsonBound.html (Echenim, Mhalla & Mori, \"The CHSH inequality: Tsirelson's upper-bound and other results\", AFP 2023-04-18; companion arXiv:2306.12535)", "relationship": "other", "note": "THE CLOSEST PRIOR FORMALIZATION, stated honestly: the AFP 2023 entry machine-checks, in Isabelle/HOL, the LHV bound 2, the separable-state CHSH bound (CHSH_expect_separable_leq, at ARBITRARY local dimensions and for arbitrary Hermitian-involution settings tuples), the state-level Tsirelson bound 2*sqrt(2) in any finite dimension, and Bell-state attainment (CHSH_expect_limit). Statement 1 here is, " }, { "id": "https://github.com/leanprover-community/mathlib4 (Algebra/Star/CHSH.lean: tsirelson_inequality, Kim Morrison 2020; Kronecker product; Matrix.PosSemidef API)", "relationship": "builds-on", "note": "Mathlib's tsirelson_inequality proves the operator-level ceiling 2^(3/2) in any ordered star-algebra — no states, no matrices — and its file declares state-level tightness (a 4x4 CHSH tuple with eigenvalue 2*sqrt(2)) as future work. The source modules behind this entry realize exactly that declared future work at matrix Born-expectation level (in-repo, not in Mathlib), and are built on Mathlib's K" }, { "id": "https://arxiv.org/abs/2604.03884 (Zhao & Yu, \"Formalizing CHSH Rigidity in Lean 4\", 2026-04-04; repo github.com/trzstony/RigidityTheorem)", "relationship": "other", "note": "The only other Lean 4 CHSH development found by the sweeps (baseline and 2026-08-31 re-run): formalizes the CHSH RIGIDITY theorem — the Tsirelson-attainment side. It states no separable-state bound, no mixed-state separability notion, and no Born/effect layer, so it does not touch the compared statements; disclosed here as the nearest Lean 4 neighbor." }, { "id": "https://github.com/leanprover-community/physlib (QuantumInfo: MState.IsSeparable, Ket.MES_isEntangled, Matrix.traceLeft/traceRight, trace_mul_kron_one_right)", "relationship": "other", "note": "Physlib's separability DEFINITION for mixed states, its pure-level entanglement of the maximally entangled state (Ket.MES_isEntangled), and the partial-trace adjunction trace_mul_kron_one_right ALL PREDATE this entry (present in the ancestor Lean-QuantumInfo by 2026-05-05); no definitional novelty is claimed for any of them, and the no-signalling / partial-trace layer of the source modules is deli" }, { "id": "https://www.isa-afp.org/entries/Isabelle_Marries_Dirac.html (Bordg, Lachnitt & He, AFP 2020-11-22)", "relationship": "other", "note": "Proves the four Bell states entangled at PURE-state/vector level (bell00_is_entangled2: the Bell kets are not tensor products of one-qubit kets) — the earliest machine-checked Bell-state entanglement found by the sweeps. Mixed-state separability and the density-matrix witness of statement 2 are outside its scope." }, { "id": "https://github.com/coq-quantum/CoqQ (Zhou, Barthe, Strub, Liu & Ying, POPL 2023, DOI 10.1145/3571222)", "relationship": "other", "note": "Holds the classical CHSH bound over finite distributions, the singlet's PURE-state expectation attaining 2*sqrt(2) (CHSH_violation), and the partial-trace adjunction at matrix level. No separability notion (zero hits), no separable bound, no mixed-state non-separability, no Born-uniqueness layer (checked at HEAD b169ea4, 2026-08-26 sweep; re-checked 2026-08-31: zero commits on any branch since — t" }, { "id": "https://github.com/stalex444/pdt-lean (parent development; pinned: commit d66ffbe813e48636aa636ffcd80070b81cbf10da)", "relationship": "other", "note": "The parent project this entry was developed in. This repository is the complete, self-contained proof development, not a wrapper: the five proof modules (PdtTsirelson, PdtBusch, PdtEntangled, PdtBornComposite, PdtWerner), the compared surface, and the pinned toolchain build here with no imports from the parent. At the pinned revision the parent holds the PdtTsirelson and PdtBusch modules of its re" } ], "classification": { "arxiv": [ "quant-ph", "math-ph" ], "msc2020": [ "81P40", "81P15", "81P16" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Anthropic Claude (Opus/Fable-class, 2026)" ], "framework": "Claude Code" }, { "method": "manual", "models": [], "framework": "n/a" } ], "strongest": "agent", "models": [ "Anthropic Claude (Opus/Fable-class, 2026)" ], "spend": "subscription-based", "notes": "Human-directed, machine-verified workflow: the author fixes targets and reviews statements; AI assistance drafts and repairs proofs; the Lean kernel is the acceptance criterion." }, "scope": "Formalized, 10 compared theorems (5 in the original package, 5 added in the 2026-08-31 extension) on the bipartite algebra (matrices over the complex numbers on the product index Fin 2 x Fin 2; three statements at arbitrary finite local dimensions). separable_chsh_le: every separable state (inline Werner definition) has normalized CHSH expectation at most 2 for the real-Pauli tuple. entanglement_exists: there is a state, positive semidefinite with unit trace, NOT separable, whose Born CHSH expectation equals 2*sqrt(2). effect_of_involution (any finite dimension): (1+W)/2 is an effect for every", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The challenge statements introduce the Pauli matrices through defining equations (X = !![0,1;1,0], Z = !![1,0;0,-1]) rather than named definitions, and spell separability, effects, and generalized probability measures inline; the source modules state the same theorems through named definitions (IsSeparable, IsEffect, IsFrameFunction), of which the challenge forms are the literal unfoldings. Party-2 observables are carried unnormalized (B± = 1 ⊗ (X ± Z)); the normalization sqrt(2) enters once, as the scalar (sqrt 2)⁻¹ on the trace, so the compared expectation is the physical CHSH value. In statement 4 the nonnegativity, additivity and normalization hypotheses on the measure mirror the source module's interface; the representation hypothesis alone suffices for the identity, so the statement is weaker than what is provable — stated this way for fidelity to the source module. The uniqueness hypothesis of statement 5 compares a candidate state's pairing against the real part of the witness state's pairing — a no-op, since the pairing of any positive-semidefinite state with an effect is real (witnessed in-module by rhoBell_represents_bellFrame). At the fixed real-Pauli tuple the separabl", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Stephanie Alexander" ], "notes": "Statements audited against the source modules and papers; the kernel and the axiom audit are the acceptance criteria. An in-house adversarial mock-referee pass, automated (AI-assisted: Anthropic Claude, in Claude Code, under the author's direction; 2026-08-30) was applied to the original five-statement surface on 2026-08-30; its findings were resolved before submission. The 2026-08-31 strengthening (statements 6-10: the general bounds and the Werner layer) was applied after an editorial research-interest review of the original five-statement surface; a second adversarial mock-referee pass (AI-assisted: Anthropic Claude, in Claude Code, under the author's direction; 2026-08-31, four independent lenses) was applied to the extended ten-statement surface; its findings were resolved before this" }, "canonical": { "repo": "stalex444/werner-window", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-01-000003", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-01-000003", "commit": "81c70fe45f81da86c25254f3eb52123d504b452c", "trust": "high", "theorems": 10, "date": "2026-09-01" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "sterraf/chain-bounding/formalization.yaml", "repo": "sterraf/chain-bounding", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/sterraf/chain-bounding/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Chain Bounding", "description": "Lean 4 formalization of the core results of \"Chain bounding, the leanest proof of Zorn's lemma, and an illustration of computerized proof formalization\" by Guillermo L. Incatasciato and Pedro Sánchez Terraf, published at the American Mathematical Monthly (preprint freely accessible at http://arxiv.org/abs/2404.11638). It formalizes the order-theoretic notion of _good chains_ for an \"expander\" of subsets of a poset: the comparability of good chains, the existence of a greatest good chain, the well-ordering of good chains for an expander of the form C ↦ C ∪ {f C}, the Chain Bounding Lemma (there", "authors": [ "Pedro Sánchez Terraf" ], "maintainers": [ "Pedro Sánchez Terraf" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Chain bounding, the leanest proof of Zorn's lemma, and an illustration of computerized proof formalization", "id": "http://dx.doi.org/10.1080/00029890.2025.2571375", "authors": [ "Guillermo L. Incatasciato", "Pedro Sánchez Terraf" ], "type": "article", "relationship": "formalizes", "endorsement": "participated" } ], "related": [ { "id": "https://github.com/sterraf/ChainBounding", "relationship": "other", "note": "The original repository of this same development, whose `main` branch contains formalizations existing up to the moment of publication, and its `palomar-test` branch formalizes the extra result `OrderSelector.wellOrdered_of_good`; the present repository is its Palomar-compliant reorganization, with identical statements and proofs." } ], "classification": { "arxiv": [ "math.LO" ], "msc2020": [ "03E25", "06A06" ] }, "automation": { "methods": [ { "method": "manual", "models": [], "framework": "n/a" }, { "method": "agent", "models": [ "GLM-5.3-Flash" ], "framework": "OpenCode" } ], "strongest": "agent", "models": [ "GLM-5.3-Flash" ], "spend": "subscription-based", "notes": "The original formalization was written by hand. In 2026 it was ported to the current toolchain and extended with prop:good-well-ordered with AI assistance under OpenCode at a branch of the original repository, and later formatted here for Palomar; every statement was checked against the paper's source and every proof was verified by Lean's kernel." }, "scope": "Formalizes the paper's core results: comparability of good chains, the existence of a greatest good chain (th:greatest-good-chain), well-ordering of good chains for an expander C ↦ C ∪ {f C} (prop:good-well-ordered, proved including the verification left to the reader in the paper's proof), Chain Bounding (lem:chain-bounding), the Unbounded Chain Lemma (lem:unbounded-chain), and the corollaries on Zorn's Lemma and the Bourbaki–Witt fixed point theorem. Not formalized: the ZFC-equivalence corollary (cor:unbounded-implies-AC), the converse remark that every well-ordered chain is good for some ex", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "(1) lem:unbounded-chain assumes AC in the paper; the Lean statement is for inhabited posets and its proof uses classical logic (Lean's Classical.choice, a permitted axiom). (2) Well-ordering in prop:good-well-ordered is rendered as the least-element property, with an equivalent WellFoundedOn rendering additionally proved in the proof module. (3) cor:unbounded-implies-AC is not formalized. No other divergences are known.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Pedro Sánchez Terraf" ], "notes": "Statements audited against the paper's TeX source; development compiled with Lean v4.32.0 against a pinned Mathlib revision; axiom footprint of the advertised results checked with `#print axioms` (the standard three axioms only)." }, "canonical": { "repo": "sterraf/chain-bounding", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-17-000002", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-17-000002", "commit": "f3eb6e0becc13d645f0809c98ed28912ee41ee6a", "trust": "high", "theorems": 6, "date": "2026-09-17" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "sweeneyde/1mfld/formalization.yaml", "repo": "sweeneyde/1mfld", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/sweeneyde/1mfld/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "The classification of compact 1-manifolds", "description": "A Lean 4 (mathlib) proof of the classification of 1-manifolds, specifically that every compact, connected, Hausdorff topological space charted on the half-line [0,∞) is homeomorphic to the circle S¹ or to the closed unit interval [0,1]. The proof follows David Gale's outline: shrink to a finite atlas of interval charts, then induct on the atlas size, merging two overlapping charts at each step; a connected overlap glues two charts into one (two boundary charts glue onto [0, 1]), while a disconnected overlap has exactly two components and yields the circle.", "authors": [], "maintainers": [ "Jim Fowler" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "The classification of 1-manifolds: a take-home exam", "id": "https://doi.org/10.2307/2322421", "authors": [ "David Gale" ], "type": "article", "relationship": "formalizes", "endorsement": "n/a" } ], "related": [], "classification": { "arxiv": [ "math.GT", "math.GN" ], "msc2020": [ "57N99", "54F65", "68V20" ] }, "automation": { "methods": [ { "method": "manual", "models": [], "framework": "n/a" }, { "method": "copilot", "models": [ "gpt-5.1" ], "framework": "ChatGPT" }, { "method": "agent", "models": [ "claude-fable-5" ], "framework": "Claude Code (Anthropic)" } ], "strongest": "agent", "models": [ "claude-fable-5", "gpt-5.1" ], "spend": "not tracked", "notes": "The skeleton of the development (the statement, the chart normalization pipeline, the induction in Classification.lean, and the classification of connected subsets of ℝ and ℝ≥0) was written by hand over 2024-2025; the remaining gluing lemmas were completed with machine assistance." }, "scope": "Formalized in full, proving that a compact connected Hausdorff space with a `ChartedSpace NNReal` instance is homeomorphic to the circle or to the closed unit interval, with the homeomorphism produced as data. The converse (that the circle and the interval themselves admit ℝ≥0-atlases, making the classification an iff) is not formalized in this repository. The setting is topological (C⁰) manifolds with boundary; the smooth case is not addressed. Noncompact 1-manifolds (the line and half-line cases) are also not treated.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "None known at the level of the statement. Relative to Gale's exposition the proof is rearranged for formalization: charts take values in ℝ≥0 rather than in ℝ with half-open model intervals, the \"outer overlap\" step is proved via a compactness/closure argument, and the gluings are piecewise constructions rather than abstract identifications.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Jim Fowler", "Dennis Sweeney" ], "notes": "No independent human review of the formalization. The statement in Challenge.lean was audited by the authors against the informal theorem. The development is machine-checked and sorry-free; `#print axioms classification` reports exactly [propext, Classical.choice, Quot.sound], and Comparator independently certifies that the library proves the Mathlib-only statement in Challenge.lean from those axioms." }, "canonical": { "repo": "sweeneyde/1mfld", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-10-000005", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-10-000005", "commit": "83e961ba28dd8182e7bae844756aba614eb1512e", "trust": "high", "theorems": 1, "date": "2026-09-10" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "TaifengYi/zeta7-irrationality/formalization.yaml", "repo": "TaifengYi/zeta7-irrationality", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/TaifengYi/zeta7-irrationality/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Irrationality of the 7-adic zeta value ζ₇(3)", "description": "A complete Lean 4 / Mathlib proof that the 7-adic zeta value ζ₇(3) is irrational. Here ζ₇(3) = L₇(3, ω⁻²) is the Kubota–Leopoldt value, equal to the 7-adic limit of the Euler-corrected Bernoulli numbers −(1 − 7^{w−1}) B_w / w over the weights w = 6·7^{n+1} − 2. The same is proved for η = ζ₇(3)/2. The proof is a determinant irrationality argument on the modular curve X₀(7), in four parts: independence of seven explicit germs attached to the weight −2 seven-adic Eisenstein series, with a uniform zero estimate; a seven-adic annular saving; an auxiliary-prime denominator estimate; and an Archimede", "authors": [ "Yi Taifeng" ], "maintainers": [ "Yi Taifeng" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Irrationality of ζ₇(3): first presented by this formalization and its companion manuscript \"A determinant argument for the irrationality of ζ₇(3)\" (Yi Taifeng, 15 September 2026, unrefereed, not otherwise published)", "id": "", "authors": [ "Yi Taifeng" ], "type": "original-proof", "relationship": "other", "endorsement": "" }, { "title": "Irrationality of certain p-adic periods for small p", "id": "doi:10.1155/IMRN.2005.1235", "authors": [ "Frank Calegari" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Irrationality of some p-adic L-values", "id": "doi:10.1007/s10114-007-1029-2", "authors": [ "Frits Beukers" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "On the irrationality of certain p-adic zeta values", "id": "doi:10.1007/s40687-025-00559-x", "authors": [ "Li Lai", "Cezar Lupu", "Johannes Sprang" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Analytic continuation of overconvergent eigenforms", "id": "doi:10.1090/S0894-0347-02-00405-8", "authors": [ "Kevin Buzzard" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "An introduction to p-adic L-functions", "id": "arXiv:2309.15692", "authors": [ "Joaquín Rodrigues Jacinto", "Chris Williams" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" } ], "related": [ { "id": "https://github.com/CBirkbeck/padic-L-functions/tree/45307015169791f1316f23a459c8e92ed5961c54", "relationship": "builds-on", "note": "The Kubota–Leopoldt measure construction (PadicLFunctions/) is adapted from this repository under Apache-2.0, with the copyright notices kept. See history/KL_CONSTRUCTION_PROVENANCE.md." }, { "id": "https://github.com/CBirkbeck/LeanModularForms/tree/512911ce1a936ac9054415c5c72701fbd5cd5ae5", "relationship": "adapts", "note": "The level-raising module (LeanModularForms/) is adapted under Apache-2.0." }, { "id": "ANR-FALSE PadicModForms (Riccardo Brasca), revision f463dbbae6d24e08538ebe49b86718f094c5026a", "relationship": "adapts", "note": "Some uniform power-series topology, Serre p-adic modular form presentation and q-expansion derivative lemmas are adapted under Apache-2.0 (history/UNIFORM_PROVENANCE.md, history/THIRD_PARTY_SECTION11.md)." }, { "id": "arXiv:2302.14491", "relationship": "other", "note": "Narayanan's Lean 3 formalization of p-adic L-functions. Consulted as a reference only; no code from it is imported." } ], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11J72", "11S40", "11F33", "11F85", "11B68", "68V20" ] }, "automation": { "methods": [ { "method": "manual", "models": [], "framework": "n/a" }, { "method": "agent", "models": [ "Claude Opus 5 (in the Claude Code sessions that completed the internal radius-49 proof, the removal of the radius axiom, the final audits and this packaging)", "model versions of earlier Claude Code sessions are not individually recorded" ], "framework": "Claude Code (Anthropic)" }, { "method": "other", "models": [ "GPT-6 Astra" ], "framework": "not recorded" } ], "strongest": "agent", "models": [ "Claude Opus 5 (in the Claude Code sessions that completed the internal radius-49 proof, the removal of the radius axiom, the final audits and this packaging)", "GPT-6 Astra", "model versions of earlier Claude Code sessions are not individually recorded" ], "spend": "not tracked", "notes": "This project was developed through substantial human–AI collaboration. GPT-6 Astra and Claude Code were used extensively for mathematical exploration, proof development, Lean implementation, debugging, refactoring, computational experimentation, certificate generation, and proof-audit work. The human author: provided detailed mathematical direction; selected the research target; imposed the proof and verification standards; decomposed the project into intermediate goals; repeatedly evaluated and redirected proposed approaches; supplied guidance; supervised the elimination of conditional assump" }, "scope": "Formalized in full: ζ₇(3) ∉ ℚ and η = ζ₇(3)/2 ∉ ℚ, for the genuine Kubota–Leopoldt value. Challenge.lean defines ζ₇(3) as the 7-adic limit of the Euler-corrected Bernoulli approximants and states their convergence (approximant_tendsto), which the Solution proves. The Solution identifies this limit with the project's measure-theoretic construction Zeta7Main.zeta7Three. Not claimed: anything about ζ_p(3) for p ≠ 7 or about other p-adic zeta values.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The companion manuscript defines ζ₇(3) = lim (1 − 7^{w_ν−1}) ζ(1 − w_ν) with w_ν = 6·7^ν − 2, ν ≥ 1. The Challenge uses the same sequence, indexed from n = ν − 1 and written with Bernoulli numbers via ζ(1 − w) = −B_w/w. The manuscript cites published results for the radius-49 continuation. The Lean proof replaces those citations by an internal proof. No other divergences are known.", "alignment": true, "original": true, "review": { "status": "author-verified", "bucket": "author-verified", "reviewers": [ "none (no independent human mathematical review yet)" ], "notes": "The author reviewed the statement and the audit outputs, and has accepted the formal proof. Mechanical checks: - `lake build` passes; - #print axioms for both endpoints gives exactly [propext, Classical.choice, Quot.sound]; - FinalInternalAudit finds no project axiom among 10,903 project declarations in the import closure; - Radius49CompleteAudit confirms the radius proof does not depend on the final theorems; - a local surrogate of Comparator (palomar_local_check/) finds the Challenge and Solution closures identical. The official Comparator could not be run locally because its Landrun sandbox is Linux-only. The mathematical argument has not been refereed." }, "canonical": { "repo": "taifengyi/zeta7-irrationality", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-19-000004", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-19-000004", "commit": "8af843a0536a8f1fc3945ef302d16fb2f6622b2a", "trust": "high", "theorems": 3, "date": "2026-09-19" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "teal-sea/zeta-lab/lean/bridge/formalization.yaml", "repo": "teal-sea/zeta-lab", "path": "lean/bridge/formalization.yaml", "directory": "lean/bridge", "origins": [ "palomar", "search" ], "url": "https://github.com/teal-sea/zeta-lab/blob/HEAD/lean/bridge/formalization.yaml", "version": "v0.4", "missing": [], "name": "Zeta Lab: the seven-point simple-zero bound, formalised to its hypotheses", "description": "Let N(T,2T) count the nontrivial zeros of the Riemann zeta function with ordinate in (T,2T], with multiplicity, and N_0^s(T,2T) the simple zeros on the critical line in that range. The Lean development accompanying arXiv:2608.13637 (anthropics/zeta-23-lean) proves, for Mathlib's riemannZeta and unconditionally, that for every eps > 0 and all large T, (H - eps) N(T,2T) <= N_0^s(T,2T) with H = 3/2 - (1/sqrt 2) cot(1/sqrt 2) = 0.6725007036794116... (its Theorem D). Ainta (github.com/ainta/zeta-simple-zeros, paper/riemann.tex, August 2026) refines this to 0.6730085279277797... by a stability term ", "authors": [ "Thomas Lince" ], "maintainers": [ "Thomas Lince" ], "license": "MIT", "role": "", "substantive": "", "sources": [ { "title": "More than 67.3% of the zeros of the Riemann zeta function are simple and lie on the critical line", "id": "", "authors": [ "Ainta" ], "type": "paper", "relationship": "adapts", "endorsement": "not-contacted" }, { "title": "More than two thirds of the zeta zeros are simple and on the critical line", "id": "arXiv:2608.13637", "authors": [ "L. Alpoge", "R. Furman" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" } ], "related": [ { "id": "https://github.com/anthropics/zeta-23-lean", "relationship": "builds-on", "note": "The Lean development accompanying arXiv:2608.13637, pinned at commit 3635e74826a4c1fcece7d1cd2b6fa75e43a00510 as a Lake dependency of the selected project. It supplies the counting functions Ncount and N0simple for Mathlib's riemannZeta, the analytic inputs PaperInputs discharged for riemannZeta, the constant HD 1 = H, the Montgomery-Taylor window and its endgame, the von Neumann trace inequality," } ], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11M26", "15A42" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Claude (Anthropic)" ], "framework": "Zeta Lab, github.com/teal-sea/zeta-lab" } ], "strongest": "agent", "models": [ "Claude (Anthropic)" ], "spend": "not tracked", "notes": "Human direction, agent proof writing, kernel checking, and an explicit obligation ledger per attack group under hunts/ainta_seven_point/bridge/. The single non-Lean input is carried as a named hypothesis of every advertised theorem so that the conditional result cannot be read as an unconditional one. The interval-arithmetic runs behind that hypothesis were executed by this laboratory and are recorded, with node and depth counts and their artifacts, in hunts/ainta_seven_point/RUNS.md and RESULTS.md; they are evidence for the hypothesis, not a proof of it. The derivations were directed and veri" }, "scope": "Formalized: the four advertised declarations, sorry-free, against pinned Mathlib v4.33.0-rc2 and anthropics/zeta-23-lean at the pinned commit, with #print axioms reporting exactly propext, Classical.choice and Quot.sound for each of the four. CONDITIONAL, AND THE CERTIFICATE IS A HYPOTHESIS, NOT A LEAN FACT. seven_point_bound takes the named hypothesis hCert, which says: for every g : Fin 6 -> R with every g i >= 0, c <= F6 p g, where F6 is Ainta's seven-point functional (the pressure term (1/p) sum g_i plus the 21 pairwise overlap weights w(y_j - y_i) = k(y_j - y_i)^2 with coefficient 2/(7-r)", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The formal argument follows the paper's structure and departs from its text in the following places, each recorded in hunts/ainta_seven_point/BRIDGE.md section 6. The kernel limit (Lemma 3.1) is proved for every pair of retained zeros at every separation; the paper's bounded-separation parameter R_0 is kept in the signature but is not used. The tail passage (Corollary 2.2) does not pass through the dependency's limit over windows lambda -> 1; it re-states two of the dependency's endgame lemmas at lambda = 1 with the defect carried as a hypothesis, which their own inputs permit. Block pinching (eq:pinching) is proved without convexity of the matrix functional or unitary invariance, from the row-stochastic mixture of the spectrum of a principal submatrix and scalar Jensen; the partition form needs no positivity of Psi, the two-block form does. The block defect lemma (Lemma 4.3) is proved for every Hermitian matrix, positivity unused. The block energy lemma (Lemma 4.2) is proved as a fibre count plus an exact telescoping identity, both needing w even. The block bound (eq:269block) uses |k| <= 1, which the paper does not state and which holds because cos(sqrt 2 t) >= 0 on [-1/2, 1/2]; ", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "No external mathematical review has been performed, and no person outside this laboratory has read the development. \"self-assessed\" means exactly that: the checks below were run by the laboratory on its own work. Packaging state. The three blockers recorded here in an earlier revision are resolved. (1) The advertised development now lives in its own Lake package, lean/bridge, which requires anthropics/zeta-23-lean at the pinned commit and whose root module imports every module in it; `lake build` at that root completes with 8860 jobs and zero errors, so a replay of the selected project builds the theorem rather than stopping on three modules of hunts/frontier_math/zeta23ext that the theorem never imports (repository issue 101, which remains open for that other package and is now unrelated " }, "canonical": { "repo": "teal-sea/zeta-lab", "directory": "lean/bridge" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-25-000005", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-25-000005", "commit": "84312e4477dfeb7e0d8a91c38897f225f5a52f19", "trust": "high", "theorems": 7, "date": "2026-08-25" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "teal-sea/zeta-lab/lean/bridge/palomar-v2/formalization.yaml", "repo": "teal-sea/zeta-lab", "path": "lean/bridge/palomar-v2/formalization.yaml", "directory": "lean/bridge/palomar-v2", "origins": [ "search" ], "url": "https://github.com/teal-sea/zeta-lab/blob/HEAD/lean/bridge/palomar-v2/formalization.yaml", "version": "v0.4", "missing": [], "name": "Zeta Lab: the n-point simple-zero bound, with unconditional three- and four-point instances", "description": "Let N(T,2T) count the nontrivial zeros of the Riemann zeta function with ordinate in (T,2T], with multiplicity, and N_0^s(T,2T) the simple zeros among them on the critical line. The Lean development accompanying arXiv:2608.13637 (anthropics/zeta-23-lean) proves, for Mathlib's riemannZeta and unconditionally, that for every eps > 0 and all large T, (H - eps) N(T,2T) <= N_0^s(T,2T), with H = 3/2 - (1/sqrt 2) cot(1/sqrt 2) = 0.67250070367941164573... (its Theorem D). Ainta (github.com/ainta/zeta-simple-zeros) refines H by carrying the spectral defect of the Gram matrix of the simple-zero vectors ", "authors": [ "Thomas Lince" ], "maintainers": [ "Thomas Lince" ], "license": "MIT", "role": "", "substantive": "", "sources": [ { "title": "More than 67.3% of the zeros of the Riemann zeta function are simple and lie on the critical line", "id": "", "authors": [ "Ainta" ], "type": "paper", "relationship": "adapts", "endorsement": "no-response" }, { "title": "More than two thirds of the zeta zeros are simple and on the critical line", "id": "arXiv:2608.13637", "authors": [ "L. Alpoge", "R. Furman" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Provisional stronger 7-point certificate: F6 >= 191/50000 gives 67.302136%", "id": "", "authors": [ "Gohms" ], "type": "web discussion", "relationship": "background", "endorsement": "no-response" } ], "related": [ { "id": "https://github.com/anthropics/zeta-23-lean", "relationship": "builds-on", "note": "Pinned at commit 3635e74826a4c1fcece7d1cd2b6fa75e43a00510 as a Lake dependency of the selected project. It supplies Ncount and N0simple for Mathlib's riemannZeta, the analytic inputs discharged for riemannZeta, the constant HD 1 = H, the Montgomery-Taylor window and its endgame, and the linear algebra. The advertised theorem is its thmD_0_simple_mult with HD 1 replaced by Phi_n n c m p. Nothing in" } ], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11M26", "15A42" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Claude (Anthropic)" ], "framework": "Zeta Lab, github.com/teal-sea/zeta-lab" } ], "strongest": "agent", "models": [ "Claude (Anthropic)" ], "spend": "not tracked", "notes": "Human direction, agent proof writing, kernel checking, within an AI-assisted computational mathematics framework operated by Thomas Lince at Zeta Lab." }, "scope": "Formalized: V2Solution proves the seven advertised declarations without sorry against pinned Mathlib v4.33.0-rc2 and anthropics/zeta-23-lean at the pinned commit. V2Challenge contains seven deliberate statement placeholders. Each proved declaration reports exactly propext, Classical.choice and Quot.sound. WHICH ARE CONDITIONAL. Three of the seven carry the named hypothesis hCert: for every g : Fin (n-1) -> R with every g i >= 0, c <= F n p g, F being the n-point functional (the pressure term (1/p) sum g_i plus the n(n-1)/2 pairwise overlap weights with coefficient 2/(n-(j-i))). n_point_bound t", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The formal argument follows the paper's structure; each place it departs from the text is recorded in hunts/ainta_seven_point/BRIDGE.md section 6. The kernel limit is proved at every separation; the tail passage carries the defect as a hypothesis at lambda = 1 rather than taking a limit over windows; block pinching is proved without convexity or unitary invariance; every o(N) in the paper is an explicit eta N in the Lean. The n-point layer and the proofs of the finite inequalities at n = 3 and n = 4 have no counterpart in the paper.