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Included as a control: a system that could not recover it would not be worth believing elsewhere.", "direction": "minimise", "unit": "multiplications", "value_key": "rank", "url": "https://github.com/google-deepmind/alphatensor", "page": "t/matmul-2x2-strassen.html", "aliases": [], "tags": [ "algorithms", "tensor decomposition" ], "records": [ { "value": 7, "display": "7 multiplications", "holder": "Strassen", "kind": "human", "date": "1969-01-01", "note": "Optimal for this case.", "source": "https://www.nature.com/articles/s41586-022-05172-4" } ], "best": { "value": 7, "display": "7 multiplications", "holder": "Strassen", "kind": "human", "date": "1969-01-01", "note": "Optimal for this case.", "source": "https://www.nature.com/articles/s41586-022-05172-4" }, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaTensor" ], "humans": [], "date": "2022-10-05", "outcome": "matched", "outcome_text": "Recovered a rank-7 decomposition, matching the optimum.", "autonomy": "A2", "autonomy_reason": "Reinforcement learning against a human-designed game.", "lean_claimed": false, "sources": [ { "label": "Nature, Discovering faster matrix multiplication algorithms", "url": "https://www.nature.com/articles/s41586-022-05172-4" } ], "details": {}, "statement": "matmul:2x2-strassen", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2022-10-05", "machine_checked": true, "checked_by": [ "qed.bot" ], "id": "cf606b0bfc04", "verification": "V3", "fidelity": null } ], "checks": [ { "target": "matmul:2x2-strassen", "evaluator": "tensor_factorization", "artifact": "https://github.com/google-deepmind/alphatensor/blob/main/algorithms/factorizations_f2.npz", "method": "evaluator", "checked": "2026-08-22T21:21:52Z", "claim": { "rank": 7 }, "measured": { "rank": 7, "multiplications": 7, "shape": [ 2, 2, 2 ], "exact": true, "mismatches": 0, "modulus": 2 }, "verdict": "verified", "detail": "2x2x2 in 7 multiplications, exact identity mod 2", "systems": [ "AlphaTensor" ] } ], "declarations": [], "fidelity": { "grade": null, "reason": "An object, checked against the problem's own constraints rather than a formal statement.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": true, "checked_by": [ "qed.bot" ] }, { "id": "matmul:4x4-mod2", "key": "4x4-mod2", "label": "Multiplications needed for 4x4 matrix multiplication over F_2", "collection": "Algorithms", "kind": "target", "name": "Multiplications needed for 4x4 matrix multiplication over F_2", "description": "A rank-R decomposition of the matrix multiplication tensor is an algorithm using R multiplications. Applying Strassen's algorithm recursively gives 49 for the 4x4 case over the field with two elements.", "direction": "minimise", "unit": "multiplications", "value_key": "rank", "url": "https://github.com/google-deepmind/alphatensor", "page": "t/matmul-4x4-mod2.html", "aliases": [], "tags": [ "algorithms", "tensor decomposition" ], "records": [ { "value": 47, "display": "47 multiplications", "holder": "AlphaTensor", "kind": "ai", "date": "2022-10-05", "note": "The first improvement on the recursive Strassen count for this case since 1969.", "source": "https://www.nature.com/articles/s41586-022-05172-4", "artifact": "https://github.com/google-deepmind/alphatensor/blob/main/algorithms/factorizations_f2.npz" }, { "value": 49, "display": "49 multiplications", "holder": "Strassen, applied recursively", "kind": "human", "date": "1969-01-01", "note": "Two levels of Strassen's 1969 algorithm.", "source": "https://www.nature.com/articles/s41586-022-05172-4" } ], "best": { "value": 47, "display": "47 multiplications", "holder": "AlphaTensor", "kind": "ai", "date": "2022-10-05", "note": "The first improvement on the recursive Strassen count for this case since 1969.", "source": "https://www.nature.com/articles/s41586-022-05172-4", "artifact": "https://github.com/google-deepmind/alphatensor/blob/main/algorithms/factorizations_f2.npz" }, "previous": { "value": 49, "display": "49 multiplications", "holder": "Strassen, applied recursively", "kind": "human", "date": "1969-01-01", "note": "Two levels of Strassen's 1969 algorithm.", "source": "https://www.nature.com/articles/s41586-022-05172-4" }, "held_by_machine": true, "claims": [ { "systems": [ "AlphaTensor" ], "humans": [], "date": "2022-10-05", "outcome": "record", "outcome_text": "A decomposition of rank 47, improving on 49.", "autonomy": "A2", "autonomy_reason": "Reinforcement learning against a game whose rules, reward and representation were designed by people; the decomposition itself was found by the system.", "lean_claimed": false, "sources": [ { "label": "Nature, Discovering faster matrix multiplication algorithms", "url": "https://www.nature.com/articles/s41586-022-05172-4" } ], "details": {}, "statement": "matmul:4x4-mod2", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2022-10-05", "machine_checked": true, "checked_by": [ "qed.bot" ], "id": "bd94040d275b", "verification": "V3", "fidelity": null } ], "checks": [ { "target": "matmul:4x4-mod2", "evaluator": "tensor_factorization", "artifact": "https://github.com/google-deepmind/alphatensor/blob/main/algorithms/factorizations_f2.npz", "method": "evaluator", "checked": "2026-08-22T21:21:52Z", "claim": { "rank": 47 }, "measured": { "rank": 47, "multiplications": 47, "shape": [ 4, 4, 4 ], "exact": true, "mismatches": 0, "modulus": 2 }, "verdict": "verified", "detail": "4x4x4 in 47 multiplications, exact identity mod 2", "systems": [ "AlphaTensor" ] } ], "declarations": [], "fidelity": { "grade": null, "reason": "An object, checked against the problem's own constraints rather than a formal statement.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": true, "checked_by": [ "qed.bot" ] }, { "id": "alphaevolve:1", "key": "1", "label": "Kakeya and Nikodym sets in finite fields", "collection": "AlphaEvolve problems", "kind": "target", "name": "Kakeya and Nikodym sets in finite fields", "description": "Let $d \\\\geq 1$, and let $q$ be a prime power. Let $\\\\mathbb{F}_q$ be a finite field of order $q$. A Kakeya set is a set $K$ that contains a line in every direction, and an Nikodym set $N$ is a set with the property that every point $x$ in $\\\\mathbb{F}_q^d$ is contained in a line that is contained in $N \\\\cup \\\\{x\\\\}$. Let $C^K(d,q), C^N(d,q)$ denote the least size of a Kakeya or Nikodym set in $\\", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/1.html", "page": "t/alphaevolve-1.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/1.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "record", "outcome_text": "Best known construction at publication", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/1.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:1", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "fc539e436017", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:10", "key": "10", "label": "3D Kakeya problem", "collection": "AlphaEvolve problems", "kind": "target", "name": "3D Kakeya problem", "description": "Let $n \\\\geq 2$. Let $C(n)$ denote the minimal volume $|\\\\bigcup_{j=1}^n \\\\bigcup_{k=1}^n P_{j,k}|$ of prisms $P_{j,k}$ with vertices $$\\\\begin{align*} \\&\\\\quad (x_{j,k},y_{j,k},0), \\\\left(x_{j,k}+\\\\frac{1}{n},y_{j,k},0\\\\right), \\\\left(x_{j,k},y_{j,k}+\\\\frac{1}{n},0\\\\right), \\\\left(x_{j,k}+\\\\frac{1}{n},y_{j,k}+\\\\frac{1}{n},0\\\\right),\\\\ &\\\\left(x_{j,k}+\\\\frac{j}{n},y_{j,k}+\\\\frac{k}{n},1\\\\right), \\", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/10.html", "page": "t/alphaevolve-10.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/10.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "record", "outcome_text": "Best known construction at publication", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/10.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:10", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "a4cdb6f2b781", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:11", "key": "11", "label": "Uncertainty principle", "collection": "AlphaEvolve problems", "kind": "target", "name": "Uncertainty principle", "description": "Given a function $f \\\\in L^1(\\\\mathbb{R})$, set $$ A(f) \\\\coloneqq \\\\inf \\\\{r > 0: f(x) \\\\geq 0 \\\\hbox{ for all } |x| \\\\geq r \\\\}.$$ Let $C$ be the largest constant for which one has $$ A(f) A(\\\\hat f) \\\\geq C$$ for all even $f$ with $f(0), \\\\hat f(0)", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/11.html", "page": "t/alphaevolve-11.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/11.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "below", "outcome_text": "Did not reach the best known construction", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/11.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:11", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "208b36e07e6e", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:12", "key": "12", "label": "Sphere Packing", "collection": "AlphaEvolve problems", "kind": "target", "name": "Sphere Packing", "description": "For any dimension $n$, let $C(n)$ denote the maximal density of a packing of $\\\\mathbb{R}^n$ by unit spheres. Establish upper and lower bounds on $C(n)$ that are as strong as possible.", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/12.html", "page": "t/alphaevolve-12.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/12.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "unstated", "outcome_text": "", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/12.