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        "note": "Site-confirmed: the computational certificate was replayed here on 26 August 2026 and it passes. Cloned mbaccaro-dev/mathematical-proofs, installed python-flint 0.9.0 into a venv per requirements.txt, ran its own reproduce.py in full (4 min 17 s), and got STATUS=PASS with claim_ceiling=Haglund_Conjecture_4_for_k_equals_1_only - via MANIFEST_PASS at 289 files, S1_STRUCTURE, NONREAL_JOIN, GLOBAL_LIF",
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      {
        "axis": "evidence",
        "stronger": "palomar",
        "stronger_evidence": "checked",
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        "weaker_evidence": "reviewed",
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    "leads": [],
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  },
  {
    "id": "palomar:PALOMAR-2026-08-23-000002",
    "node": null,
    "title": "Problem 3 of Dubickas (2006): Is $\\sqrt{3} \\in \\mathcal{Z}$?",
    "keys": [
      "palomar:PALOMAR-2026-08-23-000002",
      "repo:rwst/square-roots",
      "vibemathed:question-3-of-dubickas-2006-3"
    ],
    "registries": [
      "palomar",
      "vibemathed"
    ],
    "positions": [
      {
        "registry": "palomar",
        "id": "PALOMAR-2026-08-23-000002",
        "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-23-000002",
        "title": "rwst/Square-Roots",
        "autonomy": "unknown",
        "evidence": "checked",
        "labels": {
          "autonomy": "",
          "evidence": "registered",
          "trust": "high",
          "theorems": "10"
        },
        "systems": [],
        "note": "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 t",
        "date": "2026-09-06",
        "repos": [
          "rwst/square-roots"
        ]
      },
      {
        "registry": "vibemathed",
        "id": "question-3-of-dubickas-2006-3",
        "url": "https://vibemathed.com/problem/question-3-of-dubickas-2006-3",
        "title": "Problem 3 of Dubickas (2006): Is $\\sqrt{3} \\in \\mathcal{Z}$?",
        "autonomy": "machine-led",
        "evidence": "formal-reported",
        "labels": {
          "autonomy": "ai-discovered",
          "evidence": "lean-checked",
          "resolution": "candidate",
          "outcome": "proved",
          "publication": "preprint"
        },
        "systems": [
          "Fable 5",
          "Opus 5"
        ],
        "note": "The Lean was read here on 22 August 2026, at github.com/rwst/Square-Roots: eleven modules totalling 4415 lines with zero sorry, zero declared axioms and no native_decide, comparator.json permitting only propext, Quot.sound and Classical.choice, and the MahlerZ and S definitions faithful to the statement above. Challenge.lean's ten sorries are the placeholders a comparator challenge is meant to car",
        "date": "2026-08-21",
        "repos": [
          "rwst/square-roots"
        ]
      }
    ],
    "conflicts": [],
    "gaps": [
      {
        "axis": "evidence",
        "stronger": "palomar",
        "stronger_evidence": "checked",
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    "leads": [],
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  },
  {
    "id": "palomar:PALOMAR-2026-08-24-000002",
    "node": null,
    "title": "Mahler's $Z$-number problem in its generalized form",
    "keys": [
      "palomar:PALOMAR-2026-08-24-000002",
      "palomar:PALOMAR-2026-09-12-000003",
      "repo:rwst/confinement-certificates",
      "vibemathed:mahler-s-z-number-problem-in-its-generalized-form"
    ],
    "registries": [
      "palomar",
      "vibemathed"
    ],
    "positions": [
      {
        "registry": "palomar",
        "id": "PALOMAR-2026-09-12-000003",
        "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-12-000003",
        "title": "Confinement schemas and the reach of block certificates for powers of rational numbers modulo one",
        "autonomy": "unknown",
        "evidence": "checked",
        "labels": {
          "autonomy": "",
          "evidence": "registered",
          "trust": "high",
          "theorems": "14"
        },
        "systems": [],
        "note": "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",
        "date": "2026-09-12",
        "repos": [
          "rwst/confinement-certificates"
        ]
      },
      {
        "registry": "palomar",
        "id": "PALOMAR-2026-08-24-000002",