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "No external mathematical review has been performed and no person outside this laboratory has read the development. Deliberate sorry count: V2Challenge.lean carries seven sorry, one per advertised statement, which is what the Palomar format requires of a statement-only module. The development and V2Solution.lean carry none. A claim that this tree is sorry-free has to say which object it means, and this field is that statement. Checks: the seven advertised declarations report exactly the three standard axioms in a whole-package build; a static scan of every Lean file in the selected project finds no axiom, opaque, unsafe, admit, native_decide, implemented_by or extern; the Challenge's copies of the definitions are tied to the development's by rfl bridges in the Solution, the one exception be" }, "canonical": { "repo": "teal-sea/zeta-lab", "directory": "lean/bridge/palomar-v2" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "teal-sea/zeta-lab/lean/formalization.yaml", "repo": "teal-sea/zeta-lab", "path": "lean/formalization.yaml", "directory": "lean", "origins": [ "palomar", "search" ], "url": "https://github.com/teal-sea/zeta-lab/blob/HEAD/lean/formalization.yaml", "version": "v0.4", "missing": [], "name": "Zeta Lab: source-admissible strong closure for the F1 window functional", "description": "A variational identity for the Fredholm operator A = I + T on the interval I = [-1/2, 1/2], where T is convolution against the Farmer-Gonek-Lee pair-correlation form factor F1(x) = |x| - 4x^2 + sum_k a_k |x|^(2k+3) with a_k = 2^(2k+3) k! / (2k+2)!. Let w = A^(-1)1 and c* = <1, A^(-1)1>. Over the source-admissible class of scalar profiles v(s) = phi(Ls)^2 induced by even, radially nonincreasing C^2 windows supported exactly on [-L/2, L/2] with 0 <= phi <= 1 and uniform L^1 bounds on phi'' and (phi^2)'', the supremum of <1,v>^2 / equals c*, and the infimum of the reciprocal quotient ^2/ over the compactly supported monotone admissible class is exactly <1, A^(-1)1>", "id": "", "authors": [], "type": "original-proof", "relationship": "other", "endorsement": "" }, { "title": "More than two thirds of the zeta zeros are simple and on the critical line", "id": "arXiv:2608.13637", "authors": [ "L. Alpoge", "R. Furman" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Pub 1 source-admissible strong closure: exact-rational evidence package (in-repository working document)", "id": "", "authors": [], "type": "other", "relationship": "background", "endorsement": "" } ], "related": [ { "id": "https://github.com/anthropics/zeta-23-lean", "relationship": "independent", "note": "The Lean development accompanying the cited paper, at commit ce196b66685e96f874850a2836e183feccebef2a. It formalizes generic xi-prime window-functional and trace-assembly machinery over its own WindowProfile class and verifies the displayed flat and quartic instances. It does not contain the supremum identity proved here, and this submission neither imports from it nor depends on it." } ], "classification": { "arxiv": [ "math.NT", "math.CA" ], "msc2020": [ "45B05", "49R05", "11M26" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Claude (Anthropic)", "Aristotle (Harmonic)" ], "framework": "Zeta Lab, github.com/teal-sea/zeta-lab" } ], "strongest": "agent", "models": [ "Aristotle (Harmonic)", "Claude (Anthropic)" ], "spend": "not tracked", "notes": "Human direction, machine proof search, kernel checking, and an explicit obligation ledger. Every analytic fact the argument consumes was carried as a named field of the StrongClosureData interface until it was discharged, so that a conditional result could not be read as an unconditional one. The status of that interface is tracked in ZetaLean/Pub1/OBLIGATIONS.md. The derivations and computations supporting these results were directed and verified within an AI-assisted computational mathematics framework operated by Thomas Lince at Zeta Lab." }, "scope": "Formalized: the three advertised declarations, unconditionally against pinned Mathlib v4.33.0-rc2. pub1_strong_closure and pub1_strong_closure_reciprocal assume only IsProfile w, which says that w is a bounded continuous solution of Aw = 1, that is, it names the object the statement is about rather than imposing a further condition on it. pub1_strong_closure_exists carries no hypothesis at all, so no advertised statement can be vacuous. Not formalized, and deliberately outside the advertised statements. The attained maximum over the wider C^2(I) profile class is not itself a formalized stateme", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The formalization is not a literal transcription of the L^2 Fredholm presentation used in the informal companion note, and the differences are deliberate. It works with continuous functions on I under the pairing given by the integral over I rather than on an L^2 quotient space; completeness is never needed, since the Cauchy-Schwarz half is algebra and the limit half needs only boundedness. It obtains w by a Banach fixed point rather than by inverting an operator on that quotient space. It derives the interior second-derivative identity by splitting the integral at t = s rather than through distribution theory. These are mathematically equivalent formulations of the same argument, chosen to avoid formalizing quotient-space machinery the proof does not require. One further modelling choice is internal to the development: the defining equation for the profile is stated with a clamped kernel, clampedKernel s t = F1(clamp s - clamp t), which on I x I is exactly F1(s-t) and off I repeats the boundary values. F1 is not entire and its row mass is bounded by 4/9 only for s in I, while the Schur and Banach fixed-point lemmas require a hypothesis holding for every real s. The clamped kernel ", "alignment": true, "original": true, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "No external mathematical review has been performed. Internal checks that were performed and are recorded in the repository: the development is sorry-free and uses no theorem-specific axioms, and each advertised declaration was confirmed by #print axioms to depend on exactly propext, Classical.choice and Quot.sound; the four analytic obligations of the StrongClosureData interface are tracked individually in ZetaLean/Pub1/OBLIGATIONS.md and each is discharged; the advertised existential form is quantified over every parameter so that it cannot be vacuous; and a stale status row that understated this item's grade for three days was corrected in the open rather than swept, as recorded in docs/27-state-of-the-transplant.md. On the submission surface itself: Challenge.lean imports Mathlib alone " }, "canonical": { "repo": "teal-sea/zeta-lab", "directory": "lean" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-21-000012", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-21-000012", "commit": "097215a8413641fd3b3de137386316bf294328de", "trust": "high", "theorems": 1, "date": "2026-08-21" }, { "id": "PALOMAR-2026-08-21-000004", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-21-000004", "commit": "8a28a4faa3fb0fe68649f4f22faba7efe4bc3406", "trust": "high", "theorems": 3, "date": "2026-08-21" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "teal-sea/zeta-lab/lean/palomar-dh/formalization.yaml", "repo": "teal-sea/zeta-lab", "path": "lean/palomar-dh/formalization.yaml", "directory": "lean/palomar-dh", "origins": [ "search" ], "url": "https://github.com/teal-sea/zeta-lab/blob/HEAD/lean/palomar-dh/formalization.yaml", "version": "v0.4", "missing": [], "name": "Zeta Lab: the analytic half of Davenport-Heilbronn", "description": "Davenport and Heilbronn (1936) exhibited a Dirichlet series with real coefficients satisfying a Riemann-type functional equation which nevertheless has zeros off the critical line. It is the standard demonstration that a functional equation of Riemann type does not on its own force the Riemann Hypothesis, it is the reference counterexample against which structural explanations of the Riemann Hypothesis are tested, and the location of its zeros remains an active subject. The function is DH(s) = (1 - i*kappa)/2 * L(s, chi) + (1 + i*kappa)/2 * L(s, chi^-1) for a quartic Dirichlet character chi mo", "authors": [ "Thomas Lince" ], "maintainers": [ "Thomas Lince" ], "license": "MIT", "role": "", "substantive": "", "sources": [ { "title": "On the zeros of certain Dirichlet series", "id": "", "authors": [ "H. Davenport", "H. Heilbronn" ], "type": "paper", "relationship": "formalizes", "endorsement": "n/a" } ], "related": [], "classification": { "arxiv": [ "math.NT", "math.CV" ], "msc2020": [ "11M26", "11M41" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Claude (Anthropic)" ], "framework": "Zeta Lab, github.com/teal-sea/zeta-lab" } ], "strongest": "agent", "models": [ "Claude (Anthropic)" ], "spend": "not tracked", "notes": "The functional-equation half reduces to a root-number identity, and the convention-sensitive steps are derived in the development rather than remembered, from Mathlib's Real.cos_pi_div_five by radical algebra. The development records explicitly which half of the Davenport-Heilbronn theorem is kernel-checked and which remains a numerical obligation." }, "scope": "Formalized: the single statement advertised in DHChallenge.lean, unconditionally. That is the existence of an entire function satisfying the Davenport-Heilbronn series representation on Re z > 1 together with the completed functional equation. Not formalized, and deliberately outside the advertised statement: the existence of a zero of the Davenport-Heilbronn function off the critical line, and therefore the Davenport-Heilbronn theorem itself; any numerical enclosure, interval evaluation or certificate for any zero; and any statement about the Riemann zeta function or the Riemann Hypothesis.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "One deliberate modelling choice, documented at the point of use. The functional equation is stated with explicit guards Complex.Gamma ((z+1)/2) != 0 and Complex.Gamma ((1-z+1)/2) != 0. These exclude exactly the points at which Mathlib's junk value Gamma = 0 would make the unrestricted equation false; away from them both sides are the honest completed function. Without the guards the statement would be false for reasons having nothing to do with the mathematics.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "No external mathematical review has been performed. Internal checks recorded in the repository. The substantive development under lean/ZetaLean and the DHSolution module are sorry-free and axiom-clean against pinned Mathlib v4.33.0-rc2. This claim deliberately excludes the Challenge modules: Challenge.lean and DHChallenge.lean together carry four placeholder sorrys, one per advertised statement across this repository's two submission surfaces, because the Palomar format requires a Challenge to state its claim without proving it. Those placeholders are not steps in any proof, and Comparator checks that the corresponding Solution declarations are hole-free. Further internal checks: the definitions in DHChallenge.lean are verbatim copies of the development's own definitions and the DHSolution" }, "canonical": { "repo": "teal-sea/zeta-lab", "directory": "lean/palomar-dh" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "tejstead/heilbronn-site/proofs/palomar/heilbronn-convex/formalization.yaml", "repo": "tejstead/heilbronn-site", "path": "proofs/palomar/heilbronn-convex/formalization.yaml", "directory": "proofs/palomar/heilbronn-convex", "origins": [ "palomar" ], "url": "https://github.com/tejstead/heilbronn-site/blob/HEAD/proofs/palomar/heilbronn-convex/formalization.yaml", "version": "v0.4", "missing": [], "name": "heilbronn-convex-3-8", "description": "For n = 3 through n = 8, this formalization determines the supremal minimum triangle area among n indexed planar points whose own convex hull has Lebesgue area one. It proves a pointwise upper bound, attainment, and the exact value for every n in that range. The values are 1, 1/2, (5-sqrt(5))/10, 1/6, 1/9, and, at n = 8, the unique root in (79/1000, 81/1000) of 2060x^5 - 2332x^4 + 1064x^3 - 240x^2 + 26x - 1. For n = 3, 4, 5, 6, and 8 it also proves pairwise uniqueness of optimizers modulo arbitrary relabeling and arbitrary nonsingular affine transformations. Uniqueness is false at n = 7; inste", "authors": [ "Tej Stead" ], "maintainers": [ "Tej Stead" ], "license": "MIT", "role": "", "substantive": "", "sources": [ { "title": "Exact convex Heilbronn value and optimizer classification for five points", "id": "", "authors": [ "Tej Stead" ], "type": "other", "relationship": "other", "endorsement": "" }, { "title": "An explicit classification of seven-point optimizers", "id": "", "authors": [ "Tej Stead" ], "type": "other", "relationship": "other", "endorsement": "" }, { "title": "Exact convex Heilbronn value and optimizer classification for eight points", "id": "", "authors": [ "Tej Stead" ], "type": "other", "relationship": "other", "endorsement": "" }, { "title": "Elementary convex Heilbronn cases for three and four points", "id": "", "authors": [], "type": "folklore", "relationship": "independently-proves", "endorsement": "" }, { "title": "Heilbronn Problem for Six Points in a Planar Convex Body", "id": "https://doi.org/10.1007/978-1-4613-3557-3_13", "authors": [ "Andreas W. M. Dress", "Lu Yang", "Zhenbing Zeng" ], "type": "paper", "relationship": "independently-proves", "endorsement": "" }, { "title": "Heilbronn Problem for Seven Points in a Planar Convex Body", "id": "https://doi.org/10.1007/978-1-4613-3557-3_14", "authors": [ "Lu Yang", "Zhenbing Zeng" ], "type": "paper", "relationship": "independently-proves", "endorsement": "" }, { "title": "Affinely Regular Polygons as Extremals of Area Functionals", "id": "https://doi.org/10.1007/s00454-007-9010-5", "authors": [ "Paolo Gronchi", "Marco Longinetti" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "The Heilbronn Problem for Convex Regions", "id": "https://erich-friedman.github.io/packing/heilconvex/", "authors": [ "Erich Friedman" ], "type": "web discussion", "relationship": "background", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.CO", "math.MG" ], "msc2020": [ "52C10" ] }, "automation": { "methods": [ { "method": "agent", "models": [], "framework": "" }, { "method": "other", "models": [], "framework": "" } ], "strongest": "agent", "models": [], "spend": "", "notes": "" }, "scope": "The 29 selected declarations cover the own-convex-hull-normalized problem only for n = 3 through n = 8; they do not concern fixed-container variants. There are no known unresolved proof gaps; Challenge sorries are deliberate Comparator placeholders.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "", "alignment": false, "original": true, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [ "OpenAI Codex (independent statement and solution-surface audits)", "Anthropic Claude Sonnet (independent statement audit)" ], "notes": "Automated reviewers checked all 29 Comparator declarations for alignment between the Challenge, the Solution surface, and the narrative. No human peer review has been performed. Mechanical checking by Comparator, Lean, and NanoDa is separate from this editorial review." }, "canonical": { "repo": "tejstead/heilbronn-site", "directory": "proofs/palomar/heilbronn-convex" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-02-000012", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-02-000012", "commit": "143e09956fbbf68184cd60cfe445585a00e3b50f", "trust": "high", "theorems": 29, "date": "2026-09-02" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "teorth/IEANTN/IEANTN/Nodes/BKLNW/v2/formalization.yaml", "repo": "teorth/IEANTN", "path": "IEANTN/Nodes/BKLNW/v2/formalization.yaml", "directory": "IEANTN/Nodes/BKLNW/v2", "origins": [ "search" ], "url": "https://github.com/teorth/IEANTN/blob/HEAD/IEANTN/Nodes/BKLNW/v2/formalization.yaml", "version": "v0.4", "missing": [], "name": "BKLNW pipeline: Corollary 5.1 and the Table 8 tail", "description": "Same paper as BKLNW.v1. 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BKLNW.v1 is left unchanged.", "authors": [ "Taksh Kothari" ], "maintainers": [ "Taksh" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "Sharper bounds for the Chebyshev function theta(x)", "id": "https://doi.org/10.1090/mcom/3643", "authors": [ "Samuel Broadbent", "Habiba Kadiri", "Allysa Lumley", "Nathan Ng", "Kirsten Wilk" ], "type": "paper", "relationship": "formalizes", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11N05", "11Y70" ] }, "automation": { "methods": [ { "method": "manual", "models": [], "framework": "" } ], "strongest": "manual", "models": [], "spend": "", "notes": "" }, "scope": "", "sorry_count": null, "sorry_in_definitions": null, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "", "alignment": false, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Taksh" ], "notes": "" }, "canonical": { "repo": "teorth/ieantn", "directory": "IEANTN/Nodes/BKLNWNumerics/v1" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "teorth/IEANTN/IEANTN/Nodes/Brown1967/v1/formalization.yaml", "repo": "teorth/IEANTN", "path": "IEANTN/Nodes/Brown1967/v1/formalization.yaml", "directory": "IEANTN/Nodes/Brown1967/v1", "origins": [ "search" ], "url": "https://github.com/teorth/IEANTN/blob/HEAD/IEANTN/Nodes/Brown1967/v1/formalization.yaml", "version": "v0.4", "missing": [], "name": "Brown: the aliasing error in the sampling theorem", "description": "The aliasing bound Platt2017 uses as its Theorem 4.4. States nothing yet: the result worth stating is Brown's own Theorem 1, which is stronger than the form Platt uses, and encoding it correctly needs Fourier vocabulary the network does not otherwise have and a symmetric limit rather than a tsum. Both obstacles are recorded in the Lean file.", "authors": [ "Terence Tao" ], "maintainers": [ "Terence Tao" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "On the error in reconstructing a non-bandlimited function by means of the bandpass sampling theorem", "id": "J. Math. Anal. Appl. 18 (1967), 75-84", "authors": [ "J. L. 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What has not been checked is the Weiss abstract itself, which is not held." }, "canonical": { "repo": "teorth/ieantn", "directory": "IEANTN/Nodes/Brown1967/v1" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "teorth/IEANTN/IEANTN/Nodes/Buthe/v2/formalization.yaml", "repo": "teorth/IEANTN", "path": "IEANTN/Nodes/Buthe/v2/formalization.yaml", "directory": "IEANTN/Nodes/Buthe/v2", "origins": [ "search" ], "url": "https://github.com/teorth/IEANTN/blob/HEAD/IEANTN/Nodes/Buthe/v2/formalization.yaml", "version": "v0.4", "missing": [], "name": "Buthe's Lemma 3, the theta-to-pi transfer", "description": "The analytic transfer underlying Jan Buthe's analytic method for bounding psi(x): his Lemma 3, which carries a two-sided bound on x - theta(x) to one on li(x) - pi(x), stated for arbitrary parameters rather than at the paper's own. Two conclusions, matching the lemma's two parts -- a two-sided estimate valid above 10^7, and the implication that positivity of x - theta(x) on an interval forces positivity of li(x) - pi(x) there. Both import nothing, so a Lean proof is their whole justification. Buthe.v1 states the same paper's Theorem 2, two of whose six equations are instances of these.", "authors": [ "Terence Tao" ], "maintainers": [ "Terence Tao" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "An analytic method for bounding psi(x)", "id": "https://doi.org/10.1090/mcom/3264", "authors": [ "Jan Buthe" ], "type": "paper", "relationship": "formalizes", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11N05", "11M26", "11A41" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Claude Opus 5 (Anthropic)" ], "framework": "Claude Code" } ], "strongest": "agent", "models": [ "Claude Opus 5 (Anthropic)" ], "spend": "", "notes": "" }, "scope": "", "sorry_count": null, "sorry_in_definitions": null, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "", "alignment": false, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Terence Tao" ], "notes": "Both parts were transcribed from the published Math. Comp. version, Lemma 3 as printed. The transcription of the two-sided estimate has an independent consistency check: at C = 1.95, the lemma's middle term 2C/log x (1 + 5/log x) is 3.9/log x + 19.5/(log x)^2, which are exactly the constants printed in the paper's (1.9) -- so the lemma as transcribed does specialise to the equation the paper says it gives. 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That makes it the bridge between the network's computational nodes and its analytic ones, and it is one of the three inputs BKLNW uses to compute the eps(b) of its Table 8.", "authors": [ "Terence Tao" ], "maintainers": [ "Terence Tao" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "Estimating pi(x) and related functions under partial RH assumptions", "id": "arXiv:1410.7015v4; Math. 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The conditional shape -- quantified in T with the verification as a hypothesis -- is a deliberate choice and is what makes the node importable by any verification." }, "canonical": { "repo": "teorth/ieantn", "directory": "IEANTN/Nodes/Buthe2016/v1" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "teorth/IEANTN/IEANTN/Nodes/ButheNumerics/v1/formalization.yaml", "repo": "teorth/IEANTN", "path": "IEANTN/Nodes/ButheNumerics/v1/formalization.yaml", "directory": "IEANTN/Nodes/ButheNumerics/v1", "origins": [ "search" ], "url": "https://github.com/teorth/IEANTN/blob/HEAD/IEANTN/Nodes/ButheNumerics/v1/formalization.yaml", "version": "v0.4", "missing": [], "name": "Buthe's finite numerical checks", "description": "The auxiliary finite computations inside Jan Buthe's analytic method for bounding psi(x), separated from the analysis that consumes them. Same paper as Buthe.v1, different kind of claim. Buthe.v1 states the paper's Theorem 2, equations (1.5) to (1.10); this node holds two checks that Theorem 2's own proof uses and the paper does not display -- the sign of the constant A in its Lemma 3 at a = 1500, and equation (1.9) on the range below 10^7 where the paper checks directly rather than deriving. Stating them is what turns (1.9) from a bare assertion into a claim whose inputs are all written down.", "authors": [ "Terence Tao" ], "maintainers": [ "Terence Tao" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "An analytic method for bounding psi(x)", "id": "https://doi.org/10.1090/mcom/3264", "authors": [ "Jan Buthe" ], "type": "paper", "relationship": "formalizes", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11N05", "11Y70", "11M26" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Claude Opus 5 (Anthropic)" ], "framework": "Claude Code" } ], "strongest": "agent", "models": [ "Claude Opus 5 (Anthropic)" ], "spend": "", "notes": "" }, "scope": "", "sorry_count": null, "sorry_in_definitions": null, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "", "alignment": false, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Terence Tao" ], "notes": "Both conclusions were read from the published Math. Comp. version. The constant A and the shape of Lemma 3 were transcribed from the lemma as printed; the identification 3.9 = 2C and 19.5 = 10C with C = 1.95 from (1.6) is a consistency check on that transcription and it holds. The value of A at a = 1500 has NOT been computed independently -- the claim is that the paper's use of Lemma 3 requires A <= 0, not that this has been verified numerically here." }, "canonical": { "repo": "teorth/ieantn", "directory": "IEANTN/Nodes/ButheNumerics/v1" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "teorth/IEANTN/IEANTN/Nodes/CH2/v4/formalization.yaml", "repo": "teorth/IEANTN", "path": "IEANTN/Nodes/CH2/v4/formalization.yaml", "directory": "IEANTN/Nodes/CH2/v4", "origins": [ "search" ], "url": "https://github.com/teorth/IEANTN/blob/HEAD/IEANTN/Nodes/CH2/v4/formalization.yaml", "version": "v0.4", "missing": [], "name": "Chirre-Helfgott's contour shift, on L-shaped contours", "description": "Section 5 of Chirre-Helfgott: the identity moving an integral off the line Re s = 1 onto a contour running out to Re s = -infinity, collecting the poles in between. Two conclusions, one pole-free and one general. The paper states this for an arbitrary \"admissible contour\", a tameness condition rather than a shape constraint; this node fixes the class to L-shaped paths parameterized by a single complex number, which is entirely axis-parallel and so needs only Mathlib's rectangle Cauchy theorem. THESE STATEMENTS ARE TENTATIVE: they are a notational proposal whose confirmation is a Comparator run", "authors": [ "Terence Tao" ], "maintainers": [ "Terence Tao" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "Optimal bounds for sums of non-negative arithmetic functions", "id": "https://arxiv.org/abs/2512.15709", "authors": [ "Andres Chirre", "Harald Andres Helfgott" ], "type": "paper", "relationship": "adapts", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11M06", "30E20", "11N05" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Claude Opus 5 (Anthropic)" ], "framework": "Claude Code" } ], "strongest": "agent", "models": [ "Claude Opus 5 (Anthropic)" ], "spend": "", "notes": "" }, "scope": "", "sorry_count": null, "sorry_in_definitions": null, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "", "alignment": false, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Terence Tao" ], "notes": "THE ORIENTATION WAS CHECKED NUMERICALLY, which matters because a sign error here would make the conclusions FALSE rather than vacuous, and an earlier draft of the general statement did have the top-edge term with the wrong sign. Both identities were evaluated by quadrature on explicit examples: for the general form, f(s) = exp(2s)/(s - rho) with rho = 0.5 + 5i, T = 10, w = -3 + i, r = 0.3, where the two sides agree to 3e-9 and the circle integral reproduces 2 pi i exp(2 rho) to the same precision; for the pole-free form, three integrands including one reaching 1e7 in magnitude, agreeing to a relative 2e-9. That is evidence about the STATEMENT, not about any proof, and it is the reason to expect a Comparator run to succeed rather than to discover a sign. The junk-value trap was considered e" }, "canonical": { "repo": "teorth/ieantn", "directory": "IEANTN/Nodes/CH2/v4" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "teorth/IEANTN/IEANTN/Nodes/ChengGraham2004/v1/formalization.yaml", "repo": "teorth/IEANTN", "path": "IEANTN/Nodes/ChengGraham2004/v1/formalization.yaml", "directory": "IEANTN/Nodes/ChengGraham2004/v1", "origins": [ "search" ], "url": "https://github.com/teorth/IEANTN/blob/HEAD/IEANTN/Nodes/ChengGraham2004/v1/formalization.yaml", "version": "v0.4", "missing": [], "name": "Cheng-Graham: explicit estimates for the Riemann zeta function", "description": "The exponential-sum machinery underneath Hiary2016, and so underneath KLN and the FKS chain. States nothing yet, deliberately: the result Hiary depends on is Lemma 3, the van der Corput second-derivative test, which is the erroneous one, and transcribing it as printed would put a knowably wrong statement into the network.", "authors": [ "Terence Tao" ], "maintainers": [ "Terence Tao" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "Explicit estimates for the Riemann zeta function", "id": "Rocky Mountain J. Math. 34 (2004), no. 4, 1261-1280", "authors": [ "Yuanyou F. Cheng", "Sidney W. 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One conclusion. The node exists because Mathlib has Cauchy on a rectangle and invariance of a circle integral under a change of radius, but nothing bridging them: no residue, no Cauchy theorem for cycles, no homotopy invariance of contour integrals. That gap -- a rectangle with finitely many holes -- is what this states. It belongs to no paper, and it is deliberately not stated inside the node that first needs it, because any", "authors": [ "Terence Tao" ], "maintainers": [ "Terence Tao" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "The residue theorem", "id": "residue-theorem", "authors": [ "folklore" ], "type": "folklore", "relationship": "formalizes", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.CV" ], "msc2020": [ "30E20", "30A99" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Claude Opus 5 (Anthropic)" ], "framework": "Claude Code" } ], "strongest": "agent", "models": [ "Claude Opus 5 (Anthropic)" ], "spend": "", "notes": "" }, "scope": "", "sorry_count": null, "sorry_in_definitions": null, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "", "alignment": false, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Terence Tao" ], "notes": "The statement was designed against its first consumer and then checked numerically, on f(s) = exp(2s)/(s - rho) with rho = 0.5 + 5i over the rectangle [-40, 1] x [1, 10] and r = 0.3: the four-term boundary expression and the circle integral agree to 6e-9, and the companion pole-free rectangle [-3, 1] x [0, 1] has boundary 0 to 5e-11. That is evidence about the STATEMENT and about the orientation conventions, not about any proof. The junk-value question was considered and has a clean answer here, unlike in CH2.v4: holomorphy on the closed rectangle off P already forces continuity on every boundary edge and on every circle, hence integrability, so no separate integrability hypothesis is needed and none is hiding a vacuous case. It would be needed if the holomorphy hypothesis were ever weaken" }, "canonical": { "repo": "teorth/ieantn", "directory": "IEANTN/Nodes/ContourIntegration/v1" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "teorth/IEANTN/IEANTN/Nodes/CotangentSeries/v1/formalization.yaml", "repo": "teorth/IEANTN", "path": "IEANTN/Nodes/CotangentSeries/v1/formalization.yaml", "directory": "IEANTN/Nodes/CotangentSeries/v1", "origins": [ "search" ], "url": "https://github.com/teorth/IEANTN/blob/HEAD/IEANTN/Nodes/CotangentSeries/v1/formalization.yaml", "version": "v0.4", "missing": [], "name": "The cotangent expansion in terms of zeta values", "description": "The Taylor expansion pi cot(pi z) = 1/z - 2 sum_{n>=1} zeta(2n) z^{2n-1}, valid for 0 < |z| < 1, stated as a HasSum so that convergence is part of the claim. Mathlib supplies the two ends and not the middle. Trigonometric/Cotangent.lean has Euler's partial fractions, pi cot(pi z) = 1/z + sum (1/(z-n) + 1/(z+n)), which describes the POLES; HurwitzZetaValues has riemannZeta_two_mul_nat, giving zeta(2k) via Bernoulli numbers; and Bernoulli.lean has the generating function t/(e^t - 1). What is missing is the expansion at the origin, which is the form every argument wants that needs the COEFFICIENT", "authors": [ "Terence Tao" ], "maintainers": [ "Terence Tao" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "The Taylor expansion of the cotangent", "id": "cotangent-zeta-expansion", "authors": [ "folklore" ], "type": "folklore", "relationship": "formalizes", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.CA" ], "msc2020": [ "33B10", "11M06" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Claude Opus 5 (Anthropic)" ], "framework": "Claude Code" } ], "strongest": "agent", "models": [ "Claude Opus 5 (Anthropic)" ], "spend": "", "notes": "" }, "scope": "", "sorry_count": null, "sorry_in_definitions": null, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "", "alignment": false, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Terence Tao" ], "notes": "HasSum RATHER THAN AN EQUATION BETWEEN tsums, deliberately. A tsum of a non-summable family is 0, so an equation between two of them can hold vacuously rather than expressing convergence. HasSum carries convergence with it, which is what a consumer needs when it wants to differentiate or rearrange. THE INDEX IS SHIFTED so that n starts at 0 and the summand is 2 zeta(2n+2) z^{2n+1}. The paper's sum runs over n >= 1 with summand zeta(2n) z^{2n-1}; the two agree term by term at n and n+1. The shift avoids a subtraction in the exponent, which at n = 0 would be a natural subtraction truncating to 0 and would make the first term wrong. THE HYPOTHESES ARE BOTH NEEDED AND NEITHER IS COSMETIC. z /= 0 is where 1/z lives; ||z|| < 1 keeps z inside the disc of convergence, whose radius is set by the ne" }, "canonical": { "repo": "teorth/ieantn", "directory": "IEANTN/Nodes/CotangentSeries/v1" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "teorth/IEANTN/IEANTN/Nodes/DudekPlatt/v3/formalization.yaml", "repo": "teorth/IEANTN", "path": "IEANTN/Nodes/DudekPlatt/v3/formalization.yaml", "directory": "IEANTN/Nodes/DudekPlatt/v3", "origins": [ "search" ], "url": "https://github.com/teorth/IEANTN/blob/HEAD/IEANTN/Nodes/DudekPlatt/v3/formalization.yaml", "version": "v0.4", "missing": [], "name": "Dudek-Platt's criterion for Ramanujan's inequality, repaired", "description": "The criterion carrying a two-sided explicit estimate for pi(x) to Ramanujan's inequality pi(x)^2 < (e x / log x) pi(x/e), stated for arbitrary parameters. This is Dudek and Platt's Lemma 2.1 with a defect in its proof repaired: the paper bounds the term carrying its lower-bound constant m in the wrong direction whenever m is negative, which it always is in practice. The repair is a case split on the sign of m, and it is the difference between the two seventh-order lower-bound terms stated here. Imports nothing, so a Lean proof is its whole justification; the thresholds on DudekPlatt.v1 and Dud", "authors": [ "Terence Tao" ], "maintainers": [ "Terence Tao" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "On solving a curious inequality of Ramanujan", "id": "https://doi.org/10.1080/10586458.2014.990118", "authors": [ "Adrian W. Dudek", "David J. Platt" ], "type": "paper", "relationship": "adapts", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11N05", "11A41" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Claude Opus 5 (Anthropic)" ], "framework": "Claude Code" } ], "strongest": "agent", "models": [ "Claude Opus 5 (Anthropic)" ], "spend": "", "notes": "" }, "scope": "", "sorry_count": null, "sorry_in_definitions": null, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "", "alignment": false, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Terence Tao" ], "notes": "The statement was written against PrimeNumberTheoremAnd's `criterion`, `ε`, `ε'`, `εneg` and `εlower`, with names and shapes adapted to this repository's Vocabulary. Two independent checks on the transcription. First, the paper's own seventh-order lower-bound term reads 206 + m + 364/log x + 381/log^2 x + 238/log^3 x + 97/log^4 x + 30/log^5 x, which matches εPos here up to the 8/log^6 x tail that the PDF text layer truncates. Second, the threshold condition is a comparison against log x rather than against zero, and at the paper's parameters Mₐ = 3343.48, mₐ = -3103.33 the leading difference (72 + 2Mₐ) - (206 + mₐ) is about 9656 against a published threshold of exp(9657.8) -- so the condition is satisfied with very little room, which both corroborates the shape and explains why reading it " }, "canonical": { "repo": "teorth/ieantn", "directory": "IEANTN/Nodes/DudekPlatt/v3" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "teorth/IEANTN/IEANTN/Nodes/DudekPlattNumerics/v2/formalization.yaml", "repo": "teorth/IEANTN", "path": "IEANTN/Nodes/DudekPlattNumerics/v2/formalization.yaml", "directory": "IEANTN/Nodes/DudekPlattNumerics/v2", "origins": [ "search" ], "url": "https://github.com/teorth/IEANTN/blob/HEAD/IEANTN/Nodes/DudekPlattNumerics/v2/formalization.yaml", "version": "v0.4", "missing": [], "name": "The explicit pi estimate behind the exp(3915) threshold", "description": "The two-sided bound on pi(x) that DudekPlatt.v3's criterion needs in order to reach the exp(3915) threshold of DudekPlatt.v2, at PrimeNumberTheoremAnd's constants rather than the paper's. Same shape of claim as DudekPlattNumerics.v1 and much sharper: pi(x) lies within 1194 x/(log x)^6 and 1426 x/(log x)^6 of the fifth partial sum of li's asymptotic expansion, for x above exp(3914), where the paper's own constants are -3103.33 and 3343.48 above exp(9656.8). The improvement is the whole reason DudekPlatt.v2 states a better threshold than DudekPlatt.v1.", "authors": [ "Terence Tao" ], "maintainers": [ "Terence Tao" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "On solving a curious inequality of Ramanujan", "id": "https://doi.org/10.1080/10586458.2014.990118", "authors": [ "Adrian W. Dudek", "David J. 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The check that matters was done in Lean rather than assumed: with M = 1426, m = -1194 and log xₐ = 3914, DudekPlatt.v3's threshold condition eUpper M x - eNeg m xₐ x < log x holds at log x = 3915 by norm_num on exact rationals, clearing by 0.4989, and the gap decreases in log x while log x increases, so it holds above. That check was NOT redundant: the same condition FAILS at DudekPlattNumerics.v1's parameters at the paper's printed threshold. The constants themselves have not been recomputed." }, "canonical": { "repo": "teorth/ieantn", "directory": "IEANTN/Nodes/DudekPlattNumerics/v2" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "teorth/IEANTN/IEANTN/Nodes/Dusart2018/v1/formalization.yaml", "repo": "teorth/IEANTN", "path": "IEANTN/Nodes/Dusart2018/v1/formalization.yaml", "directory": "IEANTN/Nodes/Dusart2018/v1", "origins": [ "search" ], "url": "https://github.com/teorth/IEANTN/blob/HEAD/IEANTN/Nodes/Dusart2018/v1/formalization.yaml", "version": "v0.4", "missing": [], "name": "Dusart's explicit estimates of some functions over primes", "description": "Explicit bounds on the Chebyshev functions, the prime counting function, and the size of gaps between consecutive primes, from Dusart's 2018 paper in the Ramanujan Journal. This node currently exports only Proposition 5.4, the prime-gap estimate that downstream nodes in the network consume; the paper's other results should be added as further conclusions as they are needed.", "authors": [ "Terence Tao" ], "maintainers": [ "Terence Tao" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "Explicit estimates of some functions over primes", "id": "https://doi.org/10.1007/s11139-016-9839-4", "authors": [ "Pierre Dusart" ], "type": "paper", "relationship": "formalizes", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11N05", "11N56", "11Y70" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Claude Opus 5 (Anthropic)" ], "framework": "Claude Code" } ], "strongest": "agent", "models": [ "Claude Opus 5 (Anthropic)" ], "spend": "", "notes": "" }, "scope": "", "sorry_count": null, "sorry_in_definitions": null, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "", "alignment": false, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Terence Tao" ], "notes": "The statement was transcribed from the PNT+ formalization of the same proposition, which carries its own citation to the paper. It has not been checked against the printed paper directly." }, "canonical": { "repo": "teorth/ieantn", "directory": "IEANTN/Nodes/Dusart2018/v1" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "teorth/IEANTN/IEANTN/Nodes/FKBJ/v1/formalization.yaml", "repo": "teorth/IEANTN", "path": "IEANTN/Nodes/FKBJ/v1/formalization.yaml", "directory": "IEANTN/Nodes/FKBJ/v1", "origins": [ "search" ], "url": "https://github.com/teorth/IEANTN/blob/HEAD/IEANTN/Nodes/FKBJ/v1/formalization.yaml", "version": "v0.4", "missing": [], "name": "Franke-Kleinjung-Buthe-Jost: a practical analytic method for pi(x)", "description": "The computation of the zeros of zeta with imaginary part up to 10^11 that Buthe's analytic method consumes. Buthe.v1's Theorem 2 is the output of running that method on this data. The node states the part of the computation that is expressible as a proposition -- that every zero up to that height lies on the critical line -- and records in prose that the algorithm consumes the tabulated zeros themselves, which is data rather than a claim.", "authors": [ "Terence Tao" ], "maintainers": [ "Terence Tao" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "A practical analytic method for calculating pi(x)", "id": "Math. 