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:12", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "6dfaa4f56aad", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:13", "key": "13", "label": "A Linear Programming Bound", "collection": "AlphaEvolve problems", "kind": "target", "name": "A Linear Programming Bound", "description": "For any dimension $n$, let $C(n)$ denote the quantity $$ C(n) \\\\coloneqq \\\\frac{\\\\pi^{n/2}}{\\\\Gamma(n/2+ 1)} \\\\inf_{f} \\\\frac{(r/2)^n f(0)}{\\\\hat f(0)}$$ where $f$ ranges over integrable continuous functions $f \\\\coloneqq \\\\mathbb{R}^n \\\\to \\\\mathbb{R}$, not identically zero, with $\\\\hat f(\\\\xi) \\\\geq 0$ for all $\\\\xi$ and $f(x) \\\\leq 0$ for all $|x| \\\\geq r$ for some $r>0$. Establish upper bounds", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/13.html", "page": "t/alphaevolve-13.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/13.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "unstated", "outcome_text": "", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/13.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:13", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "343c73ff478e", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:14", "key": "14", "label": "Hausdorff-Young Inequality", "collection": "AlphaEvolve problems", "kind": "target", "name": "Hausdorff-Young Inequality", "description": "For $1 \\\\leq p \\\\leq 2$, let $C(p)$ be the best constant such that $$ \\\\| \\\\hat f \\\\|_{L^{p'}(\\\\mathbb{R})} \\\\leq C(p) \\\\| f \\\\|_{L^p(\\\\mathbb{R})} $$ holds for all test functions $f \\\\colon \\\\mathbb{R} \\\\to \\\\mathbb{R}$. Here $p' \\\\coloneqq \\\\frac{p}{p-1}$ is the dual exponent of $p$. What is $C(p)$?", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/14.html", "page": "t/alphaevolve-14.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/14.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "matched", "outcome_text": "Matched the best known construction", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/14.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:14", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "529cb9003ad2", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:15", "key": "15", "label": "Gagliardo-Nirenberg Inequality", "collection": "AlphaEvolve problems", "kind": "target", "name": "Gagliardo-Nirenberg Inequality", "description": "Let $1 \\\\leq q \\\\leq \\\\infty$, and let $j$ and $m$ be non-negative integers such that $j", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/15.html", "page": "t/alphaevolve-15.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/15.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "matched", "outcome_text": "Matched the best known construction", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/15.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:15", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "c9edeaac6b1f", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:16", "key": "16", "label": "Special Case of Gagliardo-Nirenberg Inequality", "collection": "AlphaEvolve problems", "kind": "target", "name": "Special Case of Gagliardo-Nirenberg Inequality", "description": "Let $2", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/16.html", "page": "t/alphaevolve-16.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/16.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "matched", "outcome_text": "Matched the best known construction", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/16.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:16", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "5786641cb555", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:17", "key": "17", "label": "Young's Convolution Inequality", "collection": "AlphaEvolve problems", "kind": "target", "name": "Young's Convolution Inequality", "description": "Let $1 \\\\leq p,q,r \\\\leq \\\\infty$ with $1/r + 1 = 1/p + 1/q$. Let $C(p,q,r)$ denote the supremum of the quantity $$ Q(f, g) \\\\coloneqq \\\\frac{\\\\|f * g\\\\|_r}{\\\\|f\\\\|_p \\\\|g\\\\|_q}$$ over all non-zero test functions $f,g$. What is $C(p,q,r)$?", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/17.html", "page": "t/alphaevolve-17.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/17.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "matched", "outcome_text": "Matched the best known construction", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/17.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:17", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "9ed21b65c6fd", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:18", "key": "18", "label": "Hardy-Littlewood Maximal Inequality", "collection": "AlphaEvolve problems", "kind": "target", "name": "Hardy-Littlewood Maximal Inequality", "description": "Let $C$ denote the best constant for which $$ \\\\left| \\\\{ x: \\\\sup_{h>0} \\\\frac{1}{2h} \\\\int_{x-h}^{x+h} f(y) dy \\\\geq \\\\lambda \\\\} \\\\right| \\\\leq \\\\frac{C}{\\\\lambda} \\\\int_\\\\mathbb{R} f(x) dx$$ for absolutely integrable non-negative $f \\\\colon \\\\mathbb{R} \\\\to \\\\mathbb{R}$. What is $C$?", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/18.html", "page": "t/alphaevolve-18.