        "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-24-000002",
        "title": "rwst/Confinement-Certificates",
        "autonomy": "unknown",
        "evidence": "checked",
        "labels": {
          "autonomy": "",
          "evidence": "registered",
          "trust": "high",
          "theorems": "24"
        },
        "systems": [],
        "note": "A Lean 4 / Mathlib development of a paper on the emptiness theory of 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, at and beyond the window length 1/p. The paper introduces block certificates --- finite integer witnesses, rechecked from scratch by the Lean proof kernel, whose soundness rests on the aperiodicit",
        "date": "2026-09-06",
        "repos": [
          "rwst/confinement-certificates"
        ]
      },
      {
        "registry": "vibemathed",
        "id": "mahler-s-z-number-problem-in-its-generalized-form",
        "url": "https://vibemathed.com/problem/mahler-s-z-number-problem-in-its-generalized-form",
        "title": "Mahler's $Z$-number problem in its generalized form",
        "autonomy": "machine-led",
        "evidence": "formal-reported",
        "labels": {
          "autonomy": "ai-discovered",
          "evidence": "lean-checked",
          "resolution": "partial",
          "outcome": "proved",
          "publication": "preprint"
        },
        "systems": [
          "Fable 5",
          "Opus 5"
        ],
        "note": "Filed at Lean-checked, as submitted. Challenge2.lean is a trusted statement of record that imports nothing but Mathlib, redeclares verbatim every definition occurring in the certified theorems, and states the paper's lettered results with sorry proofs; comparator2.json names fourteen of them and, as configured, checks constant identity across the challenge and solution environments, restricts axio",
        "date": "2026-09-12",
        "repos": [
          "rwst/confinement-certificates"
        ]
      }
    ],
    "conflicts": [],
    "gaps": [
      {
        "axis": "evidence",
        "stronger": "palomar",
        "stronger_evidence": "checked",
        "weaker": "vibemathed",
        "weaker_evidence": "formal-reported",
        "ours_is": "neither"
      }
    ],
    "leads": [],
    "agrees": false
  },
  {
    "id": "palomar:PALOMAR-2026-08-27-000007",
    "node": null,
    "title": "Colombo’s difference-power determinant conjecture",
    "keys": [
      "palomar:PALOMAR-2026-08-27-000007",
      "repo:hkjtsgmc79-boop/colombo-odd-lean",
      "repo:ttieli/colombo1928",
      "vibemathed:colombo-s-difference-power-determinant-conjecture"
    ],
    "registries": [
      "palomar",
      "vibemathed"
    ],
    "positions": [
      {
        "registry": "palomar",
        "id": "PALOMAR-2026-08-27-000007",
        "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-27-000007",
        "title": "hkjtsgmc79-boop/colombo-odd-lean",
        "autonomy": "unknown",
        "evidence": "checked",
        "labels": {
          "autonomy": "",
          "evidence": "registered",
          "trust": "high",
          "theorems": "1"
        },
        "systems": [],
        "note": "This project formalizes the odd-exponent branch of Colombo's determinant problem for even matrix size. For every m at least one, every strictly increasing x : Fin (2*m) → ℝ, and every r at least m - 1, it proves that the determinant of the matrix with entries (x_j - x_i)^(2*r+1) is strictly positive. The proof development passes through paired truncated-power determinants, B-spline and Marsden fac",
        "date": "2026-08-27",
        "repos": [
          "hkjtsgmc79-boop/colombo-odd-lean"
        ]
      },
      {
        "registry": "vibemathed",
        "id": "colombo-s-difference-power-determinant-conjecture",
        "url": "https://vibemathed.com/problem/colombo-s-difference-power-determinant-conjecture",
        "title": "Colombo’s difference-power determinant conjecture",
        "autonomy": "machine-led",
        "evidence": "formal-reported",
        "labels": {
          "autonomy": "ai-discovered",
          "evidence": "lean-checked",
          "resolution": "candidate",
          "outcome": "proved",
          "publication": "preprint"
        },
        "systems": [
          "WuJie AI agent",
          "DeepSeek",
          "Qwen",
          "Kimi",
          "GPT",
          "other LLMs"
        ],