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The natural first conclusion is the paper's headline admissible classical bound on `E_psi`, which is exactly the shape `FKS2`'s first pipeline consumes.", "authors": [ "Terence Tao" ], "maintainers": [ "Terence Tao" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "Sharper bounds for the Chebyshev function psi(x)", "id": "https://doi.org/10.1016/j.jmaa.2023.127426", "authors": [ "Andrew Fiori", "Habiba Kadiri", "Joshua Swidinsky" ], "type": "paper", "relationship": "formalizes", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11N05", "11M06", "11Y35" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Claude Opus 5 (Anthropic)" ], "framework": "Claude Code" } ], "strongest": "agent", "models": [ "Claude Opus 5 (Anthropic)" ], "spend": "", "notes": "" }, "scope": "", "sorry_count": null, "sorry_in_definitions": null, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "", "alignment": false, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Terence Tao" ], "notes": "Bibliographic details taken from blueprint/src/references.bib in PrimeNumberTheoremAnd rather than from the printed paper. The published JMAA version is now held in the library; verify them against it before this node is cited externally or used as the basis for a submission. That check would also settle the numbering discrepancy noted below." }, "canonical": { "repo": "teorth/ieantn", "directory": "IEANTN/Nodes/FKS/v1" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "teorth/IEANTN/IEANTN/Nodes/FKS2/v2/formalization.yaml", "repo": "teorth/IEANTN", "path": "IEANTN/Nodes/FKS2/v2/formalization.yaml", "directory": "IEANTN/Nodes/FKS2/v2", "origins": [ "search" ], "url": "https://github.com/teorth/IEANTN/blob/HEAD/IEANTN/Nodes/FKS2/v2/formalization.yaml", "version": "v0.4", "missing": [], "name": "FKS2's conversion pipelines", "description": "The two transformations underlying Fiori-Kadiri-Swidinsky's explicit prime number theorem bounds, stated for arbitrary admissible parameters rather than at the paper's named constants: Proposition 13 converting a classical bound on E_psi into one on E_theta, and Theorem 3 converting one on E_theta into one on E_pi. Both are conditional theorems importing nothing, so a Lean proof is their whole justification. FKS2.v1 states the same paper's corollaries, which are instances of these.", "authors": [ "Terence Tao" ], "maintainers": [ "Terence Tao" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "Sharper bounds for the error term in the prime number theorem", "id": "https://doi.org/10.1007/s40993-023-00454-w", "authors": [ "Andrew Fiori", "Habiba Kadiri", "Joshua Swidinsky" ], "type": "paper", "relationship": "formalizes", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11N05", "11M06", "11M26", "11N56" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Claude Opus 5 (Anthropic)" ], "framework": "Claude Code" } ], "strongest": "agent", "models": [ "Claude Opus 5 (Anthropic)" ], "spend": "", "notes": "" }, "scope": "", "sorry_count": null, "sorry_in_definitions": null, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "", "alignment": false, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Terence Tao" ], "notes": "Statements were first transcribed from PNT+'s formalization of the same paper, whose definitions of E_psi, E_theta, E_pi and the admissible bound were checked line by line against this repository's Vocabulary and agree, including the use of the offset Li rather than li. They have since been read against the paper itself. The published version -- Res. Number Theory 9 (2023), Paper No. 63, the version this node's sources[].id cites -- is now held alongside the arXiv preprint, and the two agree on every constant: extracting each decimal in document order gives 692 distinct values in the preprint and 693 in the published version, differing only in the DOI and one occurrence of the arXiv id. See docs/SOURCES.md." }, "canonical": { "repo": "teorth/ieantn", "directory": "IEANTN/Nodes/FKS2/v2" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "teorth/IEANTN/IEANTN/Nodes/FKS2Numerics/v1/formalization.yaml", "repo": "teorth/IEANTN", "path": "IEANTN/Nodes/FKS2Numerics/v1/formalization.yaml", "directory": "IEANTN/Nodes/FKS2Numerics/v1", "origins": [ "search" ], "url": "https://github.com/teorth/IEANTN/blob/HEAD/IEANTN/Nodes/FKS2Numerics/v1/formalization.yaml", "version": "v0.4", "missing": [], "name": "FKS2's finite numerical checks", "description": "The bounded-range arithmetic that FKS2's corollaries rest on, separated from the analysis that consumes it. 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They have since been read against the paper itself, and the published version -- Res. Number Theory 9 (2023), Paper No. 63 -- is now held alongside the arXiv preprint. The two agree on every constant; see docs/SOURCES.md." }, "canonical": { "repo": "teorth/ieantn", "directory": "IEANTN/Nodes/FKS2Numerics/v1" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "teorth/IEANTN/IEANTN/Nodes/GammaAsymptotics/v2/formalization.yaml", "repo": "teorth/IEANTN", "path": "IEANTN/Nodes/GammaAsymptotics/v2/formalization.yaml", "directory": "IEANTN/Nodes/GammaAsymptotics/v2", "origins": [ "search" ], "url": "https://github.com/teorth/IEANTN/blob/HEAD/IEANTN/Nodes/GammaAsymptotics/v2/formalization.yaml", "version": "v0.4", "missing": [], "name": "Growth of the Gamma function and its logarithmic derivative", "description": "The digamma asymptotic psi(w) = log w + O(1/|w|), restricted to a horizontal strip: for each height H a constant depending on H bounds ||psi w - log w|| by C/||w|| whenever Re w >= 1 and |Im w| <= H. GammaAsymptotics.v1 states the same estimate on the whole half-plane and is retained, unproved, as the harder target; removing the height restriction means proving v1. The split exists because the strip is where the elementary argument reaches. v1's route -- iterating the digamma recurrence out of a compact box -- controls Re w -> infinity at bounded height and says nothing as Im w -> infinity, so", "authors": [ "Terence Tao" ], "maintainers": [ "Terence Tao" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "The digamma asymptotic expansion", "id": "digamma-stirling", "authors": [ "folklore" ], "type": "folklore", "relationship": "formalizes", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.CA" ], "msc2020": [ "33B15", "41A60" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Claude Opus 5 (Anthropic)" ], "framework": "Claude Code" } ], "strongest": "agent", "models": [ "Claude Opus 5 (Anthropic)" ], "spend": "", "notes": "" }, "scope": "", "sorry_count": null, "sorry_in_definitions": null, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "", "alignment": false, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Terence Tao" ], "notes": "THE FORM WAS CHOSEN, NOT COPIED, and the choices are worth recording. The sharp statement is psi(w) = log w - 1/(2w) + O(1/w^2) on a sector |arg w| <= pi - delta. This node states the weaker O(1/|w|) on the half-plane Re w >= 1, for two reasons: the half-plane is what the applications use, and an explicit constant would invite the kind of numerology the node exists to avoid. A consumer wanting the -1/(2w) term, or the sector, should state that as a further conclusion here rather than sharpen this one -- they are different statements. What the applications actually consume is the corollary ||psi w|| <= ||log w|| + C, hence O(log|w|); that follows immediately and is not stated separately. Mathlib was checked rather than assumed, and RE-CHECKED after the bump, which moved it. On this reposito" }, "canonical": { "repo": "teorth/ieantn", "directory": "IEANTN/Nodes/GammaAsymptotics/v2" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "teorth/IEANTN/IEANTN/Nodes/Hiary2016/v1/formalization.yaml", "repo": "teorth/IEANTN", "path": "IEANTN/Nodes/Hiary2016/v1/formalization.yaml", "directory": "IEANTN/Nodes/Hiary2016/v1", "origins": [ "search" ], "url": "https://github.com/teorth/IEANTN/blob/HEAD/IEANTN/Nodes/Hiary2016/v1/formalization.yaml", "version": "v0.4", "missing": [], "name": "Hiary: an explicit van der Corput estimate for zeta on the critical line", "description": "The explicit subconvexity bound |zeta(1/2 + it)| <= a_1 t^(1/6) log t that KLN's Lemma 3.2 quotes and the FKS chain ultimately consumes. 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The density estimate itself, in a form that names its parameters, so that the nodes consuming it can instantiate rather than re-derive.", "authors": [ "Terence Tao" ], "maintainers": [ "Terence Tao" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "Explicit zero density for the Riemann zeta function", "id": "https://doi.org/10.1016/j.jmaa.2018.04.071", "authors": [ "Habiba Kadiri", "Allysa Lumley", "Nathan Ng" ], "type": "paper", "relationship": "formalizes", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11M06" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Claude Opus 5 (Anthropic)" ], "framework": "Claude Code" } ], "strongest": "agent", "models": [ "Claude Opus 5 (Anthropic)" ], "spend": "", "notes": "" }, "scope": "", "sorry_count": null, "sorry_in_definitions": null, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "", "alignment": false, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Terence Tao" ], "notes": "Lemma 3.2 read from the arXiv preprint 2101.12263v1 directly. The correction to its constant is FKS's, read from the published J. Math. Anal. Appl. version. The published KLN -- J. Math. Anal. 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The zero-free region, stated with its constant as a parameter rather than baked in, so that a later sharpening -- `MTY`, for instance -- can be swapped in by a downstream node without restating anything.", "authors": [ "Terence Tao" ], "maintainers": [ "Terence Tao" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "Une region explicite sans zeros pour la fonction zeta de Riemann", "id": "https://arxiv.org/abs/math/0401238", "authors": [ "Habiba Kadiri" ], "type": "paper", "relationship": "formalizes", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11M26" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Claude Opus 5 (Anthropic)" ], "framework": "Claude Code" } ], "strongest": "agent", "models": [ "Claude Opus 5 (Anthropic)" ], "spend": "", "notes": "" }, "scope": "", "sorry_count": null, "sorry_in_definitions": null, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "", "alignment": false, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Terence Tao" ], "notes": "Theoreme 1.1 was read first hand from the published Acta Arith. version, which the library holds, and its constant, threshold and closed form all agree with what is stated here. The bibliographic record -- originally taken from blueprint/src/references.bib in PrimeNumberTheoremAnd -- has since been checked against the paper itself: the article's own first page reads ACTA ARITHMETICA 117.4 (2005), and it runs 37 pages ending at 339, which is the recorded 303-339. Note that `sources[].id` still points at the arXiv preprint, which is the version that does NOT support this conclusion; the source note records why. 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Source attribution taken from the MathOverflow comment thread and flagged for confirmation above." }, "canonical": { "repo": "teorth/ieantn", "directory": "IEANTN/Nodes/Lcm/v2" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "teorth/IEANTN/IEANTN/Nodes/LowZeroes/v1/formalization.yaml", "repo": "teorth/IEANTN", "path": "IEANTN/Nodes/LowZeroes/v1/formalization.yaml", "directory": "IEANTN/Nodes/LowZeroes/v1", "origins": [ "search" ], "url": "https://github.com/teorth/IEANTN/blob/HEAD/IEANTN/Nodes/LowZeroes/v1/formalization.yaml", "version": "v0.4", "missing": [], "name": "Computed facts about the low-lying zeroes of zeta", "description": "Facts about the low zeroes of the Riemann zeta function that come from a computation over a list of them rather than from an argument. Currently two: the sum of reciprocal ordinates below height 2 x 10^4, which Chirre-Helfgott compute from David Platt's zero data and use in the proof of their Proposition on the contribution of the zeroes -- the step that produces C_T, the square-root constant in CH2.v1's corollaries; and a lower bound on the first nontrivial ordinate, which ZeroFreeHeight needs to push a classical region below exp(2/R).", "authors": [ "Terence Tao" ], "maintainers": [ "Terence Tao" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "Optimal bounds for sums of non-negative arithmetic functions", "id": "https://arxiv.org/abs/2512.15709", "authors": [ "Andres Chirre", "Harald Andres Helfgott" ], "type": "paper", "relationship": "formalizes", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11M26", "11Y35" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Claude Opus 5 (Anthropic)" ], "framework": "Claude Code" } ], "strongest": "agent", "models": [ "Claude Opus 5 (Anthropic)" ], "spend": "", "notes": "" }, "scope": "", "sorry_count": null, "sorry_in_definitions": null, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "", "alignment": false, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Terence Tao" ], "notes": "WHY THIS NODE EXISTS AT ALL is worth recording, because it was found by working backwards and contradicted an earlier plan. The route to C_T was thought to need pointwise control of CH2.v3's extremal approximant, and a sup bound was going to be added to that node. Reading section 7 from source showed otherwise: Lemma `cor:thonny` reduces the residue weights on the critical line to an explicit elementary function, F(z) = 1/pi - (1-z) cot(pi (1-z)), so no bound on the approximant is needed. What the derivation does need is zero counting -- now ZeroCount.v1 -- and this computation. Neither would have been supplied by the abandoned plan. The counting convention was checked: `Set.Ioc 0 t` matches CH2's `0 < Im rho <= t`, and it excludes the trivial zeroes, which sit at Im rho = 0, so no separat" }, "canonical": { "repo": "teorth/ieantn", "directory": "IEANTN/Nodes/LowZeroes/v1" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "teorth/IEANTN/IEANTN/Nodes/MT/v1/formalization.yaml", "repo": "teorth/IEANTN", "path": "IEANTN/Nodes/MT/v1/formalization.yaml", "directory": "IEANTN/Nodes/MT/v1", "origins": [ "search" ], "url": "https://github.com/teorth/IEANTN/blob/HEAD/IEANTN/Nodes/MT/v1/formalization.yaml", "version": "v0.4", "missing": [], "name": "Mossinghoff-Trudgian: nonnegative trigonometric polynomials and a zero-free region", "description": "The classical zero-free region for the Riemann zeta function with R = 5.573412, and the sharpened R = 5.5666305 of Section 6.1 that the FKS to FKS2 chain is calibrated to. 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What was previously flagged as a guess -- the relationship to Platt2017.v1 -- is now established by the matching height, the matching zero count, and this paper's citation of the other for its technique." }, "canonical": { "repo": "teorth/ieantn", "directory": "IEANTN/Nodes/Platt2015/v1" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "teorth/IEANTN/IEANTN/Nodes/Platt2017/v1/formalization.yaml", "repo": "teorth/IEANTN", "path": "IEANTN/Nodes/Platt2017/v1/formalization.yaml", "directory": "IEANTN/Nodes/Platt2017/v1", "origins": [ "search" ], "url": "https://github.com/teorth/IEANTN/blob/HEAD/IEANTN/Nodes/Platt2017/v1/formalization.yaml", "version": "v0.4", "missing": [], "name": "Platt: isolating some non-trivial zeros of zeta", "description": "The verification of the Riemann hypothesis to height 3.0610046e10 that KLN fixes as its H_0. 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Not read from the paper itself." }, "canonical": { "repo": "teorth/ieantn", "directory": "IEANTN/Nodes/PlattTrudgian/v1" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "teorth/IEANTN/IEANTN/Nodes/PlattTrudgian2021/v1/formalization.yaml", "repo": "teorth/IEANTN", "path": "IEANTN/Nodes/PlattTrudgian2021/v1/formalization.yaml", "directory": "IEANTN/Nodes/PlattTrudgian2021/v1", "origins": [ "search" ], "url": "https://github.com/teorth/IEANTN/blob/HEAD/IEANTN/Nodes/PlattTrudgian2021/v1/formalization.yaml", "version": "v0.4", "missing": [], "name": "Platt-Trudgian: the error term in the prime number theorem", "description": "An explicit form of a theorem of Pintz, giving a prime number theorem whose error term is roughly the square root of what was previously known. Its Theorem 1 tabulates bounds for E_psi, both in the classical asymptotic shape and as a plain numerical constant, and is where BKLNW's Table 8 gets its large-b half.", "authors": [ "Terence Tao" ], "maintainers": [ "Terence Tao" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "The error term in the prime number theorem", "id": "Math. Comp. 90 (2021), no. 328, 871-881", "authors": [ "David J. Platt", "Timothy S. Trudgian" ], "type": "paper", "relationship": "formalizes", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11N05", "11M26" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Claude Opus 5 (Anthropic)" ], "framework": "Claude Code" } ], "strongest": "agent", "models": [ "Claude Opus 5 (Anthropic)" ], "spend": "", "notes": "" }, "scope": "", "sorry_count": null, "sorry_in_definitions": null, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "", "alignment": false, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Terence Tao" ], "notes": "Worth checking that R = 5.573412 is used throughout rather than only in Theorem 1, and that Table 1's rows were read off correctly -- the X = 3000 row cross-checks against FKS's abstract, but the others were not independently confirmed." }, "canonical": { "repo": "teorth/ieantn", "directory": "IEANTN/Nodes/PlattTrudgian2021/v1" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "teorth/IEANTN/IEANTN/Nodes/PlattZeroSum/v1/formalization.yaml", "repo": "teorth/IEANTN", "path": "IEANTN/Nodes/PlattZeroSum/v1/formalization.yaml", "directory": "IEANTN/Nodes/PlattZeroSum/v1", "origins": [ "search" ], "url": "https://github.com/teorth/IEANTN/blob/HEAD/IEANTN/Nodes/PlattZeroSum/v1/formalization.yaml", "version": "v0.4", "missing": [], "name": "The reciprocal-ordinate sum of the zeroes of zeta up to height 20000", "description": "The bound 2 * sum over non-trivial zeroes rho of zeta with 0 < Im rho <= 20000, counted with multiplicity, of 1/Im rho, is at most 10.3194. The value 10.319317... is computed by Chirre and Helfgott from the ordinates of all zeroes up to that height, supplied by D. Platt, and used in their bound on the weighted sum over zeroes that gives the square-root term of their explicit psi estimate. It is a finite numerical fact that zero-counting estimates cannot replace at the precision needed, and that a Lean proof cannot reach, so it is recorded once as its own node.", "authors": [ "Terence Tao" ], "maintainers": [ "Terence Tao" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "Optimal bounds for sums of non-negative arithmetic functions", "id": "CH2", "authors": [ "Andres Chirre", "Harald Andres Helfgott" ], "type": "paper", "relationship": "formalizes", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11M06", "11M26", "11Y35" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Claude Opus 5 (Anthropic)" ], "framework": "Claude Code" } ], "strongest": "agent", "models": [ "Claude Opus 5 (Anthropic)" ], "spend": "", "notes": "" }, "scope": "", "sorry_count": null, "sorry_in_definitions": null, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "", "alignment": false, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Terence Tao" ], "notes": "Stated as an upper bound only, since prop:vihuela uses only that direction; a two-sided statement would add a claim nothing consumes. The rounding 10.319317... -> 10.3194 costs 8e-5 in a margin of about 2e-2." }, "canonical": { "repo": "teorth/ieantn", "directory": "IEANTN/Nodes/PlattZeroSum/v1" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "teorth/IEANTN/IEANTN/Nodes/PrimeInterval/v1/formalization.yaml", "repo": "teorth/IEANTN", "path": "IEANTN/Nodes/PrimeInterval/v1/formalization.yaml", "directory": "IEANTN/Nodes/PrimeInterval/v1", "origins": [ "search" ], "url": "https://github.com/teorth/IEANTN/blob/HEAD/IEANTN/Nodes/PrimeInterval/v1/formalization.yaml", "version": "v0.4", "missing": [], "name": "PrimeInterval.v1", "description": "A pipeline node carrying a bound on the normalised error term Eθ to the existence of a prime in a short interval (x, x + h]. States the θ characterisation of a prime in a short interval, the Eθ criterion it is proved through, and the numerical- and classical-bound routes that consume an Eθ bound however it was itself justified. Every conclusion imports nothing: all inputs arrive as hypotheses, so the node can sit downstream of a bound-producing node such as FKS2 without importing or reproving that node's proof.", "authors": [ "Sebastian Rodrigo" ], "maintainers": [ "Sebastian Rodrigo" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "PrimeInInterval.lean", "id": "PrimeNumberTheoremAnd/IEANTN/PrimeInInterval.lean", "authors": [ "Alex Kontorovich" ], "type": "other", "relationship": "adapts", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11N05", "11N56" ] }, "automation": { "methods": [ { "method": "ai-assisted", "models": [], "framework": "" } ], "strongest": "other", "models": [], "spend": "", "notes": "" }, "scope": "", "sorry_count": null, "sorry_in_definitions": null, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "", "alignment": false, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Terence Tao" ], "notes": "CONTRIBUTED FROM OUTSIDE, AND REVIEWED BEFORE THE VERIFICATION WAS APPROVED. This node arrived as PR #83 from Sebastian Rodrigo, the repository's first external node contribution; it was merged through #84, which carried the branch together with its receipt, because verify.yml records receipts by pushing to a branch on this repository and could not write to a fork. The reviewer named here is the maintainer, who approved the Comparator run -- which in this repository is the act of vouching for a branch, since the receipt job elaborates the branch's core Lean while holding a write token. WHAT THE REVIEW COVERED, so that a later reader is not left guessing. The four statements were read against their intended meanings; theta_characterisation is quantified over all reals with no positivity hyp" }, "canonical": { "repo": "teorth/ieantn", "directory": "IEANTN/Nodes/PrimeInterval/v1" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "teorth/IEANTN/IEANTN/Nodes/RosserSchoenfeld/v1/formalization.yaml", "repo": "teorth/IEANTN", "path": "IEANTN/Nodes/RosserSchoenfeld/v1/formalization.yaml", "directory": "IEANTN/Nodes/RosserSchoenfeld/v1", "origins": [ "search" ], "url": "https://github.com/teorth/IEANTN/blob/HEAD/IEANTN/Nodes/RosserSchoenfeld/v1/formalization.yaml", "version": "v0.4", "missing": [], "name": "Rosser-Schoenfeld: sharper bounds for the Chebyshev functions theta(x) and psi(x)", "description": "The classical source of explicit bounds on the Chebyshev functions, and the ancestor of every node in this chain. Still cited for the inequalities its successors have not superseded. Whichever inequalities downstream nodes actually cite, added one at a time. The paper contains a great many, and transcribing them speculatively would be work without a consumer.", "authors": [ "Terence Tao" ], "maintainers": [ "Terence Tao" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "Sharper bounds for the Chebyshev functions theta(x) and psi(x)", "id": "https://doi.org/10.2307/2005479", "authors": [ "J. Barkley Rosser", "Lowell Schoenfeld" ], "type": "paper", "relationship": "formalizes", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11N05" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Claude Opus 5 (Anthropic)" ], "framework": "Claude Code" } ], "strongest": "agent", "models": [ "Claude Opus 5 (Anthropic)" ], "spend": "", "notes": "" }, "scope": "", "sorry_count": null, "sorry_in_definitions": null, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "", "alignment": false, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Terence Tao" ], "notes": "Bibliographic details taken from blueprint/src/references.bib in PrimeNumberTheoremAnd rather than from the printed paper. Verify them against the paper before this node is cited externally or used as the basis for a submission." }, "canonical": { "repo": "teorth/ieantn", "directory": "IEANTN/Nodes/RosserSchoenfeld/v1" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "teorth/IEANTN/IEANTN/Nodes/Trudgian2011/v1/formalization.yaml", "repo": "teorth/IEANTN", "path": "IEANTN/Nodes/Trudgian2011/v1/formalization.yaml", "directory": "IEANTN/Nodes/Trudgian2011/v1", "origins": [ "search" ], "url": "https://github.com/teorth/IEANTN/blob/HEAD/IEANTN/Nodes/Trudgian2011/v1/formalization.yaml", "version": "v0.4", "missing": [], "name": "Trudgian: improvements to Turing's method", "description": "Turing's method confirms that a zero count is complete, and so that no zero has been missed in the range checked. Making it explicit needs a bound on the integral of S(t), and this paper's Theorem 2.2 is the bound modern verifications use. Platt2017 states it as its own Theorem 4.2.", "authors": [ "Terence Tao" ], "maintainers": [ "Terence Tao" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "Improvements to Turing's method", "id": "Math. Comp. 80 (2011), no. 276, 2259-2279", "authors": [ "Timothy S. Trudgian" ], "type": "paper", "relationship": "formalizes", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11M26", "11Y35" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Claude Opus 5 (Anthropic)" ], "framework": "Claude Code" } ], "strongest": "agent", "models": [ "Claude Opus 5 (Anthropic)" ], "spend": "", "notes": "" }, "scope": "", "sorry_count": null, "sorry_in_definitions": null, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "", "alignment": false, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Terence Tao" ], "notes": "The statement was read from the rendered page and cross-checked against Platt2017's restatement of it, which agrees exactly." }, "canonical": { "repo": "teorth/ieantn", "directory": "IEANTN/Nodes/Trudgian2011/v1" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "teorth/IEANTN/IEANTN/Nodes/Wedeniwski/v1/formalization.yaml", "repo": "teorth/IEANTN", "path": "IEANTN/Nodes/Wedeniwski/v1/formalization.yaml", "directory": "IEANTN/Nodes/Wedeniwski/v1", "origins": [ "search" ], "url": "https://github.com/teorth/IEANTN/blob/HEAD/IEANTN/Nodes/Wedeniwski/v1/formalization.yaml", "version": "v0.4", "missing": [], "name": "Wedeniwski: the ZetaGrid distributed verification", "description": "The verification height Kadiri2005 takes for T_0, and therefore the root of the chain Wedeniwski -> Kadiri2005 -> MT -> FKS -> FKS2. Its justification is `asserted` rather than `numerical`: this is a claim on a website, not a computation someone described and stood behind in a paper, and Platt and Trudgian record specific doubts about both its reliability and what height it actually claims.", "authors": [ "Terence Tao" ], "maintainers": [ "Terence Tao" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "The first 10 billion zeros of the Riemann zeta function are calculated and satisfy the Riemann hypothesis", "id": "http://www.zetagrid.net", "authors": [ "Sebastian Wedeniwski" ], "type": "web discussion", "relationship": "formalizes", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11M26", "11Y35" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Claude Opus 5 (Anthropic)" ], "framework": "Claude Code" } ], "strongest": "agent", "models": [ "Claude Opus 5 (Anthropic)" ], "spend": "", "notes": "" }, "scope": "", "sorry_count": null, "sorry_in_definitions": null, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "", "alignment": false, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Terence Tao" ], "notes": "Everything is at second hand; the website is not reachable. The height is Kadiri's quotation of it, and the doubts are Platt and Trudgian's." }, "canonical": { "repo": "teorth/ieantn", "directory": "IEANTN/Nodes/Wedeniwski/v1" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "teorth/IEANTN/IEANTN/Nodes/ZeroCount/v1/formalization.yaml", "repo": "teorth/IEANTN", "path": "IEANTN/Nodes/ZeroCount/v1/formalization.yaml", "directory": "IEANTN/Nodes/ZeroCount/v1", "origins": [ "search" ], "url": "https://github.com/teorth/IEANTN/blob/HEAD/IEANTN/Nodes/ZeroCount/v1/formalization.yaml", "version": "v0.4", "missing": [], "name": "Rosser's explicit Riemann-von Mangoldt error bound", "description": "Writing N(t) for the number of zeroes of the Riemann zeta function with 0 < Im rho <= t counted with multiplicity, and Q(t) = N(t) - ((t/2pi) log(t/2pi) - t/2pi + 7/8), the two claims are |Q(t)| < 1 for 0 < t <= 280 and |Q(t)| <= (1/5) log t + 2 for t >= 1. These are Lemma B.1 of Chirre-Helfgott, who derive them from Rosser 1941 and use them throughout their Appendix B. The node exists separately from the CH2 family because the result is not theirs and because an explicit zero count is exactly the kind of input several papers want. It is the missing input for the constant C_T in CH2.v1's corol", "authors": [ "Terence Tao" ], "maintainers": [ "Terence Tao" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "Optimal bounds for sums of non-negative arithmetic functions", "id": "https://arxiv.org/abs/2512.15709", "authors": [ "Andres Chirre", "Harald Andres Helfgott" ], "type": "paper", "relationship": "formalizes", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11M26", "11M06" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Claude Opus 5 (Anthropic)" ], "framework": "Claude Code" } ], "strongest": "agent", "models": [ "Claude Opus 5 (Anthropic)" ], "spend": "", "notes": "" }, "scope": "", "sorry_count": null, "sorry_in_definitions": null, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "", "alignment": false, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Terence Tao" ], "notes": "THE COUNTING CONVENTION WAS CHECKED RATHER THAN ASSUMED, and it is why this node does not use IEANTN.HasRvMBound, which would otherwise have been the obvious composable choice -- that predicate is literally |Q(T)| <= b1 log T + b2 log log T + b3, which is this statement at (1/5, 0, 2). It is built on `zetaN`, which counts Set.Ioo 0 T, whereas Rosser and CH2 count 0 < Im rho <= T, so the conclusions use the source's convention. A CORRECTION TO AN EARLIER VERSION OF THIS NOTE, which said the two bounds are inequivalent. They are not. That comparison was made at a SINGLE height, where they differ by m(t) and the lower bound appears to lose m(t). But the hypothesis holds at every nearby height, and the zeroes in a bounded region are finite, so there is an s < t with N_Ioc(s) = N_Ioo(t) exactly" }, "canonical": { "repo": "teorth/ieantn", "directory": "IEANTN/Nodes/ZeroCount/v1" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "teorth/IEANTN/IEANTN/Nodes/ZeroFreeHeight/v1/formalization.yaml", "repo": "teorth/IEANTN", "path": "IEANTN/Nodes/ZeroFreeHeight/v1/formalization.yaml", "directory": "IEANTN/Nodes/ZeroFreeHeight/v1", "origins": [ "search" ], "url": "https://github.com/teorth/IEANTN/blob/HEAD/IEANTN/Nodes/ZeroFreeHeight/v1/formalization.yaml", "version": "v0.4", "missing": [], "name": "Lowering the height of a classical zero-free region", "description": "A classical zero-free region is stated above some height -- 2 in Kadiri, Mossinghoff-Trudgian and Mossinghoff-Trudgian-Yang, 3 in the Korobov-Vinogradov theorem of the last, 21 in Rosser-Schoenfeld. This node supplies the step that lowers that height, given any numerical verification of the Riemann hypothesis reaching at least as far: below the old height a zero would have to sit on the critical line or right of Re s = 1, and the region excludes both. It is used without comment throughout the literature, which is what makes it folklore rather than a citation.", "authors": [ "Terence Tao" ], "maintainers": [ "Terence Tao" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "Lowering the height threshold of a classical zero-free region", "id": "unattributed; used without comment in the explicit zero-free region literature", "authors": [ "Terence Tao" ], "type": "folklore", "relationship": "formalizes", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11M26", "11M06" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Claude Opus 5 (Anthropic)" ], "framework": "Claude Code" } ], "strongest": "agent", "models": [ "Claude Opus 5 (Anthropic)" ], "spend": "", "notes": "" }, "scope": "", "sorry_count": null, "sorry_in_definitions": null, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "", "alignment": false, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Terence Tao" ], "notes": "The statement is short and the proof is four cases. What deserves scrutiny is the hypothesis exp (2 / R) < t0 -- whether it is the weakest that works, and whether the exp form really closes the junk-value hole the log form leaves open." }, "canonical": { "repo": "teorth/ieantn", "directory": "IEANTN/Nodes/ZeroFreeHeight/v1" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "teorth/IEANTN/IEANTN/Nodes/ZetaHadamard/v1/formalization.yaml", "repo": "teorth/IEANTN", "path": "IEANTN/Nodes/ZetaHadamard/v1/formalization.yaml", "directory": "IEANTN/Nodes/ZetaHadamard/v1", "origins": [ "search" ], "url": "https://github.com/teorth/IEANTN/blob/HEAD/IEANTN/Nodes/ZetaHadamard/v1/formalization.yaml", "version": "v0.4", "missing": [], "name": "The partial-fraction expansion of the logarithmic derivative of zeta", "description": "The identity zeta'/zeta(s) = sum over non-trivial zeroes rho of (1/(s-rho) + 1/rho) - psi(s/2+1)/2 - 1/(s-1) + C, which comes from the Hadamard factorisation of xi and is Corollary 10.14 of Montgomery and Vaughan. It is the only route from the zeroes of zeta to the size of zeta'/zeta inside the critical strip, and every explicit estimate of the shape \"zeta'/zeta(s) is its nearby zeroes plus O(log t)\" -- Titchmarsh Theorem 9.6(A) and its explicit descendants -- starts from it. Mathlib has the functional equation, the Dirichlet series and the discreteness of the zero set, but not the Hadamard fa", "authors": [ "Terence Tao" ], "maintainers": [ "Terence Tao" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "Multiplicative Number Theory I: Classical Theory", "id": "MontgomeryVaughan2007", "authors": [ "Hugh L. Montgomery", "Robert C. Vaughan" ], "type": "book", "relationship": "formalizes", "endorsement": "" }, { "title": "Multiplicative Number Theory", "id": "Davenport2000", "authors": [ "Harold Davenport" ], "type": "book", "relationship": "background", "endorsement": "" }, { "title": "Optimal