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/18.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "below", "outcome_text": "Did not reach the best known construction", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/18.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:18", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "c22d8545971c", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:19", "key": "19", "label": "The Ovals Problem", "collection": "AlphaEvolve problems", "kind": "target", "name": "The Ovals Problem", "description": "Let $C$ denote the infimal value of $\\\\lambda_0(\\\\gamma)$, the least eigenvalue of the Schrödinger operator $$ H_\\\\gamma = -\\\\frac{d^2}{ds^2} + \\\\kappa^2(s) $$ associated with a simple closed convex curve $\\\\gamma$ parameterized by arclength and normalized to have length $2\\\\pi$, where $\\\\kappa(s)$ is the curvature. Obtain upper and lower bounds for $C$ that are as strong as possible.", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/19.html", "page": "t/alphaevolve-19.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/19.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "matched", "outcome_text": "Matched the best known construction", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/19.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:19", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "017b2f962099", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:2", "key": "2", "label": "An autocorrelation problem related to Sidon sets", "collection": "AlphaEvolve problems", "kind": "target", "name": "An autocorrelation problem related to Sidon sets", "description": "Let $C$ denote the largest constant for which one has $$ \\\\max_{-1/2 \\\\leq t \\\\leq 1/2} \\\\int_\\\\mathbb{R} f(t-x) f(x)\\ dx \\\\geq C \\\\left(\\\\int_{-1/4}^{1/4} f(x)\\ dx\\\\right)^2 $$ for all non-negative $f \\\\colon \\\\mathbb{R} \\\\to \\\\mathbb{R}$. What is $C$?", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/2.html", "page": "t/alphaevolve-2.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/2.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "record", "outcome_text": "Best known construction at publication", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/2.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:2", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "7ed164cba2f7", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:20", "key": "20", "label": "Sendov's Conjecture", "collection": "AlphaEvolve problems", "kind": "target", "name": "Sendov's Conjecture", "description": "For each $n \\\\geq 2$, let $C(n)$ be the smallest constant such that for any complex polynomial $f$ of degree $n \\\\geq 2$ with zeros $z_1, \\\\dots, z_n$ in the unit disk and critical points $w_1, \\\\dots, w_{n-1}$, $$ \\\\max_{1 \\\\leq k \\\\leq n} \\\\min_{1 \\\\leq j \\\\leq n-1} |z_k - w_j| \\\\leq C(n). $$ Sendov conjectured that $C(n)=1$.", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/20.html", "page": "t/alphaevolve-20.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/20.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "unstated", "outcome_text": "", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/20.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:20", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "ff0f647ae754", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:21", "key": "21", "label": "Schmeisser's Conjecture", "collection": "AlphaEvolve problems", "kind": "target", "name": "Schmeisser's Conjecture", "description": "For each $n \\\\geq 2$, let $C(n)$ be the smallest constant such that for any complex polynomial $f$ of degree $n \\\\geq 2$ with zeros $z_1, \\\\dots, z_n$ in the unit disk and critical points $w_1, \\\\dots, w_{n-1}$, and for any nonnegative weights $l_1, \\\\dots, l_n \\\\geq 0$ satisfying $\\\\sum_{k=1}^n l_k = 1$, we have $$ \\\\min_{1 \\\\leq j \\\\leq n-1} \\\\left| \\\\sum_{k=1}^n l_k z_k - w_j \\\\right| \\\\leq C(n", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/21.html", "page": "t/alphaevolve-21.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/21.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "unstated", "outcome_text": "", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/21.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:21", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "0b6055489b6b", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:22", "key": "22", "label": "Borcea's Conjecture", "collection": "AlphaEvolve problems", "kind": "target", "name": "Borcea's Conjecture", "description": "For any $1 \\\\leq p < \\\\infty$ and $n \\\\geq 2$, let $C(p,n)$ be the smallest constant such that for any complex polynomial $f$ of degree $n$ with zeroes $z_1,\\\\dots,z_n$ satisfying $$ \\\\frac{1}{n} \\\\sum_{i=1}^n |z_i|^p \\\\leq 1, $$ and every zero $f(\\\\zeta)=0$ of $f$, there exists a critical point $f'(\\\\xi) = 0$ of $f$ with $|\\\\xi - \\\\zeta| \\\\leq C(p,n)$. What is $C(p,n)$?", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/22.html", "page": "t/alphaevolve-22.