        "note": "The paper’s new odd-exponent theorem is formalized in Lean 4. The public endpoint proves\n$0<\\det[(x_j-x_i)^{2r+1}]$\nfor every strictly increasing real $2m$-tuple and every $r\\ge m-1$, matching Theorem 1.2 of the paper. The repository reports a complete 2817-job build, no `sorry`, `admit`, or project-specific axioms, and an axiom audit containing only `propext`, `Classical.choice`, and `Quot.sound`",
        "date": "2026-08-18",
        "repos": [
          "hkjtsgmc79-boop/colombo-odd-lean",
          "ttieli/colombo1928"
        ]
      }
    ],
    "conflicts": [],
    "gaps": [
      {
        "axis": "evidence",
        "stronger": "palomar",
        "stronger_evidence": "checked",
        "weaker": "vibemathed",
        "weaker_evidence": "formal-reported",
        "ours_is": "neither"
      }
    ],
    "leads": [],
    "agrees": false
  },
  {
    "id": "palomar:PALOMAR-2026-08-31-000013",
    "node": null,
    "title": "$(\\xi\\alpha^n)_{n\\ge1}$ is not uniformly distributed modulo one for Pisot $\\alpha$ and $\\xi$ in the Cantor set $C(\\alpha)$",
    "keys": [
      "palomar:PALOMAR-2026-08-31-000013",
      "repo:rwst/pisot-cantor-61",
      "vibemathed:bugeaud-problem-10-61"
    ],
    "registries": [
      "palomar",
      "vibemathed"
    ],
    "positions": [
      {
        "registry": "palomar",
        "id": "PALOMAR-2026-08-31-000013",
        "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-31-000013",
        "title": "Lean formalization toward Bugeaud Problem 10.61",
        "autonomy": "unknown",
        "evidence": "checked",
        "labels": {
          "autonomy": "",
          "evidence": "registered",
          "trust": "high",
          "theorems": "17"
        },
        "systems": [],
        "note": "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 Proble",
        "date": "2026-09-06",
        "repos": [
          "rwst/pisot-cantor-61"
        ]
      },
      {
        "registry": "vibemathed",
        "id": "bugeaud-problem-10-61",
        "url": "https://vibemathed.com/problem/bugeaud-problem-10-61",
        "title": "$(\\xi\\alpha^n)_{n\\ge1}$ is not uniformly distributed modulo one for Pisot $\\alpha$ and $\\xi$ in the Cantor set $C(\\alpha)$",
        "autonomy": "machine-led",
        "evidence": "formal-reported",
        "labels": {
          "autonomy": "ai-discovered",
          "evidence": "lean-checked",
          "resolution": "partial",
          "outcome": "proved",
          "publication": "announcement"
        },
        "systems": [
          "Fable 5",
          "Opus 5"
        ],
        "note": "Lean-checked, statement unaudited. Registered in the Palomar registry as PALOMAR-2026-08-31-000013: status registered, trust high, pinned to commit d61132ff, mirrored to PalomarArchive.\n\nSeventeen statements are compared with leanprover/comparator, which checks three things per theorem - that the statement in Solution is definitionally the same statement as in the trusted Challenge module, compare",
        "date": "2026-08-31",
        "repos": [
          "rwst/pisot-cantor-61"
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      }
    ],
    "conflicts": [],
    "gaps": [
      {
        "axis": "evidence",
        "stronger": "palomar",
        "stronger_evidence": "checked",
        "weaker": "vibemathed",
        "weaker_evidence": "formal-reported",
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    ],
    "leads": [],
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  },
  {
    "id": "palomar:PALOMAR-2026-09-03-000002",
    "node": null,
    "title": "Conway's Refinement Conjecture for Omnific Integers",
    "keys": [
      "palomar:PALOMAR-2026-09-03-000002",
      "palomar:PALOMAR-2026-09-21-000005",
      "repo:gaearon/conway-refinement",
      "repo:gregorymorse/unrestricted-boolean-mul",
      "repo:mrconter1/vibemathed",
      "vibemathed:conway-s-refinement-conjecture-for-omnific-integers",
      "vibemathed:fourth-moment-conjectures-for-rademacher-sums",
      "vibemathed:unrestricted-multiplicative-complexity-mul4"
    ],
    "registries": [
      "palomar",
      "vibemathed"
    ],
    "positions": [
      {
        "registry": "palomar",
        "id": "PALOMAR-2026-09-21-000005",
        "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-21-000005",
        "title": "Unrestricted Boolean multiplicative complexity of four-term polynomial multiplication",