bounds for sums of non-negative arithmetic functions", "id": "CH2", "authors": [ "Andres Chirre", "Harald Andres Helfgott" ], "type": "paper", "relationship": "background", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11M06", "11M26" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Claude Opus 5 (Anthropic)" ], "framework": "Claude Code" } ], "strongest": "agent", "models": [ "Claude Opus 5 (Anthropic)" ], "spend": "", "notes": "" }, "scope": "", "sorry_count": null, "sorry_in_definitions": null, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "", "alignment": false, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Terence Tao" ], "notes": "THE UNPAIRED STATEMENT WOULD HAVE BEEN VACUOUS, NOT WRONG, WHICH IS WORSE. The textbook form sums (1/(s-rho) + 1/rho) over the non-trivial zeroes in order of |Im rho|; that family is not absolutely summable, and Lean's tsum of a non-summable family is 0, so a conclusion built from IEANTN.zetaZeroesSum in the obvious way would typecheck and assert nothing. The Vocabulary docstring on zetaZeroesSum flags exactly this trap and names the fix -- state a regularised form -- and that is what was done. HasSum was chosen over tsum for the same reason: it carries summability as part of the claim. THE HYPOTHESES ARE FOUR, AND TWO OF THEM DO DOUBLE DUTY. s /= 1 and w /= 1 keep the 1/(s-1) terms finite and stay off the pole of zeta. zeta(s) /= 0 and zeta(w) /= 0 make the logarithmic derivatives finite," }, "canonical": { "repo": "teorth/ieantn", "directory": "IEANTN/Nodes/ZetaHadamard/v1" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "teorth/IEANTN/IEANTN/Nodes/ZetaLogDeriv/v1/formalization.yaml", "repo": "teorth/IEANTN", "path": "IEANTN/Nodes/ZetaLogDeriv/v1/formalization.yaml", "directory": "IEANTN/Nodes/ZetaLogDeriv/v1", "origins": [ "search" ], "url": "https://github.com/teorth/IEANTN/blob/HEAD/IEANTN/Nodes/ZetaLogDeriv/v1/formalization.yaml", "version": "v0.4", "missing": [], "name": "The functional equation for the logarithmic derivative of zeta", "description": "Identities and estimates for the logarithmic derivative of the Riemann zeta function, stated once so several nodes can import them rather than each carrying a copy. At present one conclusion: the functional equation, differentiated. Mathlib supplies the two ends of this subject without the middle -- it has the functional equation for zeta itself (riemannZeta_one_sub) and the Dirichlet series -zeta'/zeta(s) = sum Lambda(n) n^{-s} on Re s > 1 (LSeries_vonMangoldt_eq_deriv_riemannZeta_div, with LSeriesSummable_vonMangoldt), but not the functional equation in logarithmic-derivative form, which is ", "authors": [ "Terence Tao" ], "maintainers": [ "Terence Tao" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "The logarithmic derivative of the functional equation", "id": "zeta-logderiv-functional-equation", "authors": [ "folklore" ], "type": "folklore", "relationship": "formalizes", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11M06" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Claude Opus 5 (Anthropic)" ], "framework": "Claude Code" } ], "strongest": "agent", "models": [ "Claude Opus 5 (Anthropic)" ], "spend": "", "notes": "" }, "scope": "", "sorry_count": null, "sorry_in_definitions": null, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "", "alignment": false, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Terence Tao" ], "notes": "THE ORIENTATION WAS CHOSEN, NOT COPIED. The textbook form is zeta'/zeta(s) = log 2pi + (pi/2) cot(pi s/2) - psi(1-s) - zeta'/zeta(1-s), which comes from the sin / Gamma(1-s) form of the functional equation. That form is unavailable as stated: MATHLIB HAS NO Complex.cot -- checked, not assumed -- and its riemannZeta_one_sub is the other way round, giving zeta(1-s) in terms of zeta(s) with a cos(pi s/2) and a Gamma(s). Differentiating Mathlib's own form yields the equivalent identity with tan and psi(s) in place of cot and psi(1-s), and that is what this node states. The two are the same identity; a consumer who wants the cot form should state it as a further conclusion rather than reshape this one. CHECKED NUMERICALLY BEFORE BEING WRITTEN, to 30 digits at five points, including s = -3 + 2i " }, "canonical": { "repo": "teorth/ieantn", "directory": "IEANTN/Nodes/ZetaLogDeriv/v1" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "teorth/IEANTN/IEANTN/Nodes/ZetaLogDerivValues/v1/formalization.yaml", "repo": "teorth/IEANTN", "path": "IEANTN/Nodes/ZetaLogDerivValues/v1/formalization.yaml", "directory": "IEANTN/Nodes/ZetaLogDerivValues/v1", "origins": [ "search" ], "url": "https://github.com/teorth/IEANTN/blob/HEAD/IEANTN/Nodes/ZetaLogDerivValues/v1/formalization.yaml", "version": "v0.4", "missing": [], "name": "Numerical values of the logarithmic derivative of zeta", "description": "Values of zeta'/zeta at real points, and the sign pattern of its Laurent coefficients at s = 1. Mathlib has zeta'/zeta and its Dirichlet series but no numerical value of either, and every explicit contour argument needs some: section 8 of Chirre-Helfgott alone reaches for zeta'/zeta(2), zeta'/zeta(3/2), zeta'/zeta(-1) and (zeta'/zeta)'(-1) in five separate places, together with the fact that the Laurent coefficients of -zeta'/zeta at 1 alternate in sign. None of these is specific to a paper: each value is a sum over Lambda(n) n^{-t} or a coefficient of a fixed expansion, and any explicit estim", "authors": [ "Terence Tao" ], "maintainers": [ "Terence Tao" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "Optimal bounds for sums of non-negative arithmetic functions", "id": "https://arxiv.org/abs/2512.15709", "authors": [ "Andres Chirre", "Harald Andres Helfgott" ], "type": "paper", "relationship": "formalizes", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11M06", "11Y60" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Claude Opus 5 (Anthropic)" ], "framework": "Claude Code" } ], "strongest": "agent", "models": [ "Claude Opus 5 (Anthropic)" ], "spend": "", "notes": "" }, "scope": "", "sorry_count": null, "sorry_in_definitions": null, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "", "alignment": false, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Terence Tao" ], "notes": "THE TOLERANCE FORM IS DELIBERATE. Each value is stated as `exists c, |c - v| <= margin 0 * 1e-6 and zeta'/zeta(p) = c` rather than as a two-sided bracket. A bracket would need the margin applied in opposite ways on its two sides depending on the signs of the endpoints, since raising a margin must always widen; the tolerance form widens whatever the sign of v, which is the property the site exists to have. It also carries \"this quantity is real\" with it, which a bound on the norm would not, and consumers need the signs: zeta'/zeta(-1) is positive while the other three values are negative. 1e-6 IS FAR WIDER THAN THE COMPUTATION AND FAR TIGHTER THAN ANY CONSUMER. Every one of the four values is within 4e-7 of the stated centre. WHAT IS NOT HERE, deliberately. The monotonicity |zeta'/zeta(t)| " }, "canonical": { "repo": "teorth/ieantn", "directory": "IEANTN/Nodes/ZetaLogDerivValues/v1" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "teorth/IEANTN/IEANTN/Nodes/ZetaZeroes/v1/formalization.yaml", "repo": "teorth/IEANTN", "path": "IEANTN/Nodes/ZetaZeroes/v1/formalization.yaml", "directory": "IEANTN/Nodes/ZetaZeroes/v1", "origins": [ "search" ], "url": "https://github.com/teorth/IEANTN/blob/HEAD/IEANTN/Nodes/ZetaZeroes/v1/formalization.yaml", "version": "v0.4", "missing": [], "name": "Location and finiteness of the zeroes of the Riemann zeta function", "description": "Six unconditional facts about where the zeroes of zeta are and how many there are, none of which Mathlib has. The trivial zeroes are the only ones in the left half-plane; a zero off the real axis lies in the closed critical strip; a compact set missing s = 1 contains finitely many zeroes; every unit interval above the origin contains a height free of zero ordinates; and zeta equals riemannZeta_1 divided by s - 1, so the two vanish at exactly the same points away from s = 1. These are the facts every contour argument in explicit analytic number theory reaches for. A contour placing vertical col", "authors": [ "Terence Tao" ], "maintainers": [ "Terence Tao" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "Location of the zeroes of the Riemann zeta function", "id": "zeta-zero-location", "authors": [ "folklore" ], "type": "folklore", "relationship": "formalizes", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11M06", "11M26" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Claude Opus 5 (Anthropic)" ], "framework": "Claude Code" } ], "strongest": "agent", "models": [ "Claude Opus 5 (Anthropic)" ], "spend": "", "notes": "" }, "scope": "", "sorry_count": null, "sorry_in_definitions": null, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "", "alignment": false, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Terence Tao" ], "notes": "THE SCOPE WAS CHOSEN BY WHAT A CONSUMER WOULD IMPORT, not by what was available. Two facts that might have been included were left out deliberately. THE RESIDUES OF zeta'/zeta ARE NOT HERE, and their absence is the sharpest decision in this node. The residue at a zero is minus the multiplicity and at s = 1 it is 1; those are what a residue calculation actually consumes, and they belong beside these statements. They are absent because MATHLIB HAS NO residue AT ALL -- checked, not assumed -- and the one available in PrimeNumberTheoremAnd's port is labelled by its own authors a placeholder valid only for simple poles. A conclusion resting on a placeholder attests to nothing. When they are added they should be stated as limits, (s - rho) * (-zeta'/zeta)(s) -> -m as s -> rho, which is Mathlib-n" }, "canonical": { "repo": "teorth/ieantn", "directory": "IEANTN/Nodes/ZetaZeroes/v1" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "teorth/pfr/formalization.yaml", "repo": "teorth/pfr", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/teorth/pfr/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Marton's conjecture (the polynomial Freiman-Ruzsa conjecture)", "description": "A Lean formalization of Marton's conjecture, widely known as the polynomial Freiman-Ruzsa conjecture, together with its principal consequences. If A is a non-empty subset of an abelian group of exponent 2 with |A + A| <= K|A|, then A is covered by fewer than 2K^12 cosets of a subgroup of cardinality at most |A|; the same conclusion holds with 2K^9 cosets. For an abelian group in which mx = 0 for every x, with m >= 2, the covering number is m * K^(256m^3 + 1). Three consequences are also recorded: a finite set of small doubling in Z^D has a subset of at least K^(-34) of its size whose affine di", "authors": [ "Aaron Anderson", "Mantas Bakšys", "Jonas Bayer", "Mauricio Collares", "Rémy Degenne", "Yaël Dillies", "Ben Eltschig", "Sébastien Gouëzel", "Kalle Kytölä", "Rob Lewis", "Paul Lezeau", "Lorenzo Luccioli", "Heather Macbeth", "Patrick Massot", "Arend Mellendijk", "Kyle Miller", "Pietro Monticone", "Kim Morrison", "Oliver Nash", "Utensil Song", "Terence Tao", "Floris van Doorn", "Sky Wilshaw", "Lawrence Wu" ], "maintainers": [ "Terence Tao", "Yaël Dillies" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "On a conjecture of Marton", "id": "arXiv:2311.05762", "authors": [ "W. T. Gowers", "Ben Green", "Freddie Manners", "Terence Tao" ], "type": "preprint", "relationship": "formalizes", "endorsement": "" }, { "title": "Marton's conjecture in abelian groups with bounded torsion", "id": "arXiv:2404.02244", "authors": [ "W. T. Gowers", "Ben Green", "Freddie Manners", "Terence Tao" ], "type": "preprint", "relationship": "formalizes", "endorsement": "" }, { "title": "Improved Exponent for Marton's Conjecture in $\\mathbb{F}_2^n$", "id": "arXiv:2404.09639", "authors": [ "Jyun-Jie Liao" ], "type": "preprint", "relationship": "formalizes", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.CO", "math.NT" ], "msc2020": [ "11B30", "11P70", "20K01", "94A17" ] }, "automation": { "methods": [ { "method": "manual", "models": [], "framework": "" }, { "method": "agent", "models": [ "Claude Opus 5 (Anthropic)" ], "framework": "Claude Code" } ], "strongest": "agent", "models": [ "Claude Opus 5 (Anthropic)" ], "spend": "", "notes": "" }, "scope": "Complete, with no unproved step, for each of the six compared theorems. Three families of results are formalized but are not among the compared declarations, because Palomar requires the Challenge module to import only Lean core and Mathlib, and Mathlib has no Shannon entropy or entropic Ruzsa distance: the entropy form of the conjecture in characteristic 2 (Theorem 1.8 of the arXiv:2311.05762 entry below, formalized as entropic_PFR_conjecture and entropic_PFR_conjecture'), the entropy form in the bounded torsion case (formalized as dist_of_X_U_H_le), and the Kullback-Leibler and rho-functiona", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "Marton.pfr_conjecture", "file": "PFRPalomar/Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Marton.pfr_conjecture_nine", "file": "PFRPalomar/Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Marton.torsion_pfr_conjecture", "file": "PFRPalomar/Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Marton.weak_pfr_int", "file": "PFRPalomar/Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Marton.homomorphism_pfr", "file": "PFRPalomar/Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Marton.approx_hom_pfr", "file": "PFRPalomar/Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "Faithful to the sources, and in three respects stronger than them. First, the two characteristic-2 statements and the bounded-torsion statement are proved for an arbitrary abelian group in which every element satisfies 2x = 0 (respectively mx = 0), with A merely finite, rather than for a finite group; the papers state them for F_2^n and for a group of torsion m. The bounded-torsion generalization is not present in the library itself, whose torsion_PFR assumes a finite ambient group; it is obtained in PFRPalomar/Solution.lean by passing to the span of A over ZMod m, which is finite because A is. Second, every constant is explicit where the papers leave one unspecified: Theorem 1.3 of arXiv:2311.05762 asserts the existence of absolute constants C_1 and C_2, and Marton.weak_pfr_int realizes C_1 = 68 and C_2 = 80/log 2; Corollary 1.4 and Corollary 1.5 of the same paper assert bounds O(K^{C_3}) and >> K^{-C_4}, and the corresponding statements here give |S|^10 and (|G|/(2^144 K^122) - 1)/2; Theorem 1.1 of arXiv:2404.02244 asserts (2K)^{O(m^3)} and Marton.torsion_pfr_conjecture gives m K^{256m^3+1}. Third, Marton.pfr_conjecture_nine implies Marton.pfr_conjecture, since a non-empty A forc", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Terence Tao" ], "notes": "The development in PFR/ was produced in the open: contributions arrived as pull requests reviewed by project participants, were tracked against a blueprint whose dependency graph is published, and were discussed on a public Zulip stream. That is ordinary open-source project review, not referee review of the formalization, and no external body has certified it. The Palomar files and the correspondence between the compared statements and the three sources were checked by the responsible maintainer. Every step is checked by Lean." }, "canonical": { "repo": "teorth/pfr", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-06-000006", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-06-000006", "commit": "3d7898164ebff70a809dce618f9082a7b39e7850", "trust": "high", "theorems": 6, "date": "2026-09-06" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "tetsuo-jp/ddg-lean/formalization.yaml", "repo": "tetsuo-jp/ddg-lean", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/tetsuo-jp/ddg-lean/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Discrete Gauss–Bonnet for closed triangulated surfaces", "description": "The combinatorial form of the discrete Gauss–Bonnet theorem: for a finite closed triangulated surface carrying angle data, the sum over vertices of the angle defect equals 2*pi times the Euler characteristic #V - #E + #F. The relation 3*#F = 2*#E is not assumed; it is proved by double counting from \"every face has three vertices\" and \"every edge lies in exactly two faces\". The tetrahedron is derived as an instance: four triangular faces force six edges, and the total defect is 4*pi. The development isolates how little is needed for the theorem — beyond \"the angles of a face sum to pi\", everyth", "authors": [ "Tetsuo Yokoyama" ], "maintainers": [ "Tetsuo Yokoyama" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Critical Points and Curvature for Embedded Polyhedral Surfaces", "id": "doi:10.1080/00029890.1970.11992523", "authors": [ "Thomas Banchoff" ], "type": "article", "relationship": "independently-proves", "endorsement": "not-contacted" }, { "title": "Descartes on Polyhedra: A Study of the De Solidorum Elementis", "id": "doi:10.1007/978-1-4612-5759-2", "authors": [ "Pasquale Joseph Federico" ], "type": "book", "relationship": "background", "endorsement": "n/a" }, { "title": "Discrete Differential Geometry: Integrable Structure", "id": "doi:10.1090/gsm/098", "authors": [ "Alexander I. Bobenko", "Yuri B. Suris" ], "type": "book", "relationship": "background", "endorsement": "n/a" } ], "related": [], "classification": { "arxiv": [ "math.MG", "math.CO" ], "msc2020": [ "53A70", "52B70", "52B05", "05C10" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Claude Opus 5 (Anthropic)" ], "framework": "Claude Code" } ], "strongest": "agent", "models": [ "Claude Opus 5 (Anthropic)" ], "spend": "subscription-based", "notes": "Human-directed, machine-written. Every proof step is checked by Lean. scripts/audit.sh enforces five conditions and fails closed: the oleans are produced; every claimed theorem depends on exactly propext, Classical.choice and Quot.sound; Solution.lean contains no sorry, native_decide, unsafe or project-defined axiom and Challenge.lean contains exactly four deliberate holes; the four claimed theorems appear in both modules; and the mutation test passes. The mutation test is the part worth flagging to a reviewer. A correct proof of a badly stated theorem passes every other check, so scripts/muta" }, "scope": "Formalized in full generality for the structure defined in Challenge.lean: the total angle defect equals 2*pi times #V - #E + #F, the double-counting identity 3*#F = 2*#E, and the two tetrahedron instances (#E = 6 and total defect 4*pi). Not formalized: any connection to Euclidean geometry or to topology. The angles are abstract real data constrained only by summing to pi on each face and vanishing off the face; they are not derived from edge lengths or from an embedding, and they are not required to be positive. The Euler characteristic is defined here as #V - #E + #F from the data of the str", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The cited source proves Gauss–Bonnet for polyhedral surfaces embedded in Euclidean space; this development proves only the combinatorial identity, with angles as abstract data and the Euler characteristic as a defined quantity rather than a topological invariant. A reader wanting \"the angle defect of a polyhedron in R^3\" obtains it from this statement only after supplying the geometric input separately. No other divergence is known.", "alignment": true, "original": true, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Tetsuo Yokoyama" ], "notes": "No external or independent review has been performed, and no mathematician other than the author has read the Lean source. The statements in Challenge.lean were read for fidelity by the author; the mutation test described above is the mechanical check that they are not vacuous. Every proof step is checked by Lean." }, "canonical": { "repo": "tetsuo-jp/ddg-lean", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "thonmay/thesis-lean/formalization.yaml", "repo": "thonmay/thesis-lean", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/thonmay/thesis-lean/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Dynamic MCS Ordering Maintenance", "description": "A fully-dynamic Maximum-Cardinality-Search (MCS) ordering algorithm that, on a single-edge insertion or deletion into a loop-free undirected graph, scans the affected window between the two endpoints for the first step at which the current ordering's choice stops being a legal MCS pick, and only then regreedies the suffix (probe-then-repair). Correctness - every order produced stays a valid MCS ordering of the updated graph, and in fact equals the from-scratch canonical ordering - is machine-checked in Lean 4 / Mathlib, together with locality (only positions inside the window can change), an e", "authors": [ "Md Thoriqul Islam Thonmay", "Gregory Morse" ], "maintainers": [ "Md Thoriqul Islam Thonmay" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Dynamic Maximum-Cardinality-Search ordering maintenance under single-edge insertions and deletions (probe-then-repair algorithm)", "id": "", "authors": [ "Md Thoriqul Islam Thonmay" ], "type": "original-proof", "relationship": "other", "endorsement": "" }, { "title": "Algorithmic Aspects of Vertex Elimination on Graphs", "id": "", "authors": [ "Donald J. Rose", "Robert Endre Tarjan", "George S. Lueker" ], "type": "", "relationship": "background", "endorsement": "" }, { "title": "Simple Linear-Time Algorithms to Test Chordality of Graphs, Test Acyclicity of Hypergraphs, and Selectively Reduce Acyclic Hypergraphs", "id": "", "authors": [ "Robert Endre Tarjan", "Mihalis Yannakakis" ], "type": "", "relationship": "background", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "cs.DS", "math.CO" ], "msc2020": [ "05C85", "68Q25" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "deepseek-ai/DeepSeek-V4-Flash-0731" ], "framework": "enlcode agent harness over Lean 4 / lean4-repl / lean-lsp" }, { "method": "agent", "models": [ "zai-org/GLM-5.2" ], "framework": "enlcode agent harness over Lean 4 / lean4-repl / lake build" } ], "strongest": "agent", "models": [ "deepseek-ai/DeepSeek-V4-Flash-0731", "zai-org/GLM-5.2" ], "spend": "not tracked", "notes": "The proof development was carried out interactively with AI assistance (the agent harnesses above) and manually checked with lake build and leanchecker on every stage. The palette of transport / bridge lemmas was written by hand and each intermediate lemma verified independently." }, "scope": "Formalized for any finite n: the abstract UGraph (loop-free, symmetric finite graph on Fin n), the MCS legality predicate IsMCSNext (max-cardinal unchosen vertex, lowest-index tie-break) and its tie-break-free variant IsMCSOrderingAny, valid-MCS-ordering, the probe-then-repair init/insert/delete updates, and the correctness theorems init_valid / insert_update_valid / delete_update_valid together with greedySuffix_full and update_from_prefix. The canonical ordering is unique (IsMCSOrdering_unique) and each update returns exactly the from-scratch ordering (insert/delete_update_eq_initOrder). Loc", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The MCS tie-breaking rule is fixed to lowest vertex index; the standard literature statement allows an arbitrary (but fixed) tie-break, so this is equivalent and explicit here. The stability results are stated on the tie-break-free predicate IsMCSOrderingAny (any legal pick at each step), which is the natural setting for the common-ordering question; the uniqueness results use the canonical IsMCSOrdering. The cost model is a unit-cost query model: each legality check and each pick step counts as one unit, measured by instrumented functions (execInsertUpdateC / execDeleteUpdateC) whose first projection equals the plain update; this is an operational count along the real executable control flow, not a bit-level or word-RAM running time. The deletion probe performs exactly one legality check (exec_delete_cost_of_no_break proves the cost is 1 when no regreedy is needed). The regreedy step (greedySuffix) is noncomputable (Classical.choice) in the abstract model; the executable layer (execRegreedyFrom via bestPick) is computable and its refinement to the abstract regreedy is proved (regreedy_eq_greedySuffix), so the cost count tracks the computable implementation.", "alignment": true, "original": true, "review": { "status": "author-verified", "bucket": "author-verified", "reviewers": [ "Md Thoriqul Islam Thonmay", "Gregory Morse" ], "notes": "Every headline theorem statement in Solution.lean is textually identical to its Challenge.lean counterpart (same namespace Challenge, same types), enforced by generating both from a single source of truth (scripts/gen_comparator_wrappers.py). Each proof builds under lake build with the leanchecker external kernel checker passing and no sorry / axiom / admit / unsafe / native_decide in the Solution module; the AxiomAudit confirms all 34 comparator theorems depend only on propext, Quot.sound and Classical.choice. The supervisor (Gregory Morse) is the independent reviewer: he reviewed the theorem statements against the intended mathematical claims and the stability-boundary statements, and confirmed the worst-case cost equals static recomputation (no asymptotic separation is claimed)." }, "canonical": { "repo": "thonmay/thesis-lean", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "Timeroot/BS_Lam/formalization.yaml", "repo": "Timeroot/BS_Lam", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/Timeroot/BS_Lam/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "BS_Lam", "description": "Formalization of the superquadratic separation of block sensitivity and spectral sensitivity. Companion to https://arxiv.org/abs/2608.00851.", "authors": [ "Alex Meiburg" ], "maintainers": [ "Alex Meiburg" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "A counterexample to bs(f) = O(lambda(f)^2)", "id": "bs_lambda.txt (this repository; unpublished, no DOI or arXiv ID)", "authors": [ "Alex Meiburg" ], "type": "manuscript", "relationship": "formalizes", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "cs.CC", "math.CO" ], "msc2020": [] }, "automation": { "methods": [ { "method": "agent", "models": [ "claude-opus-5" ], "framework": "Claude Code" }, { "method": "autonomous", "models": [], "framework": "Harmonic ATP, via the batch driver in atp/" } ], "strongest": "autonomous", "models": [ "claude-opus-5" ], "spend": "~$400 (one significant figure -- the uncertainty is large). Metered from the session transcripts: 6.03M output, 39.1M cache-write, 516M cache-read and 0.05M base input tokens, all claude-opus-5, across the main session and 88 subagent transcripts. At Opus 5 list prices ($25 / $6.25 / $0.50 / $5 per Mtok) that is ~$650 for everything the session ever did; about a third of it went to a second result that was formalized and then removed at the author's request, leaving ~$400 attributable to the work in this repository. Harmonic ATP usage is not metered by the client and is believed to be negligible against this.", "notes": "The formalization was written from the manuscript alone; there was no pre-existing Lean development to build on and no blueprint." }, "scope": "The whole manuscript: the Paley-tournament construction, the Lovasz Local Lemma argument producing the gate labelling, the spectral bound on lambda, and the composition argument, ending in an unconditional counterexample to bs(f) = O(lambda(f)^2). The manuscript's Section 14 imports multiplicativity of lambda under composition from ABKRT; here that is proved rather than assumed, so no statement in the development carries a hypothesis drawn from the literature. The one thing deliberately out of scope is a numerically exact exponent: the development certifies 2.12 rather than alpha = 2.1269738..", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "BSLambda.Final.exists_bs_gt_lam_rpow", "file": "BSLambda/Main.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "BSLambda.Final.exists_ratio_blowup", "file": "BSLambda/Main.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "BSLambda.lam_comp", "file": "BSLambda/Spectral/Multiplicative.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "(1) Strengthening: the manuscript's Section 14 cites ABKRT for lambda(f . g) = lambda(f) lambda(g); here it is proved (BSLambda.lam_comp), which is what makes the final statements hypothesis-free. (2) Added hypothesis: `card_typeB_bot_bot` (Section 8.2, BSLambda/LLL/GateExists.lean) needs `IsTournament Arc` in addition to the double-regularity hypotheses. Without asymmetry the claimed bound C(t,2)(d-2) is false -- it counts the top pair unordered, and Arc = true satisfies the degree hypotheses vacuously while overshooting by the swap factor. Asymmetry holds for any tournament and is available at the sole call site, so nothing downstream changes. (3) Numerics: the headline exponent is the round number 2.12 rather than alpha = 2 log 14011 / log(7149 + 2 sqrt 150108) = 2.1269738..., because 2.12 is certified without evaluating a logarithm: `rpow_1_06_lt` (BSLambda/Main.lean) reduces 7924 ^ 1.06 < 14011 to the integer inequality 7924 ^ 53 < 14011 ^ 50. `alpha` itself is defined, and `two_lt_alpha` is proved. (4) Naming: declaration names are Mathlib-style rather than the manuscript's labels; `alignment.statements` below is the map.", "alignment": true, "original": false, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [], "notes": "No human has line-by-line reviewed the proofs. What has been checked mechanically: `lake build` is 0 errors, 0 warnings, 0 sorries; `#print axioms` on all 18 headline declarations returns exactly [propext, Classical.choice, Quot.sound]; and leanprover/comparator accepts `comparator/Solution.lean` against `comparator/Challenge.lean`, which restates the headline result from scratch in 30 lines of definitions with no import from the development -- so the statement being proved does not depend on the development's own definitions being the right ones, only on those 30 lines being read and believed. Definitions were also cross-read against Section 1 of the manuscript during a cleanup pass." }, "canonical": { "repo": "timeroot/bs_lam", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "tomo61098/c-separated-np-complete-lean/formalization.yaml", "repo": "tomo61098/c-separated-np-complete-lean", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/tomo61098/c-separated-np-complete-lean/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "NP-hardness of Gaussian c-separation", "description": "A Lean 4 proof development of reduction idea toward NP-hardness of feature selection for relative Gaussian c-separation in the formulation of Borozan et al. The argument starts from NP-completeness of equal-cardinality partition of positive squares, giving integral Gaussian data via integer roots and an even paired total. Three Gaussians enforce partition balance, and a schedule of d+1 Gaussians enforces binary selectors. Explicit separation of the two blocks makes cross-block hinge penalties vanish. The fixed threshold is (dot s 1 / 2)*d*(d+1)/2 - 3*d/4, an integer multiple of 1/4 for natural", "authors": [ "Tomislav Prusina" ], "maintainers": [ "Tomislav Prusina" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Fixed-threshold Gaussian gadget equivalence and square-partition reduction", "id": "", "authors": [ "Tomislav Prusina" ], "type": "original-proof", "relationship": "other", "endorsement": "" }, { "title": "Reducibility Among Combinatorial Problems", "id": "https://doi.org/10.1007/978-1-4684-2001-2_9", "authors": [ "Richard M. Karp" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Optimal Marker Genes for c-Separated Cell Types", "id": "https://doi.org/10.1007/978-3-031-90252-9_53", "authors": [ "Bartol Borozan", "Luka Borozan", "Domagoj Ševerdija", "Domagoj Matijević", "Stefan Canzar" ], "type": "paper", "relationship": "background", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "cs.LO", "cs.CC" ], "msc2020": [ "03B35", "68Q17" ] }, "automation": { "methods": [ { "method": "manual", "models": [], "framework": "Lean 4 and Mathlib" }, { "method": "agent", "models": [ "GPT-6" ], "framework": "Codex" } ], "strongest": "agent", "models": [ "GPT-6" ], "spend": "not tracked", "notes": "The Lean proofs were developed with AI assistance, including parallel proof development, source review, and Lean checking. The main theorem is complete; independent human review remains pending." }, "scope": "The sole submission target is partition_gadget_schedule_partition_iff. The main theorem and its dependencies are proved, lake build passes, and main-theorem axiom inspection found only the three standard axioms. The result is a pointwise equivalence with a fixed threshold, in dimension d with d+4 Gaussian classes. For this Gaussian result, a d-bit certificate and sound and complete integer verifier are formalized, with work at most 128*(L+1)^2 in the explicit binary-arithmetic cost model, where L encodes s. The dimension is at least 4, the weights are natural squares, and their half-total is a", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "CSeparatedNPComplete.partition_gadget_schedule_partition_iff", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The development uses functions Fin n -> Real for vectors and lists for Gaussian collections. The scalar schedule uses finite sums and factored covariances. The main statement assumes d >= 4, half-total at least 1, and natural-square coordinates. The threshold is (dot s 1 / 2)*d*(d+1)/2 - 3*d/4. Its Gaussian count is d+4 in dimension d. Schedule class i > 0 uses coordinate i-1, so all d coordinates contribute to the correction. Karamata's domain convexity and conditional function convexity are bundled as convexOnPredicate, proved equivalent to Mathlib ConvexOn. Equal coordinates are handled by extending secant slopes. The paired-square lemmas use finite sums and assume K >= 1 and a_i <= K^(2*i). The square selectors choose exactly one element per pair; the original-weight sum equality does not require equal cardinality. For (1,1,1,3) at K=2, the constructed squares admit a paired equal-cardinality partition although the original input has no such partition. The separate strengthened construction uses indices 1 through n and K = 2+n+2*sum(a_i)+sum(a_i^2). It proves that every balanced square selector splits all pairs, so the reduction source is ordinary PARTITION. Correctness, positi", "alignment": true, "original": true, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [ "Codex agents" ], "notes": "Auxiliary Karamata and gadget proofs passed local Lean checks. Karamata examples covered empty vectors, identical vectors, and a closed interval with shared coordinates. Agents reviewed source fidelity and mathematical context. Main-theorem build and axiom inspection passed. A local audit imported Challenge and Solution into separate environments and checked exact equality of the main theorem types and 42 definition bodies. The Solution does not import Challenge. Palomar's pinned metadata validator and the repository submission preflight pass with the confirmed author and responsible maintainer. The paired-square results and the equal-cardinality counterexample passed Lean checks. Their axiom audits found only the three standard axioms. The unconditional reduction, positivity, injectivity," }, "canonical": { "repo": "tomo61098/c-separated-np-complete-lean", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "tristanbuckmaster/fluid_lean/affinecore/formalization.yaml", "repo": "tristanbuckmaster/fluid_lean", "path": "affinecore/formalization.yaml", "directory": "affinecore", "origins": [ "search" ], "url": "https://github.com/tristanbuckmaster/fluid_lean/blob/HEAD/affinecore/formalization.yaml", "version": "v0.4", "missing": [], "name": "Finite-time gradient blow-up for the forced two-dimensional inviscid Boussinesq system — Lean 4 formalisation", "description": "For every real A₀ ≥ 1 and every radial smooth compactly supported cut-off χ₀ on ℝ² equal to 1 on the ball of radius 2, there exist a time T > 0, smooth forcing terms f_θ, f_u on [0, T) × ℝ² with every mixed space–time derivative bounded up to T and supports in a fixed ball, and a classical solution (θ, u, p) of the two-dimensional inviscid Boussinesq system with buoyancy constant 1 and that forcing, issuing from the explicit odd initial temperature −(A₀/N₀) sin(N₀ x₂) χ₀(x), N₀ = ⌈(√A₀)/2⌉, and zero initial velocity, such that the temperature stays bounded while ‖∇θ(t, ·)‖_∞ → +∞ as t ↑ T and ", "authors": [ "Levent Alpöge" ], "maintainers": [ "Levent Alpöge" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Finite-time gradient blow-up with bounded temperature for the inviscid Boussinesq system on the plane under smooth compactly supported forcing (provisional title — the title printed on the current edition of the manuscript in preparation; to be confirmed by the authors)", "id": "", "authors": [ "Levent Alpöge", "Tristan Buckmaster" ], "type": "original-proof", "relationship": "other", "endorsement": "n/a" } ], "related": [], "classification": { "arxiv": [ "math.AP" ], "msc2020": [ "35Q31", "35B44", "76B03", "76R10" ] }, "automation": { "methods": [ { "method": "autonomous", "models": [ "Claude" ], "framework": "" } ], "strongest": "autonomous", "models": [ "Claude" ], "spend": "", "notes": "All Lean code, including the statement layer and the trusted statement file, was written by Claude (Anthropic) under human direction." }, "scope": "A complete Lean 4 proof over Mathlib of the single proposition stated in the Mathlib-only module Challenge.lean, configured for comparator against that statement with the three standard axioms permitted (comparator.json; enable_nanoda); the comparator run is pending.