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/22.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "unstated", "outcome_text": "", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/22.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:22", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "9b2b48c041ca", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:23", "key": "23", "label": "Smale's Problem", "collection": "AlphaEvolve problems", "kind": "target", "name": "Smale's Problem", "description": "For $n \\\\geq 2$, let $C(n)$ be the least constant such that for any polynomial $f$ of degree $n$, and any $z \\\\in \\\\mathbb{C}$ with $f'(z) \\\\neq 0$, there exists a critical point $f'(\\\\xi)=0$ such that $$ \\\\left|\\\\frac{f(z)-f(\\\\xi)}{z-\\\\xi}\\\\right| \\\\leq C(n) |f'(z)|. $$ Establish upper and lower bounds for $C(n)$ that are as strong as possible.", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/23.html", "page": "t/alphaevolve-23.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/23.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "unstated", "outcome_text": "", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/23.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:23", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "a1b53c5ab23e", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:24", "key": "24", "label": "de Bruin-Sharma Problem", "collection": "AlphaEvolve problems", "kind": "target", "name": "de Bruin-Sharma Problem", "description": "For $n \\\\geq 4$, let $\\\\Omega(n)$ be the set of pairs $(\\\\alpha,\\\\beta) \\\\in \\\\mathbb{R}_+^2$ such that, whenever $P$ is a degree $n$ polynomial whose roots $z_1,\\\\dots,z_n$ sum to zero, and $\\\\xi_1,\\\\dots,\\\\xi_{n-1}$ are the critical points (roots of $P'$), that $$|\\\\xi_1|^4 + \\\\dots + |\\\\xi_{n-1}|^4 \\\\leq \\\\alpha (|z_1|^4 + \\\\dots + |z_n|^4) + \\\\beta (|z_1|^2 + \\\\dots + |z_n|^2)^2.$$ What is $\\\\", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/24.html", "page": "t/alphaevolve-24.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/24.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "unstated", "outcome_text": "", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/24.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:24", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "cb8ed9a9782f", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:25", "key": "25", "label": "Crouzeix's Conjecture", "collection": "AlphaEvolve problems", "kind": "target", "name": "Crouzeix's Conjecture", "description": "Let $C$ be the smallest constant for which one has the bound $$ \\\\| p(A) \\\\|_{op} \\\\leq C \\\\sup_{z \\\\in W(A)} |p(z)| $$ for all $n \\\\times n$ square matrices $A$ and all polynomials $p$ with complex coefficients, where $\\\\| \\\\|_{op}$ is the operator norm and $$ W(A) \\\\coloneqq \\\\{ \\\\langle Ax, x \\\\rangle: \\\\|x\\\\| \\\\leq 1 \\\\}$$ is the numerical range of $A$. What is $C$? What polynomials $p$ attain", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/25.html", "page": "t/alphaevolve-25.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/25.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "unstated", "outcome_text": "", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/25.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:25", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "6116b802ca28", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:26", "key": "26", "label": "Sidorenko's Conjecture", "collection": "AlphaEvolve problems", "kind": "target", "name": "Sidorenko's Conjecture", "description": "A graphon is a symmetric measurable function $W \\\\colon [0,1]^2 \\\\to [0,1]$. Given a graphon $W$ and a finite graph $H = (V(H),E(H))$, the homomorphism density $t(H,W)$ is defined as $$ t(H,W) = \\\\int_{[0,1]^{V(H)}} \\\\prod_{\\\\{v,w\\\\} \\\\in E(H)} W(x_v,x_w)\\ \\\\prod_{v \\\\in V(H)} dx_v.$$ For a finite bipartite graph $H$, let $C(H)$ denote the least constant for which $$t(H, W) \\\\geq t(K_2, W)^{C(H)}$", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/26.html", "page": "t/alphaevolve-26.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/26.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "unstated", "outcome_text": "", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/26.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:26", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "835fc1c6e5b0", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:27", "key": "27", "label": "The Prime Number Theorem", "collection": "AlphaEvolve problems", "kind": "target", "name": "The Prime Number Theorem", "description": "Let $\\\\pi(x)$ denote the number of primes less than or equal to $x$, and let $C^- \\\\leq C^+$ denote the quantities $$ C^- \\\\coloneqq \\\\liminf_{x \\\\to \\\\infty} \\\\frac{\\\\pi(x)}{x/\\\\log x}$$ and $$ C^+ \\\\coloneqq \\\\limsup_{x \\\\to \\\\infty} \\\\frac{\\\\pi(x)}{x/\\\\log x}.