        "autonomy": "unknown",
        "evidence": "checked",
        "labels": {
          "autonomy": "",
          "evidence": "registered",
          "trust": "high",
          "theorems": "6"
        },
        "systems": [],
        "note": "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 ",
        "date": "2026-09-21",
        "repos": [
          "gregorymorse/unrestricted-boolean-mul"
        ]
      },
      {
        "registry": "palomar",
        "id": "PALOMAR-2026-09-03-000002",
        "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-03-000002",
        "title": "A proof of Conway's refinement conjecture",
        "autonomy": "unknown",
        "evidence": "checked",
        "labels": {
          "autonomy": "",
          "evidence": "registered",
          "trust": "high",
          "theorems": "1"
        },
        "systems": [],
        "note": "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-",
        "date": "2026-09-03",
        "repos": [
          "gaearon/conway-refinement"
        ]
      },
      {
        "registry": "vibemathed",
        "id": "conway-s-refinement-conjecture-for-omnific-integers",
        "url": "https://vibemathed.com/problem/conway-s-refinement-conjecture-for-omnific-integers",
        "title": "Conway's Refinement Conjecture for Omnific Integers",
        "autonomy": "joint",
        "evidence": "reviewed",
        "labels": {
          "autonomy": "ai-co-developed",
          "evidence": "site-confirmed",
          "resolution": "candidate",
          "outcome": "proved",
          "publication": "announcement"
        },
        "systems": [
          "ChatGPT",
          "Claude"
        ],
        "note": "Site-confirmed: rebuilt here on 4 September 2026, not taken from the repository's CI. This site's verify-lean workflow checked out gaearon/conway-refinement at commit 264445c9, the commit the entry cites, installed the toolchain it pins (leanprover/lean4:v4.31.0) with its CombinatorialGames dependency, ran $\\texttt{lake build}$ over every module (3145 jobs), then the project's own scripts/Axioms.l",
        "date": "2026-09-03",
        "repos": [
          "gaearon/conway-refinement",
          "mrconter1/vibemathed"
        ]
      },
      {
        "registry": "vibemathed",
        "id": "unrestricted-multiplicative-complexity-mul4",
        "url": "https://vibemathed.com/problem/unrestricted-multiplicative-complexity-mul4",
        "title": "Unrestricted Boolean multiplicative complexity of four-term binary polynomial multiplication",
        "autonomy": "human-led",
        "evidence": "reviewed",
        "labels": {
          "autonomy": "ai-assisted",
          "evidence": "site-confirmed",
          "resolution": "partial",
          "outcome": "proved",
          "publication": "preprint"
        },
        "systems": [
          "OpenAI GPT-5.6 Sol (extra-high)",
          "Anthropic Opus 5 (high",
          "referee)"
        ],
        "note": "Site-confirmed: rebuilt here on 3 September 2026, not taken from the author's CI badge. This site's verify-lean workflow checked out release n4-arxiv-v2 (commit 2ebc0cf4) of GregoryMorse/unrestricted-boolean-mul, installed the toolchain the repository pins (leanprover/lean4:v4.32.1), ran $\\texttt{lake build}$ over every module, then the project's own AxiomAudit.lean, then $\\texttt{lake env leanche",
        "date": "2026-08-31",
        "repos": [
          "gregorymorse/unrestricted-boolean-mul",
          "mrconter1/vibemathed"
        ]
      },
      {
        "registry": "vibemathed",
        "id": "fourth-moment-conjectures-for-rademacher-sums",
        "url": "https://vibemathed.com/problem/fourth-moment-conjectures-for-rademacher-sums",
        "title": "Fourth-moment conjectures for Rademacher sums",
        "autonomy": "joint",
        "evidence": "claimed",
        "labels": {
          "autonomy": "ai-co-developed",
          "evidence": "unreviewed",
          "resolution": "resolved",
          "outcome": "proved",
          "publication": "preprint"
        },
        "systems": [
          "ChatGPT 5.6 Sol"
        ],
        "note": "Unreviewed: a one-day-old arXiv preprint, unrefereed, with no independent endorsement, and none of the proofs were checked here.\n\nThe statements were, and all three survived. Every theorem here is an inequality between finite sums and one-dimensional integrals, so it can be evaluated directly and hunted for counterexamples. Theorem 1.1 was tested on 18,960 coefficient vectors - flat families, one ",