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "forced_boussinesq_affine_core_blowup", "file": "Challenge.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The certified proposition asserts boundedness of the temperature, blow-up of its gradient, unboundedness of the vorticity along a sequence of times, compact supports, the Biot–Savart representation, class membership and uniqueness. Neither sup-norm appears literally in the Lean statement: the gradient blow-up is rendered as «for every M, eventually as τ ↑ T some point x has ‖∇θ(τ, x)‖ ≥ M», and the vorticity statement as «for every M, frequently as τ ↑ T some point x has |curl u(τ, x)| ≥ M». No rate of blow-up is certified. The forcing is part of the data produced by the theorem, not prescribed. Uniqueness is stated in pressure-free weak form against all finite-energy Lipschitz competitors with the same forcing.", "alignment": true, "original": true, "review": { "status": "unreviewed", "bucket": "unchecked", "reviewers": [], "notes": "Challenge.lean has not yet been read by the responsible maintainer; status will be updated after that read." }, "canonical": { "repo": "tristanbuckmaster/fluid_lean", "directory": "affinecore" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "tristanbuckmaster/fluid_lean/boussinesq-blowup/formalization.yaml", "repo": "tristanbuckmaster/fluid_lean", "path": "boussinesq-blowup/formalization.yaml", "directory": "boussinesq-blowup", "origins": [ "search" ], "url": "https://github.com/tristanbuckmaster/fluid_lean/blob/HEAD/boussinesq-blowup/formalization.yaml", "version": "v0.4", "missing": [], "name": "Finite-time blow-up for the planar inviscid Boussinesq equations with smooth force — Lean 4 formalisation", "description": "For every buoyancy constant κ > 0 there exist a time T > 0, smooth forces f_θ, f_u on [0, T) × ℝ² with every mixed space–time derivative bounded up to T and support in a fixed ball, and a classical solution (θ, u, p) of the planar inviscid Boussinesq system with those forces, odd under x ↦ −x at every time, whose initial temperature and velocity are smooth, compactly supported and of finite energy, such that the temperature stays uniformly bounded while ‖∇θ(t, ·)‖_∞ → +∞ as t ↑ T, the vorticity ω = ∂₁u₂ − ∂₂u₁ is unbounded near T in the limsup sense, and such that for every T′ < T the solution", "authors": [ "Levent Alpöge" ], "maintainers": [ "Levent Alpöge" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Accompanying manuscript (title to be determined)", "id": "", "authors": [ "Levent Alpöge", "Tristan Buckmaster" ], "type": "original-proof", "relationship": "other", "endorsement": "n/a" } ], "related": [], "classification": { "arxiv": [ "math.AP" ], "msc2020": [ "35Q35", "35B44", "76B03" ] }, "automation": { "methods": [ { "method": "autonomous", "models": [ "Claude" ], "framework": "" } ], "strongest": "autonomous", "models": [ "Claude" ], "spend": "", "notes": "All Lean code, including the certificate generators and the statement layer, was written by Claude (Anthropic) under human direction." }, "scope": "A complete Lean 4 proof over Mathlib of the single proposition stated in the Mathlib-only module Challenge.lean, checked by comparator against that statement with the three standard axioms permitted.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "boussinesq_smooth_force_blowup", "file": "Challenge.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The certified proposition asserts, for every κ > 0, finite-time blow-up of the temperature gradient, uniform boundedness of the temperature itself, unboundedness of the vorticity near T, oddness of the solution and the forces, finite energy, class membership and uniqueness. No sup-norm appears literally in the Lean statement: unboundedness of the gradient is rendered as a minorant g(t) → +∞ attained at some point x at each instant, and the vorticity assertion as a limsup statement — for every M, frequently as t ↑ T there is a point x with |ω(t, x)| ≥ M — not as a limit. No rate of blow-up is certified, and the buoyancy constant κ is universally quantified over κ > 0 rather than fixed. Uniqueness is stated within the finite-energy Lipschitz class, the temperature equation holding almost everywhere and the momentum equation in pressure-free weak form against all competitors in that class.", "alignment": true, "original": true, "review": { "status": "unreviewed", "bucket": "unchecked", "reviewers": [], "notes": "To be confirmed at release: the responsible maintainer, Levent Alpöge (an author of the accompanying manuscript), will read the challenge module Challenge.lean at the released revision — its definition layer and the statement boussinesq_smooth_force_blowup — and confirm that it states the manuscript's blow-up theorem in the form given in project.description and fidelity.divergences; until then this file records no review." }, "canonical": { "repo": "tristanbuckmaster/fluid_lean", "directory": "boussinesq-blowup" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "tristanbuckmaster/fluid_lean/euler-blowup/formalization.yaml", "repo": "tristanbuckmaster/fluid_lean", "path": "euler-blowup/formalization.yaml", "directory": "euler-blowup", "origins": [ "search" ], "url": "https://github.com/tristanbuckmaster/fluid_lean/blob/HEAD/euler-blowup/formalization.yaml", "version": "v0.4", "missing": [], "name": "Finite-time blow-up for the three-dimensional incompressible Euler equations with smooth force — Lean 4 formalisation", "description": "There exist a time T > 0, a smooth force f on [0, T) × ℝ³ with every mixed space–time derivative bounded up to T and support in a fixed ball, and a classical solution (u, p) of the three-dimensional incompressible Euler equations with that force, whose initial velocity is smooth, compactly supported and of finite energy, such that the vorticity ω = curl u satisfies ‖ω(t, ·)‖_∞ → +∞ as t ↑ T and ∫₀^T ‖ω‖_∞ dt = +∞, and such that for every T′ < T the solution lies in the finite-energy Lipschitz class on [0, T′] and is unique there among weak solutions in that class with the same data and force. ", "authors": [ "Claude" ], "maintainers": [ "Levent Alpöge" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Accompanying manuscript (title to be determined)", "id": "", "authors": [ "Levent Alpöge", "Tristan Buckmaster" ], "type": "original-proof", "relationship": "other", "endorsement": "n/a" } ], "related": [], "classification": { "arxiv": [ "math.AP" ], "msc2020": [ "35Q31", "35B44", "76B03" ] }, "automation": { "methods": [ { "method": "autonomous", "models": [ "Claude" ], "framework": "" } ], "strongest": "autonomous", "models": [ "Claude" ], "spend": "", "notes": "All Lean code, including the certificate generators and the statement layer, was written by Claude (Anthropic) under human direction." }, "scope": "A complete Lean 4 proof over Mathlib of the single proposition stated in the Mathlib-only module Challenge.lean, checked by comparator against that statement with the three standard axioms permitted.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "euler_smooth_force_blowup", "file": "Challenge.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The certified proposition asserts finite-time blow-up of the vorticity, the divergence of ∫ ‖ω‖_∞ dt, finite energy, class membership and uniqueness. Neither sup-norm appears literally in the Lean statement: unboundedness is rendered as a minorant g(t) → +∞ attained at some point x at each instant, and the divergence of ∫₀^T ‖ω‖_∞ dt as, for every M, a non-negative integrable minorant on some [0, T′] with integral at least M. No rate of blow-up is certified. Uniqueness is stated in pressure-free weak form against all finite-energy Lipschitz competitors.", "alignment": true, "original": true, "review": { "status": "author-verified", "bucket": "author-verified", "reviewers": [ "Levent Alpöge" ], "notes": "Levent Alpöge (an author of the accompanying manuscript) read the challenge module Challenge.lean at the released revision — its definition layer and the statement euler_smooth_force_blowup — and confirmed it states the manuscript's blow-up theorem in the form given in project.description and fidelity.divergences." }, "canonical": { "repo": "tristanbuckmaster/fluid_lean", "directory": "euler-blowup" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "ukiyois/hs-virial-nbc-identity/formalization.yaml", "repo": "ukiyois/hs-virial-nbc-identity", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/ukiyois/hs-virial-nbc-identity/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Hard-sphere Mayer NBC identity and fork-packing volume bound", "description": "A kernel-checked Lean formalization of the exact no-broken-circuit volume identity for hard-sphere Mayer cluster integrals and the stronger dimension-three fork-packing upper bound for each tree-owned NBC region. The development proves finite broken-circuit cancellation, the measurable finite-region decomposition, the anchored Euclidean product-measure theorem, and its explicit flat-coordinate transport. Its geometric result is the attachment-aware estimate Vol(NBC_T) <= (17/32)^hardSphereNu T * hardSphereKappa^(k-1), where hardSphereNu T is the maximum cardinality of a packing of pairwise ver", "authors": [ "ukiyois", "OpenCode agent sessions" ], "maintainers": [ "ukiyois" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "Danus hs-virial verified fact chain: hard-sphere NBC identities", "id": "", "authors": [], "type": "other", "relationship": "adapts", "endorsement": "n/a" }, { "title": "Three-dimensional fork-packing upper bound for NBC regions", "id": "", "authors": [], "type": "other", "relationship": "other", "endorsement": "n/a" }, { "title": "Single-element extensions of matroids", "id": "https://doi.org/10.6028/jres.069b.003", "authors": [ "Henry H. Crapo" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "A logical expansion in mathematics", "id": "https://doi.org/10.1090/S0002-9904-1932-05460-X", "authors": [ "Hassler Whitney" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "A contribution to the theory of chromatic polynomials", "id": "https://doi.org/10.4153/CJM-1954-010-9", "authors": [ "W. T. Tutte" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "The Tutte polynomial", "id": "https://doi.org/10.1007/BF01817442", "authors": [ "Henry H. Crapo" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "The broken-circuit complex", "id": "https://doi.org/10.1090/S0002-9947-1977-0468931-6", "authors": [ "Tom Brylawski" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "The Statistical Mechanics of Condensing Systems. I", "id": "https://doi.org/10.1063/1.1749933", "authors": [ "Joseph E. Mayer" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Convergence of fugacity expansions for fluids and lattice gases", "id": "https://doi.org/10.1063/1.1703906", "authors": [ "O. 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Penrose" ], "type": "book", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Seventh virial coefficients for hard spheres and hard disks", "id": "https://doi.org/10.1063/1.1840521", "authors": [ "Francis H. Ree", "William G. Hoover" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Convergence of Mayer and virial expansions and the Penrose tree-graph identity", "id": "https://doi.org/10.1007/s11005-016-0918-7", "authors": [ "Aldo Procacci", "Sergio A. Yuhjtman" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" } ], "related": [], "classification": { "arxiv": [ "math.CO", "math-ph" ], "msc2020": [ "05C15", "82B05" ] }, "automation": { "methods": [ { "method": "manual", "models": [], "framework": "n/a" }, { "method": "agent", "models": [ "deepseek-v4-flash" ], "framework": "OpenCode" }, { "method": "agent", "models": [ "GLM-5.3-Flash" ], "framework": "OpenCode" }, { "method": "agent", "models": [ "openai/gpt-5.6-luna" ], "framework": "OpenCode" } ], "strongest": "agent", "models": [ "GLM-5.3-Flash", "deepseek-v4-flash", "openai/gpt-5.6-luna" ], "spend": "not tracked", "notes": "The human contributor supplied the graph-theoretic/NBC research direction and hypothesized that a positive decomposition might exist, but did not state the exact final identity. AI agent sessions implemented and repaired the Lean proof. The OpenCode alias ox-alpha is recorded only for the free GLM-5.3-Flash route described above; it does not identify DeepSeek." }, "scope": "For natural numbers k and d with 2 <= k, the project proves the exact identity (k.factorial : Real) * |hardSphereBk| as the finite sum of transported NBC-region volumes over all spanning trees of the complete graph. In dimension three it additionally proves, for every tree T, the stronger per-tree bound Vol(NBC_T) <= (17/32)^hardSphereNu T * hardSphereKappa^(k-1). Here hardSphereNu T is the maximum cardinality of a packing of pairwise vertex-disjoint, order-compatible forks. A formal fork has ordered labels a < b < c, center a, leaves b,c, and tree edges {a,b} and {a,c}. The proof is an attach", "sorry_count": 2, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "PalomarHS.nbc_region_real_volume_le_fork_factor", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "PalomarHS.main_result", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The formal statement uses natural d, so it also covers the degenerate d=0 case although the exposition emphasizes d>=1. Its primary domain is the canonical product measure on Fin (k-1) -> EuclideanSpace Real (Fin d); the flat Euclidean presentation is connected by an explicit measure-preserving equivalence rather than by definitional equality. The flat NBC region is the image under that equivalence. The theorem uses the lexicographically largest-edge broken-circuit convention. The fork-packing upper bound is specialized to d=3 and uses the formal maximum packing of pairwise vertex-disjoint, order-compatible forks, hardSphereNu T; it does not assert an all-dimension analogue. Strict positivity and convergence corollaries remain separate targets.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "ukiyois" ], "notes": "Statement scope and the product-versus-flat measure distinction were reviewed in the working session. The exact identity and the fork-packing upper bound were checked in the substantive Lean development; Lean lake build, Challenge/Solution compilation, axiom printing, and the human-readable TeX build pass. The bibliographic source audit is a self-assessment; no independent peer review has been performed." }, "canonical": { "repo": "ukiyois/hs-virial-nbc-identity", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-29-000014", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-29-000014", "commit": "6ff2cea17ea9320d8974b76f26bd502b4a4076a9", "trust": "high", "theorems": 2, "date": "2026-09-03" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "ultronozm/GGPTransversalityBounds/formalization.yaml", "repo": "ultronozm/GGPTransversalityBounds", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/ultronozm/GGPTransversalityBounds/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Gan–Gross–Prasad transversality, volume bounds, and bilinear estimates", "description": "A Lean 4 formalization of the local algebraic, counting, and Cauchy--Schwarz arguments underlying the transversality, volume bounds, and normalized bilinear estimates in Sections 5--7 of Hu--Nelson, Subconvex bounds for U_{n+1} x U_n in horizontal aspects. The formalization proves separate exact-positive-depth and residually-transverse branches. The volume bounds are normalized by the full additive centralizer, with geometric constants (n+1)^2 and (n+1)^2 (n^2+n+1), respectively. The bilinear bounds retain the exact additive-to-unit centralizer density loss.", "authors": [ "Paul D. Nelson" ], "maintainers": [ "Paul D. Nelson" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Subconvex bounds for U_{n+1} x U_n in horizontal aspects", "id": "arXiv:2309.06314v2", "authors": [ "Yueke Hu", "Paul D. 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It includes the exact-positive-HZ-depth and residually-outside-HZ branches. The elementary trivial bilinear bound is proved in the repository but is not selected for these entries. The Lean results are stated for quotients by powers of a nonzero principal ideal in a commutative domain, subject to explicit finiteness, residue-field, nontriviality, stability, and invertibility-of-2 hypotheses. They do not define the paper's local-field notation Q, Q*, d_H", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The formalization decomposes the paper's piecewise volume and bilinear estimates into exact-positive-depth and residual branches. Its arithmetic setting is an abstract commutative domain with powers of a principal ideal and explicit quotient hypotheses rather than the paper's nonarchimedean-local-field notation. The primary volume statements use the full additive centralizer and geometric constants (n+1)^2 and (n+1)^2 (n^2+n+1); the bilinear statements keep the exact additive-to-unit centralizer density ratio. Older theorems with a uniform 2^n density bound are retained as coarsened corollaries. Definitions directly corresponding to Q, Q*, d_H, and Delta(gamma), and one combined wrapper theorem in that notation, are not included. The residual volume theorem and both advertised bilinear wrappers assume n >= 2, matching the corrected rank restrictions in paper commit 4fdbe00. PAPER-RANK-SCOPE.md preserves the earlier discrepancy, counterexample, and resolution. The Lean mathematical source is unchanged by this documentation review; author review of the correspondence remains pending.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Paul D. Nelson" ], "notes": "The author has supervised development and performed earlier inspection. Detailed author review of the final five statements and their correspondence with the revised paper has been deferred. The repository has passed the recorded Lean build and kernel checks; these do not certify the mathematical interpretation of the paper's notation. There has been no independent human mathematical or formal-code review. The three Palomar Comparator configurations with NanoDa kernel tracing pass on Linux under the Landrun sandbox at the release-candidate content, and pass again at the final release-candidate commit with a native macOS rerun without sandboxing (Comparator's fake-landrun shim); the correspondence document remains awaiting the author's review." }, "canonical": { "repo": "ultronozm/ggptransversalitybounds", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "utensil/connes-rigidity/formalization.yaml", "repo": "utensil/connes-rigidity", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/utensil/connes-rigidity/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "connes-rigidity", "description": "Formalization of Zhou's Theorem A, complete relative to the cited Ershov–Jaikin-Zapirain–Kassabov property-(T) input for EL₃(𝔽₂[t]). It constructs two countable discrete ICC groups satisfying the formal property-(T) predicate that are not isomorphic, while their group von Neumann algebras are related by a trace-preserving normal star-algebra equivalence. 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The automated trust-boundary checks do not constitute mathematical review." }, "scope": "Formalization of Zhou's Theorem A, complete relative to the explicit literature premise that EL₃(𝔽₂[t]) has property (T). The project proves the EL₃ = SL₃ transport and every other certificate used by the final theorem.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "Connes.theoremA", "file": "Connes/Main.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [ { "statement": "EL₃(𝔽₂[t]) has Kazhdan's property (T)", "source": "Ershov–Jaikin-Zapirain–Kassabov, Theorem 1.1" } ] } ], "comparator": true, "literature_dependencies": 1, "divergences": "The formalization follows Zhou's §§2–7 construction but leaves the cited EJZK property-(T) result as the explicit premise of Connes.theoremA rather than formalizing that literature theorem. The property-(T) predicate quantifies over Hilbert-space carriers in the group's universe. Standard property (T) implies this predicate, but the converse equivalence bridge is not formalized. No other divergence is known.", "alignment": true, "original": false, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [ "GPT-5.6 Sol clean-room agents" ], "notes": "Internal clean-room agent reviews used the Tau Ceti rubrics and additional project-level evidence checks documented in docs/AUDITS.md. The planned internal human mathematical examination remains open. No peer review, source-author verification, or independent mathematical certification is claimed." }, "canonical": { "repo": "utensil/connes-rigidity", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "viazovska/PentagonalNumberTheorem/formalization.yaml", "repo": "viazovska/PentagonalNumberTheorem", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/viazovska/PentagonalNumberTheorem/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "PentagonalNumberTheorem", "description": "Two independent Lean 4 / Mathlib proofs of Euler's pentagonal number theorem. They share no lemmas -- neither depends on the other -- so each confirms the result on its own. The primary development is combinatorial: Franklin's sign-reversing involution on partitions into distinct parts, giving p_e(n) - p_o(n) = (-1)^|k| when n is the generalized pentagonal number k(3k-1)/2 and 0 otherwise, stated under a single integer index, together with the identification of that signed count with the coefficient of X^n in the truncated product over ZZ[[X]]. This is the focus of the project: its blueprint w", "authors": [ "Jonathan Conrad", "Paula Muermann", "Maryna Viazovska" ], "maintainers": [ "Jonathan Conrad", "Paula Muermann", "Maryna Viazovska" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "Pentagonal Number Theorem", "id": "", "authors": [ "Jonathan Conrad", "Paula Muermann", "Maryna Viazovska" ], "type": "unpublished manuscript", "relationship": "formalizes", "endorsement": "participated" }, { "title": "Introduction to the Theory of Partitions", "id": "", "authors": [ "George E. 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The overlap is with our q-seri" } ], "classification": { "arxiv": [ "math.NT", "math.CO" ], "msc2020": [ "11P81", "11P84", "05A17", "05A19", "33D15" ] }, "automation": { "methods": [ { "method": "autonomous", "models": [ "Aristotle" ], "framework": "" }, { "method": "agent", "models": [ "Aristotle", "Claude" ], "framework": "" }, { "method": "manual", "models": [], "framework": "" } ], "strongest": "autonomous", "models": [ "Aristotle", "Claude" ], "spend": "", "notes": "The Lean proofs in both routes were largely generated by Aristotle (Harmonic). What differs between the two is how much human work went into directing and checking that generation, and whether the mathematical plan behind it was written by a person. Franklin route (EulerPentagonalNumberTheorem_Franklin/, 1264 lines, 123 declarations), the primary development: the blueprint was written manually and in detail by the authors. The mathematical argument, its decomposition into lemmas and the proof plan are theirs, and the formalization follows that blueprint lemma by lemma. The machine-generated pr" }, "scope": "Complete formalization of Euler's pentagonal number theorem, primarily via Franklin's involution, with an independent q-series route through the Jacobi triple product as an accompanying extension. The Jacobi triple product is proved both as a formal-power-series identity in (LaurentPolynomial ℂ)[[X]] and analytically over ℂ. No `sorry`, no axioms beyond Lean's standard three, no unproved literature dependencies.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "PentagonalNumberTheorem.Franklin.euler_pentagonal_number_theorem_packaged", "file": "EulerPentagonalNumberTheorem_Franklin/FormalPowerSeries.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "PentagonalNumberTheorem.Franklin.coeff_prod_eq_pe_sub_po", "file": "EulerPentagonalNumberTheorem_Franklin/FormalPowerSeries.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "QSeries.jacobiTripleProduct", "file": "Qseries_Formalization/QSeries/JacobiTripleProduct.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "QSeries.euler_pentagonal_number", "file": "Qseries_Formalization/QSeries/PentagonalNumber.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "QSeries.FormalPowerSeries.jacobiTripleProduct", "file": "Qseries_Formalization/QSeries/FPSAlgebra.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The Franklin route proves the coefficient identity for the truncated product ∏_{k=1}^{n} (1 - q^k), which agrees with the infinite product in degree n; it does not itself pass to the infinite product. The q-series route proves the infinite-product form directly. The analytic Jacobi triple product is stated for ‖q‖ < 1, ‖z‖ < 1, z ≠ 0 (`QSeries.jacobiTripleProduct`) and extended to the full punctured disc z ≠ 0 in `QSeries.jacobiTripleProduct'`.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Jonathan Conrad", "Paula Muermann", "Maryna Viazovska" ], "notes": "The project is accompanied by a blueprint (leanblueprint) in which every formalized declaration is tied to a paragraph-level mathematical statement and proof sketch, and `leanblueprint checkdecls` verifies in CI that every `\\lean{}` annotation names a declaration that exists in the compiled environment. CI additionally builds with Mathlib's standard linter set enabled and fails on any warning (`lake build --wfail`). Review depth differs by route: the Franklin development was reviewed against an author-written blueprint lemma by lemma and carries the bulk of the manual intervention, whereas the q-series development was reviewed at the level of statements and library structure rather than proof by proof. The formalization has not undergone external peer review." }, "canonical": { "repo": "viazovska/pentagonalnumbertheorem", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "vlad902/left-pci-v-domain-not-right-pci-v-domain/formalization.yaml", "repo": "vlad902/left-pci-v-domain-not-right-pci-v-domain", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/vlad902/left-pci-v-domain-not-right-pci-v-domain/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "A left PCI left V-domain that is not right PCI and not a right V-domain", "description": "A countable domain R that is a left PCI ring (every cyclic left module not isomorphic to R is injective) and a left V-ring (every simple left module is injective), but is not right Ore, not a right PCI ring and not a right V-ring. Consequently the opposite ring of R is a right PCI ring that is not left PCI. This answers in the negative the question of Faith (1973), Cozzens and Faith (Simple Noetherian Rings, 1975) and Damiano (1979) whether every right PCI ring is left PCI, and the question, going back to Cozzens and Faith (1975), whether a left V-domain must be a right V-domain. The ring is a", "authors": [ "Vlad Tsyrklevich" ], "maintainers": [ "Vlad Tsyrklevich" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "A left PCI, left V-domain that is not right Ore, not right PCI and not right V (a right PCI ring that is not left PCI)", "id": "", "authors": [], "type": "original-proof", "relationship": "other", "endorsement": "" }, { "title": "PCI rings and V-domains are not left-right symmetric", "id": "paper/pci_counterexample.tex", "authors": [], "type": "paper", "relationship": "other", "endorsement": "n/a" }, { "title": "Simple Noetherian Rings", "id": "https://doi.org/10.1017/CBO9780511565700", "authors": [ "J. H. Cozzens", "C. Faith" ], "type": "book", "relationship": "background", "endorsement": "" }, { "title": "When are proper cyclics injective?", "id": "https://doi.org/10.2140/pjm.1973.45.97", "authors": [ "C. 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Vedadi" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Rings determined by the properties of cyclic modules and duals: A survey-I", "id": "https://leroy.perso.math.cnrs.fr/Talks/Jain.pdf", "authors": [ "S. K. Jain", "A. K. Srivastava" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Skew Fields: Theory of General Division Rings", "id": "https://doi.org/10.1017/CBO9781139087193", "authors": [ "P. M. Cohn" ], "type": "book", "relationship": "background", "endorsement": "" }, { "title": "Free Ideal Rings and Localization in General Rings", "id": "https://doi.org/10.1017/CBO9780511542794", "authors": [ "P. M. Cohn" ], "type": "book", "relationship": "background", "endorsement": "" }, { "title": "A construction of simple principal right ideal domains", "id": "https://doi.org/10.1090/S0002-9939-1977-0453805-2", "authors": [ "P. M. Cohn" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "A countable self-injective ring is quasi-Frobenius", "id": "https://doi.org/10.1090/S0002-9939-1977-0442025-3", "authors": [ "J. Lawrence" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Homological properties of the ring of differential polynomials", "id": "https://doi.org/10.1090/S0002-9904-1970-12370-9", "authors": [ "J. H. Cozzens" ], "type": "paper", "relationship": "background", "endorsement": "" }, { "title": "Hereditary QI-rings", "id": "https://doi.org/10.1090/S0002-9947-1974-0338075-X", "authors": [ "A. K. Boyle" ], "type": "paper", "relationship": "background", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.RA" ], "msc2020": [ "16D50", "16S36", "16K40", "16S10" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Claude Fable 5.1 (Anthropic, claude-fable-5-1)", "Claude Opus 5.5 (Anthropic, claude-opus-5-5)" ], "framework": "Claude Code (Anthropic), with the lean-lsp MCP server" } ], "strongest": "agent", "models": [ "Claude Fable 5.1 (Anthropic, claude-fable-5-1)", "Claude Opus 5.5 (Anthropic, claude-opus-5-5)" ], "spend": "not tracked (subscription usage; no per-token billing)", "notes": "All mathematics, the paper, the Lean development and this metadata were produced in Claude Code by Claude Fable 5.1 and Claude Opus 5.5 subagents (Anthropic), at the direction of the repository's maintainer, a human who wrote no mathematics or Lean, is the repository's responsible maintainer, and reviewed Challenge.lean only." }, "scope": "Fully formalized. The final theorems in Challenge.lean are: LeftPCI.exists_isLeftPCIRing_not_isRightPCIRing (the Main Theorem's example: a countable domain that is left PCI and left V but not right Ore, not right PCI and not right V), LeftPCI.not_forall_isLeftPCIRing_imp_isRightPCIRing (a left PCI ring need not be right PCI; equivalently, via the opposite ring, a right PCI ring need not be left PCI), and LeftPCI.not_forall_isLeftVRing_imp_isRightVRing (a left V-domain need not be right V), are proved in the library LeftPCI and checked by Comparator. Not among the compared statements: the Main ", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "Right-handed notions are formalized through the opposite ring: IsRightPCIRing R is IsLeftPCIRing Rᵐᵒᵖ and IsRightVRing R is IsLeftVRing Rᵐᵒᵖ, the standard Mathlib idiom. A proper cyclic left module is taken as a quotient R ⧸ I by a left ideal that is not isomorphic to R; since every cyclic module is such a quotient and injectivity is invariant under isomorphism, this is the usual notion. IsLeftVRing quantifies over simple modules in the same universe as R. IsRightOre states that any two non-zero elements have a non-zero common right multiple. The compared statements live in universe Type. The paper works over an arbitrary countable commutative field k; the formalization takes k = ℚ, which suffices for the compared statements. Not compared: the Main Theorem's claims that the ring is simple, a principal left ideal domain, left Noetherian and left hereditary. Proposition 2.2 is formalized through the Baer-criterion argument given after it (solving for every right-hand side) rather than via Jain–Lam–Leroy Theorem 3.2. \"Not right PCI\" is derived via Lemma 2.6 and Lemma 2.4 (countability) rather than via Cozzens–Faith 6.17; Lemma 2.4 is proved on the left and applied to Rᵐᵒᵖ. The tower o", "alignment": true, "original": true, "review": { "status": "human review of the challenge.lean, no human review of the mathematics, the paper or the lean", "bucket": "other", "reviewers": [ "Vlad Tsyrklevich (Challenge.lean only)" ], "notes": "The maintainer reviewed the statement surface, Challenge.lean. Mathematical review: none by a human; the mathematics, the paper and the Lean proofs have been reviewed only by AI sessions." }, "canonical": { "repo": "vlad902/left-pci-v-domain-not-right-pci-v-domain", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-25-000005", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-25-000005", "commit": "ba649de9835f4d2b6ab1c1c79ad27d5a3bb5e0c9", "trust": "high", "theorems": 3, "date": "2026-09-25" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "vo-rodrigues/rr-equals-nonmeager-paper/lean/formalization.yaml", "repo": "vo-rodrigues/rr-equals-nonmeager-paper", "path": "lean/formalization.yaml", "directory": "lean", "origins": [ "palomar", "search" ], "url": "https://github.com/vo-rodrigues/rr-equals-nonmeager-paper/blob/HEAD/lean/formalization.yaml", "version": "v0.4", "missing": [], "name": "The rearrangement number: non(M) ≤ rr", "description": "A Lean formalization of the lower bound non(M) ≤ rr, where non(M) is the least cardinality of a nonmeagre subset of the usual real line and rr is the least cardinality of a family of permutations of the natural numbers that changes the sum or destroys convergence of every conditionally convergent real series. It accompanies the manuscript *Determining the Rearrangement Number* by Vinicius de Oliveira Rodrigues. The reverse inequality was already known in the literature. 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Vinicius de Oliveira Rodrigues is the manuscript author and responsible human maintainer." }, "scope": "Complete lower bound non(M) ≤ rr and attainment of the two defining minima. Excludes rr ≤ non(M), the equality, and Section 5 corollaries.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "Only the non(M) ≤ rr inequality is formalized. The Lean proof supplies its classical ingredients and uses the proof organization described in docs/MANUSCRIPT.md. No extra hypotheses occur in the submitted inequality. Ordered convergence uses SummationFilter.conditional rather than unconditional Summable. Challenge has three intentional theorem holes; status counts refer to the proved development and exclude those holes.", "alignment": true, "original": false, "review": { "status": "author-verified", "bucket": "author-verified", "reviewers": [ "Vinicius de Oliveira Rodrigues", "OpenAI Codex" ], "notes": "Vinicius de Oliveira Rodrigues reviewed the formal statement and checked that it correctly represents the formalized mathematical result. Codex reviewed the submission statements against the existing definitions and proof entry points and added axiom guards for all compared theorems. Existing verification and manuscript correspondence are recorded in docs/VERIFICATION.md and docs/MANUSCRIPT.md. This does not claim independent peer review or a completed Palomar review. See docs/PALOMAR.md for the checks actually run and remaining external verification." }, "canonical": { "repo": "vo-rodrigues/rr-equals-nonmeager-paper", "directory": "lean" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-07-000004", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-07-000004", "commit": "a8e74a91fe0b9f3534194910172349c9926e3b3b", "trust": "high", "theorems": 3, "date": "2026-09-07" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "vo-rodrigues/wallace-problem-palomar/formalization.yaml", "repo": "vo-rodrigues/wallace-problem-palomar", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/vo-rodrigues/wallace-problem-palomar/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "The Wallace problem and countably compact torsion-free Abelian groups in ZFC — Lean 4 formalization", "description": "Lean 4 formalization of the theorem that every torsion-free Abelian group of cardinality continuum admits a Hausdorff countably compact group topology in which every convergent sequence is eventually constant, and constructs a commutative Tychonoff countably compact cancellative topological monoid that is not a group, giving a negative answer to Wallace's question.", "authors": [ "Juliane Trianon Fraga", "Vinicius de Oliveira Rodrigues" ], "maintainers": [ "Juliane Trianon Fraga", "Vinicius de Oliveira Rodrigues" ], "license": "Apache-2.0", "role": "thin-wrapper", "substantive": "vo-rodrigues/wallace-problem-zfc-paper", "sources": [ { "title": "The Wallace problem and countably compact torsion-free Abelian groups in ZFC", "id": "arXiv:2608.17317v1", "authors": [ "Juliane Trianon Fraga", "Vinicius de Oliveira Rodrigues" ], "type": "paper", "relationship": "formalizes", "endorsement": "participated" } ], "related": [], "classification": { "arxiv": [ "math.GR", "math.GN" ], "msc2020": [ "22A05", "54H11", "54D20", "22A15", "20K20", "54A20" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "gpt-5.6-sol" ], "framework": "OpenAI Codex" } ], "strongest": "agent", "models": [ "gpt-5.6-sol" ], "spend": "$0 out of pocket; access used a complimentary subscription normally priced at USD 200 per month", "notes": "The formalization was produced entirely by OpenAI Codex using GPT-5.6-sol in a process jointly directed by both authors. The subscription was provided at no charge by OpenAI. The authors made no API calls, so there was no task-specific API spend. Token counts were not retained. The authors reviewed and take responsibility for the released artifact." }, "scope": "This submission compares exactly the universal theorem for torsion-free Abelian groups of cardinality continuum and the commutative Tychonoff Wallace counterexample. No other result from the substantive development or cited paper is selected by this Comparator entry.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "No divergence is known between the two compared claims and their corresponding statements in the cited paper.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Juliane Trianon Fraga", "Vinicius de Oliveira Rodrigues", "OpenAI Codex (GPT-5.6-sol)" ], "notes": "Both authors read the compared statements, searched the development for sorry declarations and additional axioms, and performed a high-level review of the remaining Lean code. They also ran NanoDa and the verification and audit scripts prepared by Codex. OpenAI Codex, using GPT-5.6-sol, assisted with statement matching, Lean compilation, dependency inspection, and transitive axiom audits. No independent human peer review of the Lean code is claimed." }, "canonical": { "repo": "vo-rodrigues/wallace-problem-palomar", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-19-000004", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-19-000004", "commit": "1ab2fb7ed1fb106e14af45312c2a7ab5d568048b", "trust": "high", "theorems": 2, "date": "2026-08-19" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "Vulkin-prog/paper-c-lean/formalization.yaml", "repo": "Vulkin-prog/paper-c-lean", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/Vulkin-prog/paper-c-lean/blob/HEAD/formalization.yaml", "version": "v0.3", "missing": [], "name": "paper_c_lean", "description": "", "authors": [ "Brice Pouly" ], "maintainers": [], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "A critical Poisson law in a dyadic block — Starting points of long constant stretches of an extended Rademacher random completely multiplicative function", "id": "https://doi.org/10.33774/coe-2026-z3l74", "authors": [ "Brice Pouly" ], "type": "preprint", "relationship": "", "endorsement": "" }, { "title": "A formally verified proof of the prime number theorem", "id": "https://doi.org/10.1145/1297658.1297660", "authors": [ "Jeremy Avigad", "Kevin Donnelly", "David Gray", "Paul Raff" ], "type": "journal article", "relationship": "", "endorsement": "" }, { "title": "Two moments suffice for Poisson approximations: the Chen–Stein method", "id": "https://doi.org/10.1214/aop/1176991491", "authors": [ "Richard Arratia", "Larry Goldstein", "Louis Gordon" ], "type": "journal article", "relationship": "", "endorsement": "" }, { "title": "Squares in products from a block of consecutive integers", "id": "https://doi.org/10.4064/aa-65-3-213-220", "authors": [ "R. Balasubramanian", "T. N. Shorey" ], "type": "journal article", "relationship": "", "endorsement": "" }, { "title": "Uniform bounds for the number of solutions to Y^n = f(X)", "id": "https://doi.org/10.1017/S0305004100066068", "authors": [ "J.-H. Evertse", "J. H. Silverman" ], "type": "journal article", "relationship": "", "endorsement": "" }, { "title": "Quadratic Irrationals: An Introduction to Classical Number Theory", "id": "https://doi.org/10.1201/b14968", "authors": [ "Franz Halter-Koch" ], "type": "textbook", "relationship": "", "endorsement": "" }, { "title": "Number of prime divisors in a product of consecutive integers", "id": "https://doi.org/10.4064/aa113-4-3", "authors": [ "Shanta Laishram", "T. N. Shorey" ], "type": "journal article", "relationship": "", "endorsement": "" }, { "title": "Majorations explicites pour le nombre de diviseurs de N", "id": "https://doi.org/10.4153/CMB-1983-078-5", "authors": [ "J.-L. Nicolas", "G. Robin" ], "type": "journal article", "relationship": "", "endorsement": "" } ], "related": [], "classification": { "arxiv": [], "msc2020": [] }, "automation": { "methods": [ { "method": "agent", "models": [ "Claude Fable 5", "Claude Opus 4.8", "GPT 5.6 Sol (gpt-5.6-sol)" ], "framework": "Anthropic Claude and ChatGPT Work Mode / Codex" } ], "strongest": "agent", "models": [ "Claude Fable 5", "Claude Opus 4.8", "GPT 5.6 Sol (gpt-5.6-sol)" ], "spend": "", "notes": "Brice Pouly directed the research and selected the mathematical and release requirements. As declared on page 74 of the frozen English paper, Claude Fable 5, Claude Opus 4.8, and GPT 5.6 Sol provided substantial assistance with lemma exploration, drafting, Lean development, reconnaissance programs, and multi-provider counter-audits. GPT 5.6 Sol agents also assisted with the v0.48.0 Comparator work and the v0.48.1 manuscript audit remediation and public-release metadata work." }, "scope": "Formalization of the canonical endpoints of Paper C, including the critical Poisson law in a dyadic block, its exact finite-cylinder/infinite-product transfer, the homogeneous-mass estimates, and the supporting arithmetic and probabilistic development. The canonical endpoints remain conditional on six explicitly registered open external literature propositions; Theorem 1.1 consumes exactly three of them. The historical Halter--Koch bridge remains registered as external but is discharged by a public Lean construction and is not an endpoint premise.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "paper_c_theorem_one_one_finite_cylinder", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [ { "statement": "Agent counter-audit supports the registered source-to-proposition implication, without a Lean formalization or independent human review. Registered relation: finite-family specialization of Theorem 1 to closed graph neighborhoods and exact independence outside each neighborhood, hence b₃ = 0; total ", "source": "https://doi.org/10.1214/aop/1176991491" }, { "statement": "Agent counter-audit supports the registered source-to-proposition implication, without a Lean formalization or independent human review. Registered relation: specialization to K = L = ℚ, a split polynomial, n = 2, and the abscissa count before the elementary factor two", "source": "https://doi.org/10.1017/S0305004100066068" }, { "statement": "Agent counter-audit supports the registered source-to-proposition implication, without a Lean formalization or independent human review. Registered relation: source-shaped Nicolas–Robin input only: the direct logarithmic divisor inequality above n=64. 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(1) Observed variables are binary: outcomes are Bool. The manuscript's Remark 3.8 extends the classification to any fixed finite observed alphabets, and that transfer is not formalized. (2) Latent alphabets are finite: GCompatible quantifies over a GModel whose latent spaces are Fintypes, where the mathematics allows arbitrary measurable latent spaces. The formal compatible set is therefore a priori a subset of the general one, so \"not compatible\" is the weaker of the two readings, and the nonterminating half of the classification asserts the weaker one. (3) The recursively expressible test IS formalized on the graph side: dsep is the trail criterion for the depth-one inflation DAG, Expressible is the Wolfe–Spekkens–Fritz closure, and gExpFeasible_iff_gAIFeasible proves I^exp_t = I^AI_t on every pair-source scenario. The classification equivalence and strictly positive witness existence are formalized, but the mapped declarations do not assert rationality of those witnesses; that clause is analytic. The terminal declarations square_linear_witness and triangle_linear_witness prove the endpoint specialization q = 1/(16t) of Theorems 4.3 and 4.4, with the densit", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "Kernel-checked and axiom-audited in this repository by the author; no independent human review of the mathematics has been performed, and the manuscript has not been refereed or posted. The priority audit in paper/release/PRIORITY_AUDIT.md records PRIORITY_NOT_KILLED, with Navascués–Wolfe 2020 §4.1 as the strongest located predecessor, and does not certify firstness. Its addendum of 2026-09-14 records a targeted sweep of the pair-source sections: no predecessor was located for the classification beyond stars, for I^exp = I^AI, for the all-order witnesses, for the Θ(1/t) bounds, for the source-free convex-order constant, for the order conversion or for the second-order separations; attribution was corrected for exact parity rigidity (Boreiri et al. 2023, not 2025). 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A complex-valued graph parameter, normalized at the empty graph and invariant under isomorphism, has exponentially bounded edge-connection rank if and only if it is a mixed partition function. For a natural rank base R, a representing model can be chosen with k even and 2ℓ odd colours satisfying k + 2ℓ ≤ R. The development also proves the even connection-rank growth limit, minimum total colour dimension, minimal parity formulas and prescribed-dimension criterion. 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Algebra 374 (2013), 66-78)", "authors": [ "Carlo Mazza", "Charles Weibel" ], "type": "journal article", "relationship": "background", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.CO", "math.CT" ], "msc2020": [ "05C31", "05C50", "18M05", "18M25", "20C30", "68V20" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "claude-fable-5", "claude-opus-5" ], "framework": "Claude Code" }, { "method": "agent", "models": [ "gpt-6-astra" ], "framework": "OpenAI Codex" } ], "strongest": "agent", "models": [ "claude-fable-5", "claude-opus-5", "gpt-6-astra" ], "spend": "subscription-based usage; no metered API spend", "notes": "CI runs source style checks, the full Lean build, environment linters and leanprover/comparator. The default build includes pinned statement and axiom audits for the theorems of record, including growth, minimality and padding. BlueprintFactorial checks that the envelope's semisimplicity and abelianness are independent of Schur theory, trace zeta and Deligne, and that the forward summits use the factorial trace theorem. Challenge.lean imports Mathlib alone, copies the certified statement definitions and declares seven goals with sorry; Solution.lean proves them from the theorems of record. Com" }, "scope": "The Main Theorem and sharp colour budget (Theorem 1.1 and Corollary 4.10), for natural rank bases including zero, over the flag model of multigraph fragments. The Section 7 results comprise even rank growth, minimum total dimension, minimal parity formulas, padding and prescribed dimensions. The appendix results comprise trace-zeta rationality under exponential growth (Corollary A.2) and rationality for arbitrary hook-confined sequences (Lemma A.9). The development also proves Deligne's fibre-functor theorem, the symmetric-group character theory, and the auxiliary separate colour bound k, 2ℓ ≤", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "RS.regts_sevenster", "file": "RS/Summit.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "RS.regts_sevenster_characterisation", "file": "RS/Summit.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "RS.deligne_theorem", "file": "RS/Classical/Deligne/DeligneAssembly.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "RS.regts_sevenster_quant", "file": "RS/Summit.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "RS.regts_sevenster_total", "file": "RS/Summit.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "RS.regts_sevenster_minimum", "file": "RS/Summit.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "RS.regts_sevenster_rank_growth", "file": "RS/Summit.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "RS.regts_sevenster_prescribed", "file": "RS/Summit.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "RS.regts_sevenster_converse", "file": "RS/TheoremConverse.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "(1) Connection rank. Lean uses Module.rank of the connection pairing's row span. RS.edgeRankBounded_iff_submatrixRank proves equivalence with the paper's supremum of finite-submatrix ranks (RS/Novel/Skein/ConnectionRank.lean). (2) Rank base. The paper allows real R ≥ 1; EdgeRankBounded and RS.regts_sevenster_total use natural R, including zero. Section 7.6 derives the sharp real budget from the formal results: construct a model at B = ⌈R⌉, then identify its dimension with its even rank-growth limit, bounded by R. (3) Dimension estimate. For d = k + 2ℓ and d,n ≥ 1, Lemma 4.9 gives d^(2n) ≤ rk M_{f,2n} binom(n + d² − 1, d² − 1). It uses the sum of surviving block dimensions (Lemma 4.8) and Cauchy–Schwarz. Lean uses individual block faithfulness (Lemma A.4) to prove d^(2n) ≤ rk M_{f,2n} (n + 1)^(2d²). Both estimates count simultaneous word-pair orbits to bound the commutant in the ordinary endomorphism algebra of the total vector space, including parity-reversing maps. Both give d ≤ R and the same rank-growth limit. (4) Intrinsic dimension. Lean defines minimumColourDimension as the least total colour bound and proves that the even rank roots tend to it. The paper defines the growth b", "alignment": true, "original": true, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "" }, "canonical": { "repo": "willwhistler/regts-sevenster", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-19-000003", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-19-000003", "commit": "bb0a60543d48afac3f36938104b5c85c7c0396c7", "trust": "high", "theorems": 7, "date": "2026-09-06" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "wstrinz/configuration-23-4/formalization.yaml", "repo": "wstrinz/configuration-23-4", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/wstrinz/configuration-23-4/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "A real affine (23_4) configuration", "description": "An explicit construction of 23 distinct points and 23 distinct straight lines in the real affine plane, with exactly four selected incidences at every point and every line. Lean verifies the construction using rational coordinates interpreted in the real numbers.", "authors": [ "Will Strinz" ], "maintainers": [ "Will Strinz" ], "license": "MIT", "role": "substantive-development", "substantive": "", "sources": [ { "title": "The explicit rational (23_4) construction supplied in this development", "id": "", "authors": [], "type": "original-proof", "relationship": "other", "endorsement": "" }, { "title": "(22_4) and (26_4) configurations of lines", "id": "https://doi.org/10.26493/1855-3974.1402.733", "authors": [ "Michael Cuntz" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Eventually, geometric (n_k) configurations exist for all n", "id": "https://arxiv.org/abs/2104.00045", "authors": [ "Leah Wrenn Berman", "Gábor Gévay", "Tomaž Pisanski" ], "type": "preprint", "relationship": "background", "endorsement": "not-contacted" } ], "related": [], "classification": { "arxiv": [ "math.CO", "math.MG" ], "msc2020": [ "51A45", "52C30" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "OpenAI Codex (GPT-6)" ], "framework": "Codex" } ], "strongest": "agent", "models": [ "OpenAI Codex (GPT-6)" ], "spend": "not tracked", "notes": "Will Strinz directed the research and requested independent exact checks. Codex generated and audited the formalization and packaging. Search and computer algebra aided discovery; the formal theorem depends only on the explicit coordinate proof. See VERIFICATION.md for executed checks." }, "scope": "The compared theorem is unconditional existence over the real affine plane, expressed by finite lists of coordinate pairs and nondegenerate, pairwise nonproportional affine line equations. Counts include every selected point-line pair. Rationality is visible in the explicit coordinate expressions but is not a separate quantified theorem. No census classification, uniqueness theorem or publication-priority theorem is asserted. Challenge has one deliberate theorem hole; the proof-bearing Solution has none.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "Config23.exists_configuration", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "Uses the finite real affine plane rather than homogeneous projective coordinates. A projective change puts all selected points in this chart. The finite affine construction is sufficient for projective existence. Affine lines are represented by coefficient triples with explicit nondegeneracy and nonproportionality conditions; see STATEMENT_AUDIT.md.", "alignment": true, "original": true, "review": { "status": "self-assessed with automated adversarial checks", "bucket": "self-assessed", "reviewers": [], "notes": "No independent human peer review or source-author endorsement is claimed. The maintainer approved authorship and licensing. Mechanical evidence and its limits are recorded in VERIFICATION.md. Palomar review and registration have not occurred." }, "canonical": { "repo": "wstrinz/configuration-23-4", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-08-000004", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-08-000004", "commit": "94fc8964562fa9e246c8c7b1657e2e135bda35f6", "trust": "high", "theorems": 1, "date": "2026-09-08" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "wstrinz/plane-jacobian-72-108/lean/formalization.yaml", "repo": "wstrinz/plane-jacobian-72-108", "path": "lean/formalization.yaml", "directory": "lean", "origins": [ "search" ], "url": "https://github.com/wstrinz/plane-jacobian-72-108/blob/HEAD/lean/formalization.yaml", "version": "v0.4", "missing": [], "name": "The f37 resultant-excess branch in the plane (72,108) Jacobian case", "description": "A Lean 4 formalization of one exact component theorem from the plane (72,108) Jacobian-candidate analysis: an explicit degree-31 polynomial f31 in eight variables over the rationals belongs to the ideal generated by four explicit pre-resultant equations. Consequently f31 vanishes at every common rational zero of those equations. In the source reduction this identifies the f37 resultant branch as excess. The registered statements do not formalize or claim the full exclusion of the (72,108) degree case.", "authors": [ "Will Strinz" ], "maintainers": [ "Will Strinz" ], "license": "MIT", "role": "", "substantive": "", "sources": [ { "title": "The f37 branch is a resultant artifact", "id": "https://doi.org/10.5281/zenodo.21534895", "authors": [ "Will Strinz" ], "type": "other", "relationship": "formalizes", "endorsement": "participated" }, { "title": "Exact Computer-Assisted Exclusion of the (72,108) Frontier in the Two-Dimensional Jacobian Problem", "id": "https://doi.org/10.5281/zenodo.21479814", "authors": [ "Billel Helali" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Increasing the degree of a possible counterexample to the Jacobian Conjecture from 100 to 108", "id": "https://arxiv.org/abs/2204.14178", "authors": [ "Jorge A. Guccione", "Juan J. Guccione", "Rodrigo Horruitiner", "Christian Valqui" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" } ], "related": [], "classification": { "arxiv": [ "math.AG" ], "msc2020": [ "14R15" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "OpenAI Codex", "Anthropic Claude" ], "framework": "Codex and Claude Code" } ], "strongest": "agent", "models": [ "Anthropic Claude", "OpenAI Codex" ], "spend": "not separately metered", "notes": "The historical certificate used native_decide and is not the submitted proof. The Palomar Solution must use only proof-producing kernel-checked steps and the permitted axioms listed in comparator.json." }, "scope": "The compared declarations are exactly the rational ideal-membership theorem f31 in span {G1,G2,G3,G4} and its common-rational-zero consequence. They do not formalize the upstream Newton-polygon reduction, the other branches of the (72,108) proof, or the full degree exclusion. The Challenge contains deliberate placeholders. The proof-bearing Solution is complete and its compared declarations build with exactly propext, Classical.choice, and Quot.sound. A clean full Comparator, Lean4Export/Landrun, NanoDa, and Lean-kernel replay has passed on the public release candidate recorded in PALOMAR_REPL", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "PalomarF37.f31_mem_preResultant", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "PalomarF37.f31_eq_zero_of_preResultant_zero", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The formalization deliberately isolates one exact ideal-membership component from a much larger source argument. It does not formalize the interpretation of this component as a complete exclusion of the (72,108) degree case. The common-zero corollary quantifies over rational points; the ideal-membership theorem itself is the stronger algebraic statement over Q.", "alignment": true, "original": false, "review": { "status": "full-local-replay-passed", "bucket": "other", "reviewers": [ "OpenAI Codex adversarial statement and packaging review", "Grand Portage publication audit" ], "notes": "This status records pre-submission automated and maintainer-directed review, not peer review and not Palomar editorial review. The clean Solution build, axiom census, Comparator, Lean4Export/Landrun, NanoDa, and Lean default kernel pass. Palomar intake and editorial review remain external." }, "canonical": { "repo": "wstrinz/plane-jacobian-72-108", "directory": "lean" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "wstrinz/plane-jacobian-75-125/formalization/formalization.yaml", "repo": "wstrinz/plane-jacobian-75-125", "path": "formalization/formalization.yaml", "directory": "formalization", "origins": [ "search" ], "url": "https://github.com/wstrinz/plane-jacobian-75-125/blob/HEAD/formalization/formalization.yaml", "version": "v0.4", "missing": [], "name": "A local K5 valuation criterion for the degree-(75,125) Jacobian frontier", "description": "A Lean proof of the corrected local K5 carrier criterion used in an exact study of the degree-(75,125) plane Jacobian frontier. For the displayed denominator-cleared K5 numerator, guarded double-root order hypotheses imply that order-twenty vanishing is equivalent to vanishing of P12 at the root. After factorization, one exceptional order-19 term controls the result.", "authors": [ "Will Strinz" ], "maintainers": [ "Will Strinz" ], "license": "MIT", "role": "", "substantive": "", "sources": [ { "title": "The local K5 carrier criterion presented in this repository", "id": "https://github.com/wstrinz/plane-jacobian-75-125", "authors": [], "type": "original-proof", "relationship": "other", "endorsement": "participated" }, { "title": "On the shape of possible counterexamples to the Jacobian Conjecture", "id": "arXiv:1401.1784", "authors": [ "Jorge A. Guccione", "Juan J. Guccione", "Christian Valqui" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" } ], "related": [], "classification": { "arxiv": [ "math.AG", "math.RA" ], "msc2020": [ "14R15", "13P10", "03B35" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "OpenAI Codex", "Anthropic Claude Sonnet", "Anthropic Claude Opus" ], "framework": "Codex desktop and local research-agent orchestration" } ], "strongest": "agent", "models": [ "Anthropic Claude Opus", "Anthropic Claude Sonnet", "OpenAI Codex" ], "spend": "subscription-based and not separately tracked", "notes": "The final theorem was selected during publication hardening after an adversarial audit corrected an earlier overclaim on the m>=3,k=1 stratum." }, "scope": "Proves only the displayed local polynomial criterion over an arbitrary characteristic-zero field. It does not formalize the reduction of an actual degree-(75,125) Keller pair to the carrier model, does not prove the missing substantive actual-source/carrier attachment, does not close the six open Gate-B cells, and does not exclude or construct a degree-(75,125) counterexample.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "PlaneJacobian75125.k5_vanishes_to_order_twenty_iff", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The formal theorem isolates the denominator-cleared local polynomial criterion. It does not formalize the surrounding carrier derivation that identifies this numerator inside the Family-F2 model, nor the open attachment from that model to an actual Keller pair.", "alignment": true, "original": true, "review": { "status": "self-assessed with automated adversarial review; no external human peer review", "bucket": "self-assessed", "reviewers": [ "Will Strinz" ], "notes": "Multiple independent agent audits checked theorem scope, found and repaired the missing P12(rho)=0 survivor branch, and replayed exact positive and hostile fixtures. The curated release battery passes in the maintained release checkout; a final replay from the eventual pinned public commit is still pending. No independent mathematician has yet refereed the Lean statement or proof." }, "canonical": { "repo": "wstrinz/plane-jacobian-75-125", "directory": "formalization" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "x67ai/riemann-rh-program/rh-program/lean/formalization.yaml", "repo": "x67ai/riemann-rh-program", "path": "rh-program/lean/formalization.yaml", "directory": "rh-program/lean", "origins": [ "search" ], "url": "https://github.com/x67ai/riemann-rh-program/blob/HEAD/rh-program/lean/formalization.yaml", "version": "v0.4", "missing": [], "name": "rh-program Lean additions (de Bruijn–Newman M2a, W1 checker, A4 grid theorems, Theorem M2 Lemma G)", "description": "The Lean 4 / Mathlib files this program adds on top of the Zeta23 library. Headline unit: the de Bruijn–Newman milestone M2a — the ray form of Λ ≤ 0.2 at Polymath15 Table 1 row 2, \"kernel-checked modulo H1, H2 (H2-B, H2-A, H-TAIL), H3\" (never \"fully machine-checked\"): two independent producers' barrier and asymptotic transcripts are kernel-checked integer certificates (`decide +kernel`, no `native_decide`), the soundness theorems and the glue are Lean proofs, and the five hypotheses named in the label are DISPLAYED, never discharged. Λ ≤ 0.2 is not proved; the bracket of record stays 0 ≤ Λ ≤ 0", "authors": [ "Kunal Tyagi (sponsor; directs the program, writes no Lean)", "Claude (Anthropic) — every Lean file, under the sponsor's direction" ], "maintainers": [ "Kunal Tyagi" ], "license": "Apache-2.0", "role": "substantive-development", "substantive": "", "sources": [ { "title": "D1 M2a contract: SPEC.md — the certificate, the two lanes, the Lean contract (§1.1 target theorem; §3.7 label; §8 shapes)", "id": "rh-program/results/d1-m2a/SPEC.md", "authors": [ "rh-program (Claude under Kunal Tyagi's direction)" ], "type": "other", "relationship": "formalizes", "endorsement": "n/a" }, { "title": "M2a/M2b design note (m2a-m2b-design.md, the D-R2 repair note): ray form §1.2, trust model, work breakdown", "id": "rh-program/results/d1-m0/m2a-m2b-design.md", "authors": [ "rh-program (Claude under Kunal Tyagi's direction)" ], "type": "other", "relationship": "formalizes", "endorsement": "n/a" }, { "title": "Effective approximation of heat flow evolution of the Riemann ξ function, and a new upper bound for the de Bruijn–Newman constant", "id": "https://doi.org/10.1007/s40687-019-0193-1", "authors": [ "D. H. J. Polymath" ], "type": "article", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "The Riemann hypothesis is true up to 3·10¹²", "id": "https://doi.org/10.1112/blms.12460", "authors": [ "Dave Platt", "Tim Trudgian" ], "type": "article", "relationship": "background", "endorsement": "not-contacted" }, { "title": "A4 no-go paper: grid-Parseval decoupling and the corner theorem (rh-program/results/a4-no-go/paper.md)", "id": "rh-program/results/a4-no-go/paper.md", "authors": [ "rh-program (Claude under Kunal Tyagi's direction)" ], "type": "other", "relationship": "formalizes", "endorsement": "n/a" }, { "title": "D1 M1 v1 W1 transcript format and checker (FORMAT.md; the v1.1 discharge of the argument principle)", "id": "rh-program/results/d1-m1/FORMAT.md", "authors": [ "rh-program (Claude under Kunal Tyagi's direction)" ], "type": "other", "relationship": "formalizes", "endorsement": "n/a" } ], "related": [ { "id": "https://github.com/anthropics/zeta-23-lean", "relationship": "builds-on", "note": "The parent library (Copyright 2026 Anthropic, PBC, Apache-2.0; Lean v4.33.0-rc2, Mathlib 51e6992efd06126df61a496bebf8f49482a4e129). These files are additions to it and are not part of it; its Comparator layout (comparator/README.md, \"one topic per file\") is the pattern the topic `DBN` follows." }, { "id": "https://github.com/judegomila/dbn-lambda-01787854-candidate-audit", "relationship": "adapts", "note": "Zeta23/W1/ArgPrinciple/{Rect,General}.lean are ported, statement for statement, from that repository's branch lean/certificate-and-argument-principle (commit ea09b2f; Copyright 2026 Jude Gomila, MIT; generated with Harmonic Aristotle); Zeta23/W1/ArgPrincipleBridge.lean adapts seven of its lemmas. Record: rh-program/results/d1-m1/v11/port-notes.md, NOTICE." } ], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11M26", "11Y35", "68V20" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Claude Fable 5.1", "Claude Opus 5" ], "framework": "Claude Code (Workflow/Agent)" } ], "strongest": "agent", "models": [ "Claude Fable 5.1", "Claude Opus 5" ], "spend": "", "notes": "Independent checks of the packaging: an Opus 5 clean-clone check (results/d1-m2a/packaging/CHECK-O.md) and the Comparator run with the nanoda kernel (results/d1-m2a/packaging/COMPARATOR-RUN.md) are the next two jobs of the same queue item; their outcomes are to be appended to `review.notes` verbatim." }, "scope": "Complete for the stated units: the M2a ray theorem modulo its five displayed hypotheses (both producer legs), the W1 checker soundness modulo H-ENCL, the A4 grid theorems. Nothing under Zeta23/DBN, Zeta23/W1 or Zeta23/PairCeiling contains a `sorry`. The only `sorry`s in the program's files are the deliberate placeholders of the Comparator CHALLENGE side, which states the theorems with `sorry` by design (as the parent's Challenge files do): the seven of comparator/Challenge/DBN.lean (proved by comparator/Solution/DBN.lean) and, from Session 23, the one of comparator/Challenge/SeparationG1.lean ", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "dbn_ray_le_point2_mp", "file": "comparator/Solution/DBN.lean", "description": "(I) Λ ≤ 0.2 in ray form, mpmath-ball leg — the referee's statement: from the five displayed hypotheses H1, H2-B (`row2BarrierMP`), H2-A and H-TAIL (`row2AsymMP`), H3, every H_t with t ≥ 1/5 has only real zeros. Kernel-checked modulo H1, H2 (H2-B, H2-A, H-TAIL), H3.", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [ { "statement": "H1: no zero of ζ with 116733/200000 ≤ Re s ≤ 1, 0 ≤ Im s ≤ 2 500 000 097 429 (displayed hypothesis)", "source": "Platt–Trudgian, Theorem 1" }, { "statement": "H3: Polymath15 Theorem 1.2 in the form `Polymath15Bridge'` (SPEC §3.3, derivation D-H3), and the entirety of H_t (displayed hypotheses)", "source": "Polymath15, Theorem 1.2, p2–3, Proposition 3.3, Theorem 3.2" }, { "statement": "H2-B, H2-A, H-TAIL: the producers' enclosure claims for the kernel-checked transcripts (displayed hypotheses; Lemma T's tail reduction in prose)", "source": "SPEC §4–§5; results/d1-m2a/{INSTANCE-REPORT.md, lane-a/}" } ] }, { "declaration": "dbn_ray_le_point2_arb", "file": "comparator/Solution/DBN.lean", "description": "(I) Λ ≤ 0.2 in ray form, Arb/FLINT leg — the same from the independent producer's transcripts `row2BarrierARB`, `row2AsymARB` (the legs are never merged).", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "dbn_ray_le_point2_of_certificates", "file": "comparator/Solution/DBN.lean", "description": "(G) Generic soundness, literal-free: for ANY barrier data on the instance rectangle/final time that `checkBarrier` accepts and ANY asymptotic data with the instance's t₀, y₀, yA and a row at or below N_start = 630783 that `checkAsym` accepts, the five displayed hypotheses imply the ray conclusion. Derived on the solution side (Zeta23 proves only