        "date": "2026-08-18",
        "repos": [
          "mrconter1/vibemathed"
        ]
      }
    ],
    "conflicts": [],
    "gaps": [
      {
        "axis": "evidence",
        "stronger": "palomar",
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        "weaker": "vibemathed",
        "weaker_evidence": "claimed",
        "ours_is": "neither"
      }
    ],
    "leads": [],
    "agrees": false
  },
  {
    "id": "zeta:critical-line-proportion",
    "node": "zeta:critical-line-proportion",
    "title": "Proportion of zeta zeros proved simple and on the critical line",
    "keys": [
      "file:anthropics/formal-math/zeta23",
      "repo:anthropics/formal-math",
      "repo:anthropics/zeta-23-lean",
      "vibemathed:absence-of-critical-bernoulli-bond-percolation-on-z-d-in-every-dimension-d-2",
      "vibemathed:more-than-67-of-riemann-zeta-zeros-are-on-the-critical-line",
      "zeta:critical-line-proportion"
    ],
    "registries": [
      "qed.bot",
      "vibemathed"
    ],
    "positions": [
      {
        "registry": "qed.bot",
        "id": "zeta:critical-line-proportion",
        "url": "https://qed.bot/t/zeta-critical-line-proportion.html",
        "title": "Proportion of zeta zeros proved simple and on the critical line",
        "autonomy": "machine-led",
        "evidence": "checked",
        "labels": {
          "grades": "A3/V3",
          "primary claims": "1",
          "claims": "1"
        },
        "systems": [
          "Claude"
        ],
        "note": "",
        "date": "",
        "nonstandard_axioms": [],
        "rebuilt": true,
        "repos": [
          "anthropics/formal-math",
          "anthropics/zeta-23-lean"
        ]
      },
      {
        "registry": "vibemathed",
        "id": "absence-of-critical-bernoulli-bond-percolation-on-z-d-in-every-dimension-d-2",
        "url": "https://vibemathed.com/problem/absence-of-critical-bernoulli-bond-percolation-on-z-d-in-every-dimension-d-2",
        "title": "Absence of critical Bernoulli bond percolation on $\\mathbb Z^d$ in every dimension $d \\ge 2$",
        "autonomy": "machine-led",
        "evidence": "checked",
        "labels": {
          "autonomy": "ai-discovered",
          "evidence": "lean-verified",
          "resolution": "candidate",
          "outcome": "proved",
          "publication": "announcement"
        },
        "systems": [
          "Anthropic Claude models (specific version undisclosed)"
        ],
        "note": "Formalised in Lean 4 in anthropics/formal-math at commit 795efb8. Audited here on 8 September 2026 at four levels. Statement: Challenge.lean read against Grimmett §§1.3-1.4 - the lattice is SimpleGraph.hasse on Fin d → ℤ, whose symmetrised covering relation is nearest-neighbour adjacency; the measure is ProbabilityTheory.setBernoulli on the edge set, whose Mathlib definition was read at source and",
        "date": "2026-09-03",
        "repos": [
          "anthropics/formal-math"
        ]
      },
      {
        "registry": "vibemathed",
        "id": "more-than-67-of-riemann-zeta-zeros-are-on-the-critical-line",
        "url": "https://vibemathed.com/problem/more-than-67-of-riemann-zeta-zeros-are-on-the-critical-line",
        "title": "The Proportion of Zeta Zeros on the Critical Line",
        "autonomy": "machine-led",
        "evidence": "formal-reported",
        "labels": {
          "autonomy": "ai-discovered",
          "evidence": "lean-checked",
          "resolution": "partial",
          "outcome": "proved",
          "publication": "preprint"
        },
        "systems": [
          "Claude (unreleased research version)"
        ],
        "note": "A sorry-free Lean 4 / Mathlib formalization accompanies the paper, with the statements built from Mathlib's own riemannZeta and analyticOrderAt rather than an assumed form of the result. This site audited the sources at tag v1.0: no sorry under Zeta23/, the 27 real ones all in the comparator/Challenge files that state the theorems with placeholder proofs by design; no native_decide; no axiom decla",
        "date": "2026-08-10",
        "repos": [
          "anthropics/zeta-23-lean"
        ]
      }
    ],
    "conflicts": [],
    "gaps": [
      {
        "axis": "evidence",
        "stronger": "qed.bot",
        "stronger_evidence": "checked",