the instance form; fidelity item (f)).", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "dbn_row2BarrierMP_checked, dbn_row2BarrierARB_checked", "file": "comparator/Solution/DBN.lean", "description": "(K) The trusted copies of the two barrier transcripts pass the integer checker `checkBarrier` (39 + 72 prisms, 17 947 rows); integer facts on literals, nothing analytic.", "axioms": [ "propext", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "dbn_row2AsymMP_checked, dbn_row2AsymARB_checked", "file": "comparator/Solution/DBN.lean", "description": "(K) The trusted copies of the two Lane A transcripts pass the integer checker `checkAsym` (C-A1…C-A6; C-A6 checked, consumed by no proof).", "axioms": [ "propext" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "Zeta23.DBN.Instance02.lambda_le_point2, Zeta23.DBN.Instance02.lambda_le_point2_arb", "file": "Zeta23/DBN/Instance02.lean", "description": "The Zeta23-side originals of (I): the ray theorem at row 2 with the five displayed hypotheses, one theorem per producer leg (results/d1-m2a/lane-a/final-axioms.log).", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "Zeta23.DBN.cert_of_checkBarrier", "file": "Zeta23/DBN/BarrierCert.lean", "description": "Lane B soundness: an accepted barrier certificate (chain + per-prism checker facts) with H2-B and the holomorphy of G t near R gives G t ≠ 0 on the closed rectangle for every t ∈ [0, t₀] — the argument principle on rectangles is a theorem here, no Rouché, no zero-continuity in t (results/d1-m2a/barriercert-axioms.log).", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "Zeta23.DBN.cert_of_checkAsym", "file": "Zeta23/DBN/Asym.lean", "description": "Lane A soundness: an accepted asymptotic certificate with H2-A and H-TAIL gives g ≠ 0 for N(x) at or beyond some row's Nlo and y ∈ [y₀, yA] (coverage kernel-checked; results/d1-m2a/lane-a/asym-axioms.log).", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "Zeta23.W1.cert_of_checkW1_ap", "file": "Zeta23/W1/ArgPrincipleBridge.lean", "description": "W1 checker soundness with the argument principle DISCHARGED (v1.1): an accepted ζ transcript gives the exclusion conclusion modulo the single displayed hypothesis H-ENCL (results/d1-m1/v11/audit/audit-print-axioms.log).", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "Zeta23.W1.cert_of_checkW1_fDH, Zeta23.W1.mpDH_zero, Zeta23.W1.arbDH_zero, Zeta23.W1.differentiable_fDH", "file": "Zeta23/W1/FDH.lean", "description": "D-R8 (2026-09-10): W1 checker soundness for the Davenport–Heilbronn function f_DH — the twin of `cert_of_checkW1_ap` with `fDH` for `riemannZeta`, modulo the single displayed hypothesis H-ENCL_DH (`W1EnclOK fDH d`); `fDH` is FORMAT §9.2's 5^{-s}[ζ(s,1/5) + κζ(s,2/5) − κζ(s,3/5) − ζ(s,4/5)] on Mathlib's `hurwitzZeta`, entire (`differentiable_fDH`). With the live-fire instance corollaries `mpDH_zero", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "Zeta23.W1.cert_of_checkW1_of_diffOn, Zeta23.W1.continuousOn_logDeriv_seg_of_diffOn", "file": "Zeta23/W1/Soundness.lean", "description": "D-R8 (2026-09-10): the W1 soundness theorem generic in the function — for any f differentiable on {Re s < 1}, modulo H-ENCL (`W1EnclOK f d`) and H-AP (`RectArgPrinciple f`, a theorem for every f since v1.1); `cert_of_checkW1` (ζ) is its instance with the v1 statement unchanged (results/d1-m2a/dr8/no-regression.log).", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "Zeta23.W1.cert_of_checkW1_of_diffOn'", "file": "Zeta23/W1/Soundness.lean", "description": "Session 21 (2026-09-10), the σ-strong sibling owed by D-R8 (results/d1-m2a/dr8/CHECK-fDH-O.md §12 FIX-FIRST 1): the W1 soundness theorem generic in the function with the witness branch keeping the rectangle bounds — for m ≥ 1 a zero ρ of f with sigma1 d < Re ρ < sigma2 d and T1 d < Im ρ < T2 d (the open box of the transcript), same m = 0 branch, same hypotheses as `cert_of_checkW1_of_diffOn`, whos", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "Zeta23.W1.cert_of_checkW1_fDH'", "file": "Zeta23/W1/FDH.lean", "description": "Session 21 (2026-09-10): W1 checker soundness for f_DH in σ-strong form — `cert_of_checkW1_of_diffOn'` at f = fDH with H-AP from `rectArgPrinciple_of_local fDH`; modulo the single displayed hypothesis H-ENCL_DH, an accepted transcript with m ≥ 1 gives a zero of f_DH in the open box (sigma1 d, sigma2 d) × (T1 d, T2 d) (results/d1-m2a/dr8/sigma-strong-axioms.log).", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "Zeta23.W1.mpDH_zero'", "file": "Zeta23/W1/FDH.lean", "description": "Session 21 (2026-09-10): the box-form live-fire corollary, mpmath-ball leg — from the kernel-decided `mpDH_check` and H-ENCL_DH for the unchanged literal `mpDH`: a zero ρ of f_DH with 4/5 < Re ρ < 41/50, 1/2 < Re ρ < 1 and 85.69 < Im ρ < 85.71 (unfolding lemmas `mpDH_sigma1`, `mpDH_sigma2`, `mpDH_T1`, `mpDH_T2`). Label, verbatim and for the primed theorems only (dr8/PRICING-fDH.md §3.2): \"f_DH has", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "Zeta23.W1.arbDH_zero'", "file": "Zeta23/W1/FDH.lean", "description": "Session 21 (2026-09-10): the box-form live-fire corollary, Arb/FLINT leg — the same statement from `arbDH_check` and H-ENCL_DH for the unchanged literal `arbDH` (same rectangle, m = 1; the two legs are never merged, D-R3). Same label as `mpDH_zero'`; the unprimed `arbDH_zero` keeps the half-strip label.", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "Zeta23.W1.mpNullT100_exclusion, Zeta23.W1.arbNullT100_exclusion, Zeta23.W1.mpNullT1000_exclusion, Zeta23.W1.arbNullT1000_exclusion, Zeta23.W1.mpNullT10000_exclusion, Zeta23.W1.arbNullT10000_exclusion, Zeta23.W1.mpNullDeepT100_exclusion, Zeta23.W1.arbNullDeepT100_exclusion", "file": "Zeta23/W1/Ledger.lean", "description": "M3 exclusion-ledger seed (2026-09-10): the eight 3-line corollaries of `cert_of_checkW1_ap` (m = 0 branch) on the eight M1 acceptance null literals — one per leg of the four seed rows of results/d1-m3/ (R1 [3/5, 9/10] × [100, 101], R2 × [1000, 1001], R3 × [10000, 10001], R4 [21/40, 39/40] × [100, 101]): `W1EnclOK riemannZeta d → ∀ s ∈ W1Rect d, riemannZeta s ≠ 0`. Label per row: \"no zeros of ζ in ", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "separation_braw_iteratedDeriv_le (root, Challenge.SeparationG1 / Solution.SeparationG1); Zeta23.Separation.abs_iteratedDeriv_Braw_le, Zeta23.Separation.Braw_eq_mul", "file": "comparator/Solution/SeparationG1.lean", "description": "Session 23 (2026-09-17), M4 (i) dress-rehearsal rung — Lemma G1 of Theorem M2's separation note (results/c2-m2/separation-note.md §2): for every k ≥ 1 and every real v, |iteratedDeriv k Braw v| ≤ (k + 1)·(72/e)^k·k^{2k}, with Braw v = exp(−1/(1 − 4v²)) on |v| < 1/2 and 0 elsewhere (the trusted definition in comparator/ChallengeDeps/Separation.lean, over Mathlib alone). NO displayed hypothesis. Pro", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "separation_bump_fourier_decay, separation_bump_fourier_decay_complex, separation_bump_fourier_decay_pure_exp (root, Challenge.Separation / Solution.Separation); Zeta23.Separation.norm_paperFT_Bc_ofReal_le, Zeta23.Separation.norm_paperFT_Bc_le, Zeta23.Separation.norm_paperFT_Bc_ofReal_le', Zeta23.Separation.paperFT_iteratedDeriv", "file": "comparator/Solution/Separation.lean", "description": "Session 23 (2026-09-17), M4 (i) — Lemma G of Theorem M2's separation note (results/c2-m2/separation-note.md §2), three statements over the trusted layer comparator/ChallengeDeps/Separation.lean (Braw, Z = ∫_{−1/2}^{1/2} Braw, B = Braw/Z, ft = the paper's transform ∫ f(u)e^{izu}du, cB = 2/√(72e), CB = e²/Z, all from Mathlib alone): (G1) ∀ η : ℝ, ‖ft B η‖ ≤ CB·(1 + (cB/2)√|η|)·exp(−cB√|η|); the comp", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "separation_clause4_in_window_noise (root, Challenge.SeparationClause4 / Solution.SeparationClause4); Zeta23.Separation.inWindowNoise_le, Zeta23.Separation.shell_sum_le, Zeta23.Separation.paperFT_ftest", "file": "comparator/Solution/SeparationClause4.lean", "description": "Session 23 (2026-09-17), M4 (iii) dress-rehearsal rung — clause 4 of Theorem M2's separation note (results/c2-m2/separation-note.md §5), the in-window on-line noise, over the trusted class 𝒞(C₁) (comparator/ChallengeDeps/SepConfig.lean, `Separation.SepConfig C₁`): for C₁ ≥ 1, t ≥ 3, L ≥ 50, R ≥ 1 and every Z ∈ 𝒞(C₁), 0 ≤ N_Z ≤ 2·(87/10)·C₁·log(4 + t + R)/L², N_Z = Σ_{ρ ∈ Z, Re ρ = 1/2, |Im ρ − t| ", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "separation_clause6_assembly (root, Challenge.Separation6 / Solution.Separation6); Zeta23.Separation.clause6_assembly, Zeta23.Separation.W_decomp, Zeta23.Separation.re_tsum_offline_le, Zeta23.Separation.rstar_exp, Zeta23.Separation.edge_sq_ge, Zeta23.Separation.absorb", "file": "comparator/Solution/Separation6.lean", "description": "Session 23 (2026-09-17), M4 (iii) — the clause-6 assembly of Theorem M2 (results/c2-m2/separation-note.md §7.1, hypotheses as the addendum A1–A3 restates them) over the trusted class 𝒞(C₁): for C₁ ≥ 1, t ≥ 3, 0 < δ ≤ 1/2, L ≥ 25/δ, L ≥ (4/δ)(log log(3 + t) + 2log(1/δ) + log(2·(87/10)·C₁)), the reflection condition (R*) (clause 1′, in the note's logarithmic form), Z, Z′ ∈ 𝒞(C₁) with the orbit 1/2 ±", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "separation_clause4_in_window_noise_b (root, Challenge.SeparationClause4b / Solution.SeparationClause4b); Zeta23.Separation.inWindowNoise_le_b, Zeta23.Separation.hb1sym, Zeta23.Separation.integral_norm_deriv_Bc, Zeta23.Separation.four_le_b1sym", "file": "comparator/Solution/SeparationClause4b.lean", "description": "Session 24 item 3, Unit A (2026-09-24) — clause 4 of Theorem M2 over 𝒞(C₁) with b₁ PROVED: for C₁ ≥ 1, t ≥ 3, L ≥ 50, R ≥ 1 and Z : `Separation.SepConfig C₁`, 0 ≤ N_Z ≤ 2·b₁sym·C₁·log(4 + t + R)/L² with b₁sym = (2e^{−1}/Z)² = ‖B′‖₁² — MODULO the displayed H-B‴ only. H-b₁ of the M4 (iii) rung is gone: the solution proves (|η|·‖B̂(η)‖)² ≤ b₁sym for every real η (Zeta23/Separation/B1Sym.lean: `abs_mu", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "separation_clause6_assembly_b, separation_clause7_tight_pair, rstar_of_21L (root, Challenge.Separation6b / Solution.Separation6b); Zeta23.Separation.clause6_assembly_b, Zeta23.Separation.absorb_b, Zeta23.Separation.clause7_tight_pair, Zeta23.Separation.wsum_double_plus, Zeta23.Separation.wsum_double_minus, Zeta23.Separation.rstar_of_21L, Zeta23.Separation.one_div_eighteen_le_Z", "file": "comparator/Solution/Separation6b.lean", "description": "Session 24 item 3, Unit A (2026-09-24) — the clause-6 assembly of Theorem M2 with b₁ PROVED, the clause-7 tight pair, and \"t ≥ 21L ⟹ (R*)\" (results/c2-m2/separation-note.md §7.1, §7.2, addendum A1; PRICING-RESIDUE §1 Pieces 2, 3 (3-sym), 6). (1) `separation_clause6_assembly_b`: the M4 (iii) assembly statement with `hb1 : Hb1` DELETED and the L-hypothesis L ≥ (4/δ)(log log(3 + t) + 2log(1/δ) + log(", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "separation_clause5_summable (root, Challenge.SeparationClause5Sum / Solution.SeparationClause5Sum); Zeta23.Separation.clause5_summable, Zeta23.Separation.norm_wsum_le_strip, Zeta23.Separation.out_finset_sum_le, Zeta23.Separation.out_window_norm_le, Zeta23.Separation.crude_bound", "file": "comparator/Solution/SeparationClause5Sum.lean", "description": "Session 25 item 2, Unit B, the dress-rehearsal rung (2026-09-24) — clause 5's SUMMABILITY (results/c2-m2/separation-note.md §6 (b), absolute convergence; PRICING-RESIDUE §1 Piece 1, the rung): for every Z ∈ 𝒞(C₁), t ≥ 3, L > 0 the out-window family ρ ↦ ‖m_ρ h_f(γ_ρ) conj(h_f(conj γ_ρ))‖ over |Im ρ − t| > 73L, f = f_{t,L}, is summable — the FIRST conjunct of the displayed `SepConfig.Hout` at R = 73", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "separation_clause5_out_window, separation_clause6_assembly_c (root, Challenge.Separation5 / Solution.Separation5); Zeta23.Separation.clause5_out_window, Zeta23.Separation.clause6_assembly_c, Zeta23.Separation.shell_series_le, Zeta23.Separation.integral_pow_mul_exp_Ioi, Zeta23.Separation.hasDerivAt_exp_mul_gammaPoly, Zeta23.Separation.integral_phi_Ioi, Zeta23.Separation.sum_phi_le, Zeta23.Separation.absorb_once, Zeta23.Separation.F13_le_F13_50", "file": "comparator/Solution/Separation5.lean", "description": "Session 25 item 2, Unit B, leg 1(a) (2026-09-24) — clause 5 of Theorem M2 modulo H-R₀ (results/c2-m2/separation-note.md §6, addendum A3 with R₀ = 73 and the L-hypothesis spent once; PRICING-RESIDUE §1 Piece 1 items 1–10), and the clause-6 assembly with H-out REPLACED by H-R₀. (1) `separation_clause5_out_window`: for Z ∈ 𝒞(C₁), C₁ ≥ 1, t ≥ 3, L ≥ 50, 8(log log(3 + t) + log(2b₁C₁)) ≤ L and the displ", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "separation_clause5_out_window_c, separation_clause6_assembly_d (root, Challenge.Separation5c / Solution.Separation5c); Zeta23.Separation.F13_50_neg, Zeta23.Separation.Z_enclosure, Zeta23.Separation.cert_numeric, Zeta23.Separation.cB_bounds, Zeta23.Separation.sqrt73_bounds, Zeta23.Separation.CB_le_CBHi, Zeta23.Separation.b1Lo_le_b1sym, Zeta23.Separation.Prelax_50_le, Zeta23.Separation.riemann_bracket, Zeta23.Separation.Braw_antitoneOn", "file": "comparator/Solution/Separation5c.lean", "description": "Session 25 item 2, Unit B, leg 1(b) + Piece 8 (2026-09-24) — H-R₀ DISCHARGED: clause 5 of Theorem M2 with no displayed hypothesis, and the clause-6 assembly modulo H-edge and H-B‴ only (PRICING-RESIDUE §1 Piece 1 item 11, Piece 8 at n = 16). (1) `separation_clause5_out_window_c`: the 1(a) statement with `hR0` REMOVED — for Z ∈ 𝒞(C₁), C₁ ≥ 1, t ≥ 3, L ≥ 50, 8(log log(3 + t) + log(2b₁C₁)) ≤ L, `Z.Ho", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "Zeta23.PairCeiling.GridParseval.flat_band_trace_sq, Zeta23.PairCeiling.GridParseval.trace_sq_grid", "file": "Zeta23/PairCeiling/GridParseval.lean", "description": "A4 grid-Parseval decoupling: the paper's Lemma 3.8 (`flat_band_trace_sq`) and Theorem 3.9 (`trace_sq_grid`) (results/a4-no-go/formalization-status.md).", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] }, { "declaration": "Zeta23.PairCeiling.GridCorner.mark_one_count_ge, Zeta23.PairCeiling.GridCorner.grid_corner_pointwise, Zeta23.PairCeiling.GridCorner.grid_corner_law, Zeta23.PairCeiling.GridCorner.grid_corner_attained", "file": "Zeta23/PairCeiling/GridCorner.lean", "description": "A4 corner theorem: the paper's Lemma 4.2 (`mark_one_count_ge`), Theorem 4.3 pointwise (`grid_corner_pointwise`), in law form (`grid_corner_law`) and with exact attainment (`grid_corner_attained`) (results/a4-no-go/formalization-status.md).", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": false, "literature": [] } ], "comparator": true, "literature_dependencies": 3, "divergences": "Where the formal statements differ from the prose one (\"Λ ≤ 0.2 from Polymath15 Theorem 1.2 at Table 1 row 2 with the numerical hypotheses verified\") and from what Zeta23 proves; full ledger with source pointers in results/d1-m2a/packaging/FIDELITY.md. (a) y-band asymmetry: the Lane A hypotheses `AsymEnclOK`/`TailOK` quantify y over [y₀, yA] with yA = 3962323/5000000 = 0.7924646, which is yA − √(157/250) = 8.975·10⁻⁸ ≈ 9.0·10⁻⁸ wider than the range y² ≤ 1 − 2t₀ = 157/250 that (ii′) needs (1.42·10⁻⁷ in the squares); so `TailOK` is \"the tail region N ≥ N₁, y ∈ [y₀, yA]\", a hair stronger than the prose (ii), not literally a sub-statement of it (EMIT-NOTES §4/§9; PLAN-REVIEW §6). (b) C-A6 (the tail row's Q₁ + Q₂ + Q₃ + Q₄ + E₁ < 2K) is kernel-checked on both literals but consumed by no proof; `cert_of_checkAsym` uses only K ≥ 1, C-A3 (only E < T per row), C-A4, C-A5; Lemma T's reduction \"Σ < 2 ⟹ nonvanishing for N ≥ N₁\" (SPEC §5.4) is prose and H-TAIL is displayed in conclusion form — \"C-A6 is kernel-checked\" must not be read as \"the tail reduction is kernel-checked\". (c) The producers are untrusted and five hypotheses are displayed — H1, H2-B (the seam rows enclose SOME holomorphic ap", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [], "notes": "No human has read comparator/Challenge/DBN.lean (or any trusted file) against the prose statement: the sponsor does not review Lean, and no third party has been asked. What stands behind \"self-assessed\": the statement-identity check (results/d1-m2a/packaging/statement-identity.log), `#print axioms` on all seven statements (print-axioms.log), the trust greps with comments stripped (trust-greps-packaging.log), the byte-for-byte comparison of the trusted literal copy with the Zeta23 modules (cmp-literal-blocks.log), and the earlier second-model audits of the underlying Zeta23 files (results/d1-m2a/lane-a/AUDIT-3d.md, results/d1-m2a/v11/AUDIT.md, results/d1-m1/v11/AUDIT.md). Pending, same queue item: the Opus 5 clean-clone check (CHECK-O.md) and the Comparator run with nanoda (COMPARATOR-RUN.m" }, "canonical": { "repo": "x67ai/riemann-rh-program", "directory": "rh-program/lean" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "xiangyazi24/palomar-hahn-hecke-appell/formalization.yaml", "repo": "xiangyazi24/palomar-hahn-hecke-appell", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/xiangyazi24/palomar-hahn-hecke-appell/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "A Hahn-series coefficientwise proof of the Hickerson--Mortenson Hecke--Appell identities", "description": "A Lean 4 verification of the coefficientwise Kronecker partial-fraction identity and the general boundary-cleared Hickerson--Mortenson Hecke--Appell identity for f_{n,n+p,n}, for all positive integers n and p.", "authors": [ "Xiang Huang" ], "maintainers": [ "Xiang Huang" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "A Hahn-Series Coefficientwise Proof of the Hickerson--Mortenson Hecke--Appell Identities", "id": "", "authors": [ "Xiang Huang" ], "type": "paper", "relationship": "formalizes", "endorsement": "" }, { "title": "Hecke-type double sums, Appell--Lerch sums, and mock theta functions, I", "id": "doi:10.1112/plms/pdu007", "authors": [ "Dean Hickerson", "Eric Mortenson" ], "type": "paper", "relationship": "independently-proves", "endorsement": "" } ], "related": [], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11F27", "33D15" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "Claude", "ChatGPT", "OpenAI Codex" ], "framework": "Lean 4 with remote build and multi-model review" } ], "strongest": "agent", "models": [ "ChatGPT", "Claude", "OpenAI Codex" ], "spend": "subscription-based", "notes": "The final submission record will retain only claims supported by the pinned public source and Comparator result." }, "scope": "Comparator checks two declarations: the cleared coefficientwise Kronecker identity used in Appendix A, and the explicit general Hickerson--Mortenson coefficient identity for every positive n,p. The proof includes signed Hahn inverse coefficients, coefficient-local finiteness, row-exponent collapse, both finite residue families, the five-index expansion of the printed correction, and its identification with the wall term.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "The formal statement records equality of every q,x,y coefficient after clearing the common Jacobi factor. It proves local finiteness at each coefficient but does not assert a global well-ordered-support theorem or a series-valued Hahn wrapper for the assembled families. The registered coefficient functions are rational-valued; the paper's integer-valued interpretation, integrality, and arbitrary-ring base change are not separate registered Lean theorems.", "alignment": true, "original": true, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [ "OpenAI Codex", "OpenAI GPT-5.6 Sol (independent max-effort audit)" ], "notes": "The paper and Lean definitions were compared claim by claim. The extracted proof closure was rebuilt on uisai2 and both selected declarations were checked to use only Palomar-permitted axioms. The independent audit found no mismatch in the quantified parameters, residue ranges, signs, parity, exponents, or five-index Hickerson--Mortenson comparison; its scope correction about global Hahn support is incorporated here." }, "canonical": { "repo": "xiangyazi24/palomar-hahn-hecke-appell", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-22-000003", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-22-000003", "commit": "b9d84fd9f905605b3956f974f50ed41ee59d54e2", "trust": "high", "theorems": 2, "date": "2026-08-22" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "xiangyazi24/palomar-qseries-prime-coefficients/formalization.yaml", "repo": "xiangyazi24/palomar-qseries-prime-coefficients", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/xiangyazi24/palomar-qseries-prime-coefficients/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Prime magnitudes and nonvanishing for a nonmultiplicative indefinite theta series over Q(sqrt(5))", "description": "For an explicitly defined difference B of two signed cone theta series over Q(sqrt(5)), Lean proves a closed nonmultiplicativity witness and a complete prime-index classification. For every rational prime p congruent to 1 modulo 10, B((p-1)/10) is one of -2, -1, 1, 2; a norm-negative golden integer generator and its strict fundamental-window normalization exist; and the coefficient has magnitude two exactly when the resulting geometric residue invariant is one modulo three. The invariant is also proved independent of the generator within the explicitly defined signed fundamental-unit orbit.", "authors": [ "Xiang Huang" ], "maintainers": [ "Xiang Huang" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Prime magnitudes and nonvanishing for a nonmultiplicative indefinite theta series over Q(sqrt(5))", "id": "", "authors": [ "Xiang Huang" ], "type": "original-proof", "relationship": "other", "endorsement": "participated" }, { "title": "Ramanujan's cubic continued fraction and an analog of his most beautiful identity", "id": "doi:10.1142/S1793042110003150", "authors": [ "Hei-Chi Chan" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "An Invitation to q-Series: From Jacobi's Triple Product Identity to Ramanujan's 'Most Beautiful Identity'", "id": "", "authors": [ "Hei-Chi Chan" ], "type": "book", "relationship": "background", "endorsement": "not-contacted" }, { "title": "Notes on Mitsui's Prime Number Theorem with Siegel zeros", "id": "arXiv:2209.11816v2", "authors": [ "Wataru Kai" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" } ], "related": [ { "id": "xiangyazi24/Q-series-and-Chan-s-work", "relationship": "other", "note": "Private canonical Lean development from which the 28-file QseriesFormalization/ transitive proof closure in this public submission was extracted at commit 95573179f143e85dfc551d896fe9be6ca472250e. It is not a build dependency; this public repository contains the copied substantive proof closure." } ], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11F27", "11R11" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "OpenAI Codex", "Claude Fable", "ChatGPT" ], "framework": "Codex, Claude Code, and the ChatGPT web bridge" } ], "strongest": "agent", "models": [ "ChatGPT", "Claude Fable", "OpenAI Codex" ], "spend": "subscription-based usage", "notes": "Model agreement was not treated as proof evidence; the Lean kernel, Comparator, NanoDa, exact source checks, and paper/source audits were separate gates." }, "scope": "Comparator checks three declarations: a closed nonmultiplicativity witness; an unconditional four-value prime classification together with norm-negative generator existence, geometric normalization for every such generator, and the universal magnitude/iota equivalence; and invariance of geometric iota under the exact signed fundamental-unit orbit relation.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "PalomarQseriesPrimeCoefficients.BCoeff_not_multiplicative_witness", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "PalomarQseriesPrimeCoefficients.prime_BCoeff_complete_classification", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "PalomarQseriesPrimeCoefficients.geometric_iota_eq_of_sameIdeal", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The registered Lean surface covers the paper's finite algebraic and geometric prime theorem. It does not formalize the specialized Hecke--Mitsui prime-element asymptotic, the resulting density corollary, or the computational Chan-kernel comparison. The public relation named sameIdeal is exactly y = +/- eps^m x; no claim equates it with Mathlib Associated or with principal-ideal equality. The invariant determines the coefficient magnitude, not its sign.", "alignment": true, "original": true, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [ "OpenAI Codex (Sol Max)", "Claude Fable", "ChatGPT" ], "notes": "Sol Max reported PASS on the full Lean/manuscript audit. Claude Fable's independent cold audit passed after its findings were repaired; a later post-fix rerun was unavailable because its quota was exhausted. ChatGPT independently reviewed the density specialization, the nonvacuous public theorem surface, the Challenge import architecture, the signed-unit orbit theorem, the metadata scope, and the strict-window obstruction. All final declarations are subject to exact-commit kernel and Comparator checks." }, "canonical": { "repo": "xiangyazi24/palomar-qseries-prime-coefficients", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-22-000009", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-22-000009", "commit": "0df90a92002da988b4cbc439f5a1c76b2759a14c", "trust": "high", "theorems": 3, "date": "2026-08-22" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "xiangyazi24/palomar-qseries-row-factorization/formalization.yaml", "repo": "xiangyazi24/palomar-qseries-row-factorization", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/xiangyazi24/palomar-qseries-row-factorization/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "A shifted indefinite theta completion over Q(sqrt(5))", "description": "A Lean 4 verification of an explicit rank-two indefinite theta completion over Q(sqrt(5)). The registered results cover the coefficientwise row-model factorization; the exact holomorphic-sign-part bridge; summability of the defining completed-theta lattice series; a specialization of Zwegers' Lemma 2.8; the weight-one T and S transformations, including the five-to-two characteristic reduction; the exact mixed boundary derivative and Bruinier--Funke image; and a kernel-checked nonharmonicity witness.", "authors": [ "Xiang Huang" ], "maintainers": [ "Xiang Huang" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "A shifted indefinite theta completion over Q(sqrt(5)): an asymmetric boundary derivative and a row-model factorization", "id": "", "authors": [ "Xiang Huang" ], "type": "paper", "relationship": "formalizes", "endorsement": "participated" }, { "title": "Mock theta functions", "id": "", "authors": [ "Sander Zwegers" ], "type": "other", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "Ramanujan's cubic continued fraction and an analog of his most beautiful identity", "id": "doi:10.1142/S1793042110003150", "authors": [ "Hei-Chi Chan" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" } ], "related": [ { "id": "xiangyazi24/Q-series-and-Chan-s-work", "relationship": "other", "note": "Private canonical development from which the 39-file transitive proof closure was extracted at commit 93ade4af0234cdd007215842f8b4fbbbb9535985. It is not a build dependency." } ], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11F27", "33D15" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "OpenAI Codex", "Claude Opus", "ChatGPT" ], "framework": "Lean 4 with remote builds and independent statement review" } ], "strongest": "agent", "models": [ "ChatGPT", "Claude Opus", "OpenAI Codex" ], "spend": "subscription-based", "notes": "The submission record retains only claims supported by the pinned public source, Comparator, and the attached manuscript." }, "scope": "Comparator checks ten Solution declarations: three finite row-model coefficient theorems, the exact completed-theta/sign-part bridge, summability of the completed theta's defining lattice series, the specialized Lemma 2.8, the T transformation, the five-term and reduced two-term S laws, the exact dbar and xi images, and the explicit nonharmonicity/nonvanishing certificate.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "PalomarQseriesRowFactorization.mk_factorization", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "PalomarQseriesRowFactorization.coneDiffH_two_mul", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "PalomarQseriesRowFactorization.coneDiffH_odd", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "PalomarQseriesRowFactorization.exact_completion_bridge", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "PalomarQseriesRowFactorization.completedTheta_summable", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "PalomarQseriesRowFactorization.zwegers_lemma28", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "PalomarQseriesRowFactorization.completedTheta_add_one", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "PalomarQseriesRowFactorization.completedTheta_S", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "PalomarQseriesRowFactorization.exact_differential_image", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "PalomarQseriesRowFactorization.completedTheta_not_harmonic", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The formalization proves the explicit convergent completed theta and its transformation and differential formulas. It does not assert uniqueness or canonicity among all possible completions, does not construct a closed vector-valued representation, and does not identify the paper's row model with an older Chan-project source object. The paper states the same limits.", "alignment": true, "original": false, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [ "OpenAI Codex (Sol max)" ], "notes": "The independent Sol max audit passed the mathematical and manuscript alignment after two publication-infrastructure repairs. The exact public commit also passes the pinned Comparator under NanoDa and Lean's default kernel." }, "canonical": { "repo": "xiangyazi24/palomar-qseries-row-factorization", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-24-000007", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-24-000007", "commit": "702c524cd8bc70867c918c23039680e7adf48bf8", "trust": "high", "theorems": 10, "date": "2026-08-24" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "xinjiegit/ben_27_formalization/formalization.yaml", "repo": "xinjiegit/ben_27_formalization", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/xinjiegit/ben_27_formalization/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "NUSLean", "description": "A Lean 4 and Mathlib proof that there are absolute constants c > 0 and p₀ such that, for every prime p ≥ p₀, the minimum size m(p) of a set A ⊆ 𝔽_p with |A| ≥ 2 and no uniquely represented sum satisfies m(p) ≥ c (log p / log log p)². Here representations are unordered two-element multisets from A, repeated summands are allowed, and \"no unique sum\" means that every such multiset has a distinct equal-sum mate.", "authors": [ "Xinjie He" ], "maintainers": [ "Xinjie He" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "A near-quadratic lower bound for sets with no unique sums", "id": "", "authors": [ "Xinjie He" ], "type": "paper", "relationship": "formalizes", "endorsement": "participated" }, { "title": "On unique sums in Abelian groups", "id": "https://doi.org/10.1007/s00493-023-00069-w", "authors": [ "Benjamin Bedert" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" }, { "title": "100 Open Problems", "id": "https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf", "authors": [ "Ben Green" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" } ], "related": [], "classification": { "arxiv": [ "math.CO", "math.NT" ], "msc2020": [ "11B30", "11B34" ] }, "automation": { "methods": [ { "method": "manual", "models": [], "framework": "n/a" }, { "method": "agent", "models": [ "OpenAI Codex (GPT-5 family)" ], "framework": "Codex desktop" } ], "strongest": "agent", "models": [ "OpenAI Codex (GPT-5 family)" ], "spend": "not tracked", "notes": "The work combined human mathematical direction with AI-assisted proof engineering. Lean's kernel checks the final proof; AI review is not a substitute for independent mathematical peer review." }, "scope": "The compared declaration NUS.m_lower_bound says exactly that there exist a real constant c with 0 < c and a natural-number threshold p₀ such that, for every natural number p, if p is prime and p₀ ≤ p, then c * (Real.log p / Real.log (Real.log p))^2 ≤ m p. Natural numbers are coerced to reals in this inequality. The definition m p is the infimum in ℕ of cardinalities k for which there is a finite A ⊆ ZMod p with A.card = k, 2 ≤ k, and no unique sum. The project also proves the stronger setwise statement for every such A and formalizes all intermediate collision-lattice, clean-rectangle, common-", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "No divergence is known in the lower-bound theorem. ZMod p represents the prime field 𝔽_p. Real.log is the natural logarithm; changing the logarithm base only changes the absolute constant. The formal definition uses the paper's stated equivalent multiset formulation: every unordered two-element multiset from A has a different unordered multiset from A with the same sum, and a repeated pair {a,a} is allowed. The existential phrase \"sufficiently large\" is represented by ∃ p₀ and the proof can take p₀ = 64. The definition uses Nat.sInf, but the development proves that the defining set is nonempty and the infimum is attained for odd primes. Bedert's external upper bound is cited rather than formalized, so the full asymptotic equality in the paper's corollary is outside the proved scope.", "alignment": true, "original": false, "review": { "status": "self-assessed and agent-reviewed", "bucket": "self-assessed", "reviewers": [ "Xinjie He", "OpenAI Codex" ], "notes": "The repository has been built with the pinned Lean toolchain, searched for proof holes and prohibited proof mechanisms, and checked with #print axioms on both headline declarations. No independent human peer review of the complete Lean development is claimed." }, "canonical": { "repo": "xinjiegit/ben_27_formalization", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-18-000003", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-18-000003", "commit": "f70352964007662e97a810c880abbfea1e2ea262", "trust": "high", "theorems": 1, "date": "2026-08-18" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "yangdabei/ttn-loss-landscape/formalization.yaml", "repo": "yangdabei/ttn-loss-landscape", "path": "formalization.yaml", "directory": "", "origins": [ "search" ], "url": "https://github.com/yangdabei/ttn-loss-landscape/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "TTN Loss Landscapes", "description": "A Lean 4 and Mathlib formalization of the principal loss-landscape results for tree tensor networks accompanying Benign Loss Landscapes Can Coexist with Worst-Case Hardness.", "authors": [ "Yangda Bei" ], "maintainers": [ "Yangda Bei" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Benign Loss Landscapes Can Coexist with Worst-Case Hardness", "id": "https://arxiv.org/abs/2609.13057", "authors": [ "Zach Furman", "Stephan Wäldchen", "Yangda Bei", "Liam Hodgkinson" ], "type": "manuscript", "relationship": "formalizes", "endorsement": "endorsed" } ], "related": [], "classification": { "arxiv": [], "msc2020": [] }, "automation": { "methods": [ { "method": "agent", "models": [ "Anthropic Claude Fable 5", "Anthropic Claude Opus 4.8" ], "framework": "Claude Code" }, { "method": "agent", "models": [ "OpenAI GPT-5.5", "OpenAI GPT-5.6-Sol" ], "framework": "Codex" }, { "method": "manual", "models": [], "framework": "" } ], "strongest": "agent", "models": [ "Anthropic Claude Fable 5", "Anthropic Claude Opus 4.8", "OpenAI GPT-5.5", "OpenAI GPT-5.6-Sol" ], "spend": "subscription-based usage", "notes": "A complete record of token use, wall time, and API-equivalent cost was not retained." }, "scope": "The development formalizes the main minimum-norm and full-Tucker-rank loss landscape theorems and their supporting realizability, dormant-subspace, compression, and descent results. The escape-path corollary, separate stripping construction, minimal two-factor factorization results, parity example, computational-hardness, and Boolean-expressivity arguments are outside the release scope; further qualifications are recorded in docs/deviations.md.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "TTN.Arch.minNorm_isLocalMin_isGlobalMin", "file": "TTN/Landscape/LocalMin.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "TTN.Arch.critical_fullRank_isGlobalMin", "file": "TTN/Landscape/FullRank.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The directed root-based and undirected contraction presentations are identified after coordinate expansion by TTN.Arch.representedFromRoot_eq_represented. External modes are aggregated per node in the main development; the realizability/rank characterization alone is bridged to literal individual modes. The full-environment-rank lemma is proved in the leaf case consumed by the main argument. The proof uses connected-component dormant perturbations rather than the manuscript's iterative stripping narrative. The escape-path corollary is outside the release scope. The detailed qualifications and all specialization relations are recorded in docs/deviations.md and docs/correspondence.md.", "alignment": true, "original": false, "review": { "status": "author-verified", "bucket": "author-verified", "reviewers": [], "notes": "The manuscript authors checked the faithfulness of the mapped Lean statements. Comparator checks the two main formal statements and proof terms; AxiomAudit.lean reports their Lean-kernel axiom dependencies." }, "canonical": { "repo": "yangdabei/ttn-loss-landscape", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 }, { "id": "ykanoria/FD1d-Lean/formalization.yaml", "repo": "ykanoria/FD1d-Lean", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/ykanoria/FD1d-Lean/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "A Tight Upper Bound for Fully Dynamic Matching on the Line", "description": "There are `m` labeled supplies in `[0,1]`. In each period, an independent uniform demand arrives; an online policy selects one live supply, pays the distance between them, and replaces that supply by an independent uniform point. This is the one-dimensional fully dynamic model studied in Kanoria's [*Dynamic spatial matching*](https://doi.org/10.1214/25-AAP2154). That paper established a lower bound of order `log(m)/m` and an upper bound of order `log(m)^2/m`, and posed closing this logarithmic gap as an open problem. We construct a policy and show that its performance matches the prior known l", "authors": [ "Yash Kanoria" ], "maintainers": [ "Yash Kanoria" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Optimal fully dynamic matching on the line", "id": "", "authors": [ "Yash Kanoria" ], "type": "paper", "relationship": "formalizes", "endorsement": "participated" }, { "title": "Dynamic spatial matching", "id": "https://doi.org/10.1214/25-AAP2154", "authors": [ "Yash Kanoria" ], "type": "paper", "relationship": "background", "endorsement": "participated" } ], "related": [], "classification": { "arxiv": [ "math.OC", "math.PR" ], "msc2020": [ "68W27", "90C40", "60J20" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "OpenAI GPT-5 family" ], "framework": "ChatGPT and Codex" } ], "strongest": "agent", "models": [ "OpenAI GPT-5 family" ], "spend": "not tracked", "notes": "The manuscript itself discloses that OpenAI ChatGPT and Codex contributed to policy and proof development, algebra checking, and drafting. The Lean formalization was produced iteratively under human direction. Exact historical model versions, total wall time, and compute usage were not fully recorded. The final repository was rebuilt and mechanically audited for placeholders, custom axioms, schema validity, and statement equality." }, "scope": "The compared theorem covers the two quantitative upper-bound clauses of the manuscript's main theorem: arbitrary-initial-state long-run RMS cost and initialized finite-horizon expected average cost. It constructs the policy and path-space law and proves explicit constants. The source manuscript's implementation-complexity sentence, lower-bound-based Theta corollary, optimal long-run corollary, balanced-initial-law corollary, and non-compared supporting propositions are not advertised as Palomar results.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "FD1D.V5.Palomar.optimalDynamicMatchingUpperBound", "file": "FD1D/V5/PaperStatements.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The Lean target fixes C to 72000/log(2), strengthening the manuscript's existential universal constant. Its policy is memoryless in the current inventory and demand, which is a special case of the history-dependent online policies allowed by the manuscript. The initialization demands are universally quantified as fixed points and the independent post-refresh trajectory is modeled separately; this implies the manuscript's expected initialization guarantee because each scheduled match costs at most one. The implementation-complexity sentence is omitted: internal modules prove asymptotic inequalities for declared operation and memory accounting functions, but there is no executable operational semantics proving that evaluation of OnlinePolicy consumes those resources. The lower-bound-based optimality corollary and balanced-initial-law corollary are also omitted from the compared result.", "alignment": true, "original": false, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [ "OpenAI Codex" ], "notes": "The source theorem and Lean signature were compared clause by clause. Challenge.lean is independently Mathlib-only. Automated checks cover the target's dependency closure, all proof placeholders, permitted axioms, source checksums, and Comparator/NanoDa replay. No independent human mathematical peer review of the Lean development is recorded here." }, "canonical": { "repo": "ykanoria/fd1d-lean", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-12-000002", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-12-000002", "commit": "26a5b4e2fd83fa86053239e09e92af370c62f18c", "trust": "high", "theorems": 1, "date": "2026-09-14" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "yoshito-ishiki-math/lean-scottish-book-155/formalization.yaml", "repo": "yoshito-ishiki-math/lean-scottish-book-155", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/yoshito-ishiki-math/lean-scottish-book-155/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "A counterexample to Scottish Book Problem 155", "description": "A Lean proof that there are real Banach spaces X and Y and a bijection U from X to Y which preserves every pairwise distance on every closed ball of radius one quarter while failing to be a global isometry. This gives a counterexample to Scottish Book Problem 155.", "authors": [ "Yoshito Ishiki" ], "maintainers": [ "Yoshito Ishiki" ], "license": "MIT", "role": "", "substantive": "", "sources": [ { "title": "AI-Assisted Research Report: A counterexample to Scottish Book Problem 155", "id": "doi:10.5281/zenodo.22168214", "authors": [ "Yoshito Ishiki" ], "type": "research report", "relationship": "formalizes", "endorsement": "participated" } ], "related": [], "classification": { "arxiv": [ "math.FA", "math.MG" ], "msc2020": [ "46B20", "54E40" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "OpenAI Codex" ], "framework": "OpenAI Codex" } ], "strongest": "agent", "models": [ "OpenAI Codex" ], "spend": "subscription-based; not tracked", "notes": "Yoshito Ishiki selected the theorem, checked the informal claims, and is responsible for the published report. OpenAI Codex produced the Lean development and performed a non-independent alignment audit. No external solver or native proof oracle is used by the selected theorem." }, "scope": "The selected declaration existentially constructs two real Banach spaces and a bijection preserving all distances on every closed ball of radius one quarter while failing to be a global isometry. The development proves the bent seed, the protected one-point extension, coherent completed limit stages, cardinal bounds, delayed scheduling, final surjectivity and injectivity, and the closed-ball conclusion. The selected theorem is unconditional. It does not claim that the final spaces are separable or that the inverse bijection is continuous.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "ScottishBook155.claim14", "file": "TransfiniteConstruction.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The report constructs the protected target with a Lipschitz-free quotient. The Lean proof uses an alternative max-product Banach envelope whose dual-evaluation coordinate proves the same short-distance preservation and whose first-coordinate projection supplies the required contractive linear retraction. The final spaces are represented by completed direct limits rather than literal unions, and idle successor steps use the uniform protected extension at an already represented point. The Lean theorem has the exact existential conclusion of Theorem 1.1, but no isomorphism between the two internal envelope realizations is asserted.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "OpenAI Codex" ], "notes": "Yoshito Ishiki checked the informal mathematical claims and authorized publication of the report. The same agent that produced the Lean proof compared its objects, hypotheses, quantifiers, fixed scales, bijectivity, and non-isometry conclusion with Theorem 1.1. No independent human Lean code review or second-formalizer audit was completed before this snapshot." }, "canonical": { "repo": "yoshito-ishiki-math/lean-scottish-book-155", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-31-000008", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-31-000008", "commit": "233eff4af7901eadee7927a1eb1f814ad0f3c118", "trust": "high", "theorems": 1, "date": "2026-08-31" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "yuanchenyang/lean_low_rank_univariate_sos/formalization.yaml", "repo": "yuanchenyang/lean_low_rank_univariate_sos", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/yuanchenyang/lean_low_rank_univariate_sos/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Low-rank univariate sum of squares has no spurious second-order critical points (rank 2)", "description": "A Lean 4 formalization of the main theorem of Legat, Yuan and Parrilo, \"Low-Rank Univariate Sum of Squares Has No Spurious Local Minima\" (SIAM J. Optim. 33(3), 2023). For a real univariate polynomial p and a pair of polynomials u = (u₁, u₂), consider the quadratically penalized Burer–Monteiro objective f_p(u) = ‖σ(u) - p‖² with σ(u) = u₁² + u₂², the norm coming from any inner product on ℝ[x]. The formalized theorem states that if p is a sum of two squares (which is the case for any nonnegative univariate polynomial) and u is a second-order critical point of f_p (vanishing gradient and positive", "authors": [ "Chenyang Yuan" ], "maintainers": [ "Chenyang Yuan" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "Low-Rank Univariate Sum of Squares Has No Spurious Local Minima", "id": "doi:10.1137/22M1516208", "authors": [ "Benoît Legat", "Chenyang Yuan", "Pablo A. Parrilo" ], "type": "paper", "relationship": "formalizes", "endorsement": "participated" } ], "related": [], "classification": { "arxiv": [ "math.OC", "math.AG" ], "msc2020": [ "90C23", "90C26", "90C22", "12D15" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "GPT-5.4" ], "framework": "OpenAI Codex" }, { "method": "manual", "models": [], "framework": "n/a" } ], "strongest": "agent", "models": [ "GPT-5.4" ], "spend": "not tracked", "notes": "The formal statements (PolynomialModel.lean, Socp.lean, Challenge.lean) were written and checked by hand against Definition 3.1 and equations (1.3), (3.3), (3.4) of the paper by the formalizer, who is also an author of the paper. The proofs were developed with the help of GPT-5.4 acting as a coding agent and were accepted only when they compiled against the pinned Mathlib with no sorry and no additional axioms. The Palomar submission files (Challenge.lean, Solution.lean, comparator.json, formalization.yaml) were prepared with the help of Claude (Anthropic) and reviewed by the formalizer." }, "scope": "Formalized: the rank-2 case of Theorem 1.1 for an arbitrary positive-definite real bilinear form on ℝ[x] (the paper's statement is for an arbitrary inner product; the proof never uses a specific one). The target polynomial p is assumed to be a sum of two squares (IsSOS p), which for real univariate polynomials is classically equivalent to nonnegativity on ℝ, but that equivalence is not part of this development. The pair u = (u₁, u₂) ranges over the full polynomial ring and the first- and second-order conditions are required for every polynomial direction v (a stronger criticality hypothesis th", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [], "nonstandard_axioms": [], "results": [], "comparator": false, "literature_dependencies": 0, "divergences": "(1) Rank: only r = 2 is formalized; Theorem 1.1 is stated for all r ≥ 2. (2) Hypothesis on p: \"p is nonnegative on ℝ\" is replaced by \"p is a sum of two squares\" (IsSOS p); the two are classically equivalent for real univariate polynomials but the equivalence is not formalized. (3) Degrees: the paper fixes deg p ≤ 2d and optimizes over pairs of polynomials of degree at most d, testing criticality against degree-d directions; the formalization works in the full polynomial ring and requires the first- and second-order conditions for all polynomial directions v. This is a stronger SOCP hypothesis, not a change to the optimization problem. (4) Inner product: the formalization is stated for an arbitrary positive-definite bilinear form B (Fact B.toQuadraticMap.PosDef) on ℝ[x], which includes every inner product of the paper. (5) Conclusion: the formal theorem concludes σ(u) = p, which is equivalent to the paper's f_p(u) = 0 for a positive-definite B (objective_eq_zero_iff_residual_eq_zero in Socp.lean).", "alignment": true, "original": false, "review": { "status": "author-verified", "bucket": "author-verified", "reviewers": [ "Chenyang Yuan" ], "notes": "The formalizer is a coauthor of the source paper and checked the formal definitions and the advertised statement against the paper. All proofs compile with Lean 4.30.0 / Mathlib v4.30.0 with no sorry; the advertised theorem depends only on propext, Classical.choice and Quot.sound. No independent external review has been performed." }, "canonical": { "repo": "yuanchenyang/lean_low_rank_univariate_sos", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-20-000002", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-20-000002", "commit": "1fccd921a4530a088ab6e230f832041ab3c5c7f3", "trust": "high", "theorems": 1, "date": "2026-08-20" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "yuhangshi888/sun-cotangent-permanent-counterexample-lean/formalization.yaml", "repo": "yuhangshi888/sun-cotangent-permanent-counterexample-lean", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/yuhangshi888/sun-cotangent-permanent-counterexample-lean/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "A Lean formalization of a counterexample to Sun's sign conjecture for cotangent permanents", "description": "A Lean 4 and Mathlib formalization of the exact evaluation of the normalized 14 by 14 cotangent permanent at p = 29. The formalization proves that its value is 84643985981440 and hence gives a counterexample to the cotangent sign inequality in Zhi-Wei Sun's Conjecture 4.7(ii). It includes the analytic-to-cyclotomic bridge, two separately kernel-checked finite permanent certificates, Chinese-remainder and coefficientwise integer lifting, the Section 3 Archimedean uniqueness argument, and equation (4.1) in the integral cyclotomic quotient.", "authors": [ "Yuhang Shi" ], "maintainers": [ "Yuhang Shi" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "A counterexample to Sun's sign conjecture for cotangent permanents", "id": "doi:10.5281/zenodo.22105063", "authors": [ "Yuhang Shi" ], "type": "paper", "relationship": "formalizes", "endorsement": "participated" }, { "title": "Arithmetic properties of some permanents", "id": "arXiv:2108.07723v7", "authors": [ "Zhi-Wei Sun" ], "type": "paper", "relationship": "background", "endorsement": "not-contacted" } ], "related": [], "classification": { "arxiv": [ "math.NT", "math.CO" ], "msc2020": [ "11C20", "15A15", "11A07" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "OpenAI Codex" ], "framework": "OpenAI Codex" } ], "strongest": "agent", "models": [ "OpenAI Codex" ], "spend": "subscription-based", "notes": "OpenAI Codex assisted throughout; Yuhang Shi selected the target, reviewed the mathematical claims, and is responsible for the submitted snapshot." }, "scope": "The substantive Lean development proves the exact p = 29 cotangent permanent value and its sign consequence without sorry, custom axioms, or native_decide. The Comparator-facing proof uses a three-scalar 61-bit certificate independent of the six-small-modulus residue computation. A separate six-small-modulus certificate proves the same canonical integer and supplies the literal finite-field table and Section 3 CRT route. The development also formalizes the analytic bridge, generic permanent identities, the Archimedean uniqueness estimate, equation (4.1) in the integral polynomial quotient, and", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "SunCotangent.cotangentPermanent29_exact", "file": "SunCotangent/AdvertisedTheorem.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "SunCotangent.sun_cotangent_sign_counterexample", "file": "SunCotangent/AdvertisedTheorem.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The Comparator-facing certificate uses three scalar 61-bit residue rings: two new verification moduli and the middle modulus from the paper. A separate certificate uses six 31-bit residue rings. Both employ a kernel-checked Gray recurrence. The paper's original first and third extension-valued 61-bit C++ scans are not replayed instruction by instruction, although their stated moduli, product, bounds, and coefficient-congruence conclusions are formalized and the same integral equation is proved independently. The paper's localization language is represented by the exact inverse and specialization identities used in the proof rather than a separate localization object. Ryser and Glynn are both proved correct, but no independent numerical Glynn checkpoint trace is claimed. Sun's general odd-prime integrality theorem is cited only for the optional Section 3 route and is not a dependency of the advertised exact-value proof.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Yuhang Shi" ], "notes": "The author and Codex checked the advertised statement against the preprint's normalization and audited the proof dependency chain. No external mathematical peer review is claimed." }, "canonical": { "repo": "yuhangshi888/sun-cotangent-permanent-counterexample-lean", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-27-000019", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-27-000019", "commit": "8db902a56070f0d90866bf8a44d485a20cb817af", "trust": "high", "theorems": 2, "date": "2026-08-27" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "yuhangshi888/zeta-simple-zeros-673316977/lean/formalization.yaml", "repo": "yuhangshi888/zeta-simple-zeros-673316977", "path": "lean/formalization.yaml", "directory": "lean", "origins": [ "palomar" ], "url": "https://github.com/yuhangshi888/zeta-simple-zeros-673316977/blob/HEAD/lean/formalization.yaml", "version": "v0.4", "missing": [], "name": "A Lean verification of the two-certificate supporting-plane deduction at block length 219", "description": "A Lean 4 and Mathlib formalization of the supporting-plane and exact-arithmetic portion of the finite-dimensional argument in Yuhang Shi's research draft on simple zeros of the Riemann zeta function. It proves a conditional scalar two-certificate supporting-plane implication, its explicit piecewise-square-root specialization at block length 219, and the exact strict comparison above 673316977/10^9 for the resulting local assembly expression. The spectral derivation of the trace-envelope alternative, the upstream seven- and nine-point interval certificates, and the analytic passage from finite ", "authors": [ "Yuhang Shi" ], "maintainers": [ "Yuhang Shi" ], "license": "MIT", "role": "", "substantive": "", "sources": [ { "title": "A two-certificate trace-energy deduction for simple zeros of the Riemann zeta function", "id": "doi:10.5281/zenodo.21926962", "authors": [ "Yuhang Shi" ], "type": "paper", "relationship": "formalizes", "endorsement": "participated" }, { "title": "A 67.3312742272% lower-bound candidate for simple zeros of zeta", "id": "https://github.com/trmdy/zeta-simple-zeros-673137/tree/1610b97b7895ff34982260f8dcaf04a0f7b82cf7", "authors": [], "type": "other", "relationship": "background", "endorsement": "not-contacted" } ], "related": [], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11M26" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "OpenAI Codex (model identifier not exposed)" ], "framework": "OpenAI Codex" } ], "strongest": "agent", "models": [ "OpenAI Codex (model identifier not exposed)" ], "spend": "subscription-based", "notes": "Yuhang Shi supplied and directed the mathematical argument and is responsible for the claims. OpenAI Codex assisted with the Lean formalization and repository preparation." }, "scope": "The substantive Lean modules and Solution.lean contain no sorry, admit, custom axiom, native_decide, or Lean.ofReduceBool. The compared statements cover the abstract supporting-plane theorem, the concrete Phi_219 specialization with all certificate and trace-envelope assumptions shown, and the exact final comparison. Challenge.lean contains three deliberate statement holes permitted by Comparator; they are not proof dependencies and are excluded from the counts below. No end-to-end formalization of a lower bound for zeta zeros is claimed.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "TwoCertificatePalomar.two_certificate_supporting_plane", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "TwoCertificatePalomar.phi219_supporting_plane", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "TwoCertificatePalomar.exact_final_comparison", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Quot.sound", "Classical.choice" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The abstract Lean theorem isolates a conditional scalar implication from Lemma 2.2 of the manuscript and assumes the trace-envelope alternative R <= D or phi(E) <= D. Within that isolated implication, the span-comparison hypothesis and two auxiliary slope hypotheses are not needed. The spectral argument producing the trace-envelope alternative is outside the formalized scope. The concrete theorem retains the certificate inequalities and that alternative as explicit assumptions. The Lean development proves the local expression exceeds 673316977/10^9, but it does not prove that the imported analytic and computational inputs hold or formalize the deduction of the global zeta-zero proportion from those inputs.", "alignment": true, "original": false, "review": { "status": "self-assessed", "bucket": "self-assessed", "reviewers": [ "Yuhang Shi" ], "notes": "The author and OpenAI Codex checked the compared statements against the research draft, built the pinned Lean project, and audited the axiom and proof boundary. No external mathematical peer review is claimed." }, "canonical": { "repo": "yuhangshi888/zeta-simple-zeros-673316977", "directory": "lean" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-08-29-000004", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-29-000004", "commit": "7cfd0ce9a9ef1615889b5c1191fac5de41f13960", "trust": "high", "theorems": 3, "date": "2026-08-29" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "yuma-mizuno/KR-mod9-lean/formalization.yaml", "repo": "yuma-mizuno/KR-mod9-lean", "path": "formalization.yaml", "directory": "", "origins": [ "palomar", "search" ], "url": "https://github.com/yuma-mizuno/KR-mod9-lean/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "KR-mod9-lean", "description": "Lean 4 proofs of the three Kanade-Russell identities modulo nine as identities of integer formal power series, including coefficientwise summability of the defining double sums. 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This reports pre-submission automated review, not a Palomar editorial outcome, human peer review, novelty certification, or external endorsement." }, "canonical": { "repo": "yuningyang19/rerandsrht_prob_5_6_simons_workshop", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-07-000001", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-07-000001", "commit": "158a716c07e16ec4fca14e36836b14d8d20ee7c8", "trust": "high", "theorems": 2, "date": "2026-09-07" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "zakharov2k/EGZ-formal-proof/formalization.yaml", "repo": "zakharov2k/EGZ-formal-proof", "path": "formalization.yaml", "directory": "", "origins": [ "palomar" ], "url": "https://github.com/zakharov2k/EGZ-formal-proof/blob/HEAD/formalization.yaml", "version": "v0.4", "missing": [], "name": "Convex geometry and the Erdős–Ginzburg–Ziv problem: Theorem 1.2", "description": "A Lean 4 formalization of Theorem 1.2 of Dmitrii Zakharov's Convex geometry and the Erdős–Ginzburg–Ziv problem. For every fixed positive dimension d, as p tends to infinity through the primes, the Erdős–Ginzburg–Ziv constant s(F_p^d) satisfies s(F_p^d) = p w(F_p^d) + o(p), where w is the weak Erdős–Ginzburg–Ziv constant. Supporting proofs include the convex-flag Helly and centerpoint theorems, flag decomposition lemma, balanced convex combinations, relative set expansion, and the polynomial method bound for w. This submission covers Theorem 1.2 and its upper-bound formulation, not every result", "authors": [ "Dmitrii Zakharov" ], "maintainers": [ "Dmitrii Zakharov" ], "license": "MIT", "role": "", "substantive": "", "sources": [ { "title": "Convex geometry and the Erdős–Ginzburg–Ziv problem", "id": "https://doi.org/10.19086/da.165216", "authors": [ "Dmitrii Zakharov" ], "type": "paper", "relationship": "formalizes", "endorsement": "" } ], "related": [ { "id": "EGZ/ZeroSum/DimensionOne.lean", "relationship": "other", "note": "An optional one-dimensional regression was migrated from the user-supplied ErdosGinzburgZivPrime.lean and uses Mathlib's ZMod.erdos_ginzburg_ziv. Its earlier external provenance is not established here. It is not imported by the main theorem. No claim of being the first formalization is made." } ], "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "11B30", "11H06" ] }, "automation": { "methods": [ { "method": "agent", "models": [ "GPT-6" ], "framework": "OpenAI Codex, including collaborating agents" } ], "strongest": "agent", "models": [ "GPT-6" ], "spend": "", "notes": "AI assistance was substantial and included proof design, implementation, debugging, mathematical audits, documentation, and submission packaging. GPT-6 use and multiple collaborating Codex agents are documented for the current work. Earlier model identities and a complete earlier interaction history have not been established. No reconstructed cost, hardware, or wall-clock accounting is claimed." }, "scope": "Theorem 1.2 for every positive dimension, together with its eventual upper-bound formulation. The two Palomar comparison declarations are proved in Solution.lean from the unconditional project theorems. Challenge.lean contains exactly two deliberate theorem placeholders; these are excluded from the proof sorry counts below. Its definitions are fully specified and import only Mathlib and its dependencies. Supporting proof modules are complete. Ordinary Lean build and axiom audit results do not constitute a Palomar acceptance or a completed independent Comparator/NanoDa verification.", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "PalomarEGZ.theorem_1_2", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "PalomarEGZ.main_upper_bound", "file": "Solution.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "The compared statements retain Theorem 1.2's quantifiers: dimension is fixed and positive, and the limit is along primes. The proof does not assume the balanced-combination or relative-expansion statements; both are proved. It establishes existence of sufficiently large thresholds, without optimizing their numerical sizes. Rational polytopes and their flags are modeled in real affine spaces with explicit lattice data. Proper generating points use positive local weights; cumulative weights define support polytopes and fibre masses. For the auxiliary polytope centerpoint theorem, rationality of the measure's finite support is explicit in the real-space model. The flag application proves that condition, so it adds no assumption to Theorem 1.2. The auxiliary large-face counting bound uses positive initial mass, which is established from nonzero input in the actual flag construction. A zero-measure version would not suffice. The flag iteration folds minimalization and reduction into refinement operations and permits any valid event rather than imposing the paper's event priority. A finite capacity argument proves termination, with radii and prime thresholds fixed uniformly before the ru", "alignment": true, "original": false, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [ "OpenAI Codex collaborating agents (GPT-6)" ], "notes": "Agents reviewed statement fidelity, quantifier order, degenerate cases, normalization and termination, analytic proof interfaces, and the main theorem assembly. The unconditional project results have passed Lean compilation and axiom audits reporting only propext, Classical.choice, and Quot.sound. Human review of the complete Lean development has not been confirmed. No source-author endorsement, Palomar acceptance, or completed independent verification is claimed." }, "canonical": { "repo": "zakharov2k/egz-formal-proof", "directory": "" }, "nodes": [], "container": false, "anchors": {}, "palomar": [ { "id": "PALOMAR-2026-09-06-000001", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-06-000001", "commit": "7c82267c43a51dd5cb3e440412780ff82aff7500", "trust": "high", "theorems": 2, "date": "2026-09-06" } ], "checks": [], "checked_by": [ "Palomar" ], "contradictions": [], "confirmations": 0 }, { "id": "Zhaoyi-Tian/zeta-simple-zeros-lean/zeta23/formalization.yaml", "repo": "Zhaoyi-Tian/zeta-simple-zeros-lean", "path": "zeta23/formalization.yaml", "directory": "zeta23", "origins": [ "search" ], "url": "https://github.com/Zhaoyi-Tian/zeta-simple-zeros-lean/blob/HEAD/zeta23/formalization.yaml", "version": "v0.4", "missing": [], "name": "Zeta23 — more than two-thirds of the nontrivial zeros of the Riemann zeta function are simple and on the critical line", "description": "", "authors": [ "Claude" ], "maintainers": [ "Ralph Furman" ], "license": "Apache-2.0", "role": "", "substantive": "", "sources": [ { "title": "More than two thirds of the zeros of the Riemann zeta function are simple and on the critical line", "id": "arXiv:2608.13637", "authors": [ "Levent Alpöge", "Ralph Furman" ], "type": "article", "relationship": "formalizes", "endorsement": "participated" } ], "related": [ { "id": "https://github.com/AlexKontorovich/PrimeNumberTheoremAnd", "relationship": "adapts", "note": "Files under Zeta23/FromPNTPlus/ are derived from PrimeNumberTheoremAnd with local modifications; each carries its upstream notice. See NOTICE." } ], "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11M06", "11M26", "15A42" ] }, "automation": { "methods": [ { "method": "autonomous", "models": [ "Claude" ], "framework": "" } ], "strongest": "autonomous", "models": [ "Claude" ], "spend": "", "notes": "The formalization was entirely automated: Claude produced all Lean code. Humans directed the project and reviewed its outputs; see the project blog post and AUDIT.md." }, "scope": "Complete, sorry-free Lean 4 proofs over Mathlib alone (axioms propext/Classical.choice/Quot.sound) of the paper's headline theorems: 23 comparator-checked statements in two Mathlib-only challenge modules — Challenge.lean (17 statements: Theorems A and B at the constants stated in the paper, including the Montgomery-Taylor-window forms and the Dirichlet analogues, with their on-the-line companion statements) and Challenge/XiPrime.lean (6 statements on the zeros of xi').", "sorry_count": 0, "sorry_in_definitions": 0, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "results": [ { "declaration": "two_thirds_on_critical_line", "file": "comparator/Challenge.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "two_thirds_simple_on_critical_line", "file": "comparator/Challenge.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "five_sixths_distinct", "file": "comparator/Challenge.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "montgomery_taylor_simple_on_critical_line_mult", "file": "comparator/Challenge.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "dirichlet_two_thirds_simple_on_critical_line", "file": "comparator/Challenge.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] }, { "declaration": "xiPrime_simple_zeros_on_critical_line", "file": "comparator/Challenge/XiPrime.lean", "description": "", "axioms": [ "propext", "Classical.choice", "Quot.sound" ], "sorry_count": 0, "comparator": true, "literature": [] } ], "comparator": true, "literature_dependencies": 0, "divergences": "liminf bounds are rendered as: for all eps > 0 there is T0 such that for all T >= T0, (c - eps)*N <= X; the xi-prime proportion statements instead carry fixed decimal constants (no eps). 'Nontrivial zero' is rendered as a zero with 0 < Re rho < 1. Windows are T1 < Im rho <= T2 (positive ordinates). Left sides count with multiplicity; N0*, N0^s, N_d count distinct points (the strong direction). The repository states the theorems at the paper's constants; the weaker Cauchy-Schwarz-form variants that earlier revisions also certified are implied by these and are no longer separately stated. See comparator/README.md 'Reading notes'.", "alignment": true, "original": false, "review": { "status": "author-verified", "bucket": "author-verified", "reviewers": [ "Ralph Furman" ], "notes": "Ralph Furman (paper author) read the main challenge module (comparator/Challenge.lean, the 17 statements and their definition layer) and confirmed it matches the paper; the xi-prime challenge module is agent-reviewed. Recorded in AUDIT.md at the pinned commit: #print axioms checks for all 23 statements (exactly [propext, Classical.choice, Quot.sound]) and comparator + NanoDa kernel runs for both configurations. The underlying paper was reviewed and endorsed by a human analytic number theorist, who also produced an independent condensed proof of the main result; journal peer review is pending. The Lean statements were additionally cold-read by Claude instances with no prior project context." }, "canonical": { "repo": "zhaoyi-tian/zeta-simple-zeros-lean", "directory": "zeta23" }, "nodes": [], "container": false, "anchors": {}, "palomar": [], "checks": [], "checked_by": [], "contradictions": [], "confirmations": 0 } ]