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        "weaker_evidence": "formal-reported",
        "ours_is": "stronger"
      }
    ],
    "leads": [],
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        "title": "Erdős Problem #960",
        "autonomy": "machine-led",
        "evidence": "reviewed",
        "labels": {
          "autonomy": "ai-discovered",
          "evidence": "site-confirmed",
          "resolution": "resolved",
          "outcome": "disproved",
          "publication": "announcement"
        },
        "systems": [
          "OpenAI internal model"
        ],
        "note": "Marked solved by erdosproblems.com's official status. Solve credited via Terence Tao's AI-contributions wiki.",
        "date": "2026-04-09",
        "repos": []
      }
    ],
    "conflicts": [],
    "gaps": [],
    "leads": [],
    "agrees": true
  },
  {
    "id": "erdos:987",
    "node": "erdos:987",
    "title": "Erdős Problem 987",
    "keys": [
      "erdos:987",
      "file:marti2203/formal-conjectures/formalconjectures/erdosproblems/987.lean",
      "repo:marti2203/formal-conjectures",
      "vibemathed:erdos-987"
    ],
    "registries": [
      "qed.bot",
      "vibemathed"
    ],
    "positions": [
      {
        "registry": "qed.bot",
        "id": "erdos:987",
        "url": "https://qed.bot/s/erdos-987.html",
        "title": "Erdős Problem 987",
        "autonomy": "machine-led",
        "evidence": "reviewed",
        "labels": {
          "grades": "A3/V1",
          "primary claims": "1",
          "claims": "1"
        },
        "systems": [
          "OpenAI internal model"
        ],
        "note": "",
        "date": "",
        "nonstandard_axioms": [],
        "rebuilt": false,
        "repos": [
          "marti2203/formal-conjectures"
        ]
      },
      {
        "registry": "vibemathed",
        "id": "erdos-987",
        "url": "https://vibemathed.com/problem/erdos-987",
        "title": "Erdős Problem #987",
        "autonomy": "machine-led",
        "evidence": "reviewed",
        "labels": {
          "autonomy": "ai-discovered",
          "evidence": "site-confirmed",
          "resolution": "resolved",
          "outcome": "proved",
          "publication": "announcement"
        },
        "systems": [
          "OpenAI internal model"
        ],
        "note": "Marked solved by erdosproblems.com's official status. Solve credited via Terence Tao's AI-contributions wiki.",
        "date": "2026-04-09",
        "repos": []
      }
    ],
    "conflicts": [],
    "gaps": [],
    "leads": [],
    "agrees": true
  },
  {
    "id": "palomar:PALOMAR-2026-08-13-000001",
    "node": null,
    "title": "Phelps–Rodriguez Conjecture",
    "keys": [
      "palomar:PALOMAR-2026-08-13-000001",
      "repo:teorth/sendov",
      "vibemathed:phelps-rodriguez-conjecture"
    ],
    "registries": [
      "palomar",
      "vibemathed"
    ],
    "positions": [
      {
        "registry": "palomar",
        "id": "PALOMAR-2026-08-13-000001",
        "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-13-000001",
        "title": "teorth/sendov",
        "autonomy": "unknown",
        "evidence": "checked",
        "labels": {
          "autonomy": "",
          "evidence": "registered",
          "trust": "high",
          "theorems": "2"
        },
        "systems": [],
        "note": "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. Deg",
        "date": "2026-08-20",
        "repos": [
          "teorth/sendov"
        ]
      },
      {
        "registry": "vibemathed",
        "id": "phelps-rodriguez-conjecture",
        "url": "https://vibemathed.com/problem/phelps-rodriguez-conjecture",
        "title": "Phelps–Rodriguez Conjecture",
        "autonomy": "joint",
        "evidence": "checked",
        "labels": {
          "autonomy": "ai-co-developed",
          "evidence": "lean-verified",
          "resolution": "resolved",
          "outcome": "proved",
          "publication": "announcement"
        },
        "systems": [
          "GPT-5.6 Pro",
          "Claude Opus 5"
        ],
        "note": "Audited here on 13 August 2026, which is what lifts this above the submitter's conservative Lean-checked classification. The gap they identified was that nobody had checked the correspondence between Tao's formal statement and the historical conjecture, so that check was performed. Sendov.phelps_rodriguez in Sendov/Conjecture.lean reads: for $n \\ge 2$ and $p$ of natDegree $n$ with every root in th",
        "date": "2026-08-12",
        "repos": [
          "teorth/sendov"
        ]
      }
    ],
    "conflicts": [],
    "gaps": [],
    "leads": [],
    "agrees": true
  },
  {
    "id": "palomar:PALOMAR-2026-08-30-000002",
    "node": null,
    "title": "Feige's Conjecture",
    "keys": [
      "palomar:PALOMAR-2026-08-30-000002",
      "repo:pengzhang91/feige",
      "vibemathed:feiges-conjecture"
    ],
    "registries": [
      "palomar",
      "vibemathed"
    ],
    "positions": [
      {
        "registry": "palomar",
        "id": "PALOMAR-2026-08-30-000002",
        "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-30-000002",
        "title": "pengzhang91/Feige",
        "autonomy": "unknown",
        "evidence": "checked",
        "labels": {
          "autonomy": "",
          "evidence": "registered",
          "trust": "high",
          "theorems": "1"
        },
        "systems": [],
        "note": "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 con",
        "date": "2026-08-30",
        "repos": [
          "pengzhang91/feige"
        ]
      },
      {
        "registry": "vibemathed",
        "id": "feiges-conjecture",
        "url": "https://vibemathed.com/problem/feiges-conjecture",
        "title": "Feige's Conjecture",
        "autonomy": "machine-led",
        "evidence": "checked",
        "labels": {
          "autonomy": "ai-discovered",
          "evidence": "lean-verified",
          "resolution": "resolved",
          "outcome": "proved",
          "publication": "preprint"
        },
        "systems": [
          "ChatGPT 5.6 Pro",
          "GPT-5.6 Sol",
          "Codex"
        ],
        "note": "An end-to-end Lean formalization of the $e^{-1}$ conjecture accompanies the primary paper, formalizing the Vlassis-Thomas theorem, Grünbaum's centroid theorem and the combining argument. Three independent AI-assisted proofs appeared within days; neither preprint is peer-reviewed yet.",
        "date": "2026-07-27",
        "repos": [
          "pengzhang91/feige"
        ]
      }
    ],
    "conflicts": [],
    "gaps": [],
    "leads": [],
    "agrees": true
  },
  {
    "id": "palomar:PALOMAR-2026-09-07-000002",
    "node": null,
    "title": "Bollobás–Nikiforov conjecture",
    "keys": [
      "palomar:PALOMAR-2026-09-07-000002",
      "repo:shengtongzhang-alt/bn",
      "vibemathed:bollobas-nikiforov-conjecture"
    ],
    "registries": [
      "palomar",
      "vibemathed"
    ],
    "positions": [
      {
        "registry": "palomar",
        "id": "PALOMAR-2026-09-07-000002",
        "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-07-000002",
        "title": "A proof of the Bollobás–Nikiforov conjecture in Lean",
        "autonomy": "unknown",
        "evidence": "checked",
        "labels": {
          "autonomy": "",
          "evidence": "registered",
          "trust": "high",
          "theorems": "5"
        },
        "systems": [],
        "note": "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 formalizat",
        "date": "2026-09-07",
        "repos": [
          "shengtongzhang-alt/bn"
        ]
      },
      {
        "registry": "vibemathed",
        "id": "bollobas-nikiforov-conjecture",
        "url": "https://vibemathed.com/problem/bollobas-nikiforov-conjecture",
        "title": "Bollobás–Nikiforov conjecture",
        "autonomy": "joint",
        "evidence": "checked",
        "labels": {
          "autonomy": "ai-co-developed",
          "evidence": "lean-verified",
          "resolution": "candidate",
          "outcome": "proved",
          "publication": "announcement"
        },
        "systems": [
          "GPT-6 Astra",
          "Grok 4.6",
          "Claude Fable 5.1"
        ],
        "note": "Lifted from the submitted Lean-checked to Lean-verified, because the correspondence the submitter said nobody had audited was audited here, on 12 September 2026, at commit edb5259. Challenge.lean imports only Mathlib and states the headline theorem entirely in Mathlib vocabulary: $G$ a SimpleGraph on a finite vertex type, $G\\ne\\top$ for non-complete, Nontrivial for at least two vertices, G.cliqueN",
        "date": "2026-09-05",
        "repos": [
          "shengtongzhang-alt/bn"
        ]
      }
    ],
    "conflicts": [],
    "gaps": [],
    "leads": [],
    "agrees": true
  }
]