$$ What are $C^-$ and $C^+$?", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/27.html", "page": "t/alphaevolve-27.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/27.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "below", "outcome_text": "Did not reach the best known construction", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/27.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:27", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "741db9beec41", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:28", "key": "28", "label": "Golay's Merit Factor", "collection": "AlphaEvolve problems", "kind": "target", "name": "Golay's Merit Factor", "description": "For $n \\\\geq 1$, let $\\\\mathbb{U}_{n}$ denote the set of polynomials $p(z)$ of degree $n$ with coefficients $\\\\pm 1$. Define $$ C^-(n) \\\\coloneqq \\\\max_{p \\\\in \\\\mathbb{U}_{n}} \\\\left(\\min_{|z|=1}\\\\frac{|p(z)|}{\\\\sqrt{n+1}}\\\\right) $$ $$ C^+(n) \\\\coloneqq \\\\min_{p \\\\in \\\\mathbb{U}_{n}}\\\\left(\\\\max_{|z|=1}\\\\frac{|p(z)|}{\\\\sqrt{n+1}}\\\\right) $$ $$ C^w(n) \\\\coloneqq \\\\min_{p \\\\in \\\\mathbb{U}_{n}}\\\\le", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/28.html", "page": "t/alphaevolve-28.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/28.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "below", "outcome_text": "Did not reach the best known construction", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/28.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:28", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "3801cedc2613", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:29", "key": "29", "label": "Block Stacking Problem", "collection": "AlphaEvolve problems", "kind": "target", "name": "Block Stacking Problem", "description": "Let $n \\\\geq 1$. Let $C(n)$ be the largest displacement that the $n^{\\\\mathrm{th}}$ block in a stack of identical rigid rectangular blocks of width $1$ can be displaced horizontally over the edge of a table, with the stack remaining stable. More mathematically, $C(n)$ is the supremum of $x_n$ where $0 = x_0 \\\\leq x_1 \\\\leq \\\\dots\\\\leq x_n$ are real numbers subject to the constraints $$ \\\\frac{x_{i", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/29.html", "page": "t/alphaevolve-29.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/29.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "matched", "outcome_text": "Matched the best known construction", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/29.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:29", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "58f89e2147d8", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:3", "key": "3", "label": "How much can an autocorrelation of a non-negative function resemble an indicator function?", "collection": "AlphaEvolve problems", "kind": "target", "name": "How much can an autocorrelation of a non-negative function resemble an indicator function?", "description": "Let $C$ be the best constant for which one has $$ \\\\|f\\ * f\\\\|_{L^2(\\\\mathbb{R})}^2 \\\\leq C \\\\|f*f\\\\|_{L^1(\\\\mathbb{R})} \\\\|f * f\\\\|_{L^\\\\infty(\\\\mathbb{R})}$$ for non-negative $f \\\\colon \\\\mathbb{R} \\\\to \\\\mathbb{R}$. What is $C$?", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/3.html", "page": "t/alphaevolve-3.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/3.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "superseded", "outcome_text": "Held the record at publication, since beaten", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/3.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:3", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "8e2fe50f1d98", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:30", "key": "30", "label": "The Arithmetic Kakeya Conjecture", "collection": "AlphaEvolve problems", "kind": "target", "name": "The Arithmetic Kakeya Conjecture", "description": "For each slope $r \\in \\\\mathbb{R} \\\\cup \\\\{\\\\infty\\\\}$ define the projection $\\\\pi_r : \\\\mathbb{R}^2 \\\\to \\\\mathbb{R}$ by $\\\\pi_r(a,b) = a + rb$ for $r \\\\neq \\\\infty$ and $\\\\pi_\\\\infty(a,b)=b$. Given a set $r_1,\\\\dots,r_k, r_\\\\infty$ of distinct slopes, we let $C(\\\\{r_1,\\\\dots,r_k\\\\}; r_\\\\infty)$ be the smallest constant for which the following is true: if $X,Y$ are discrete random variables (not ", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/30.html", "page": "t/alphaevolve-30.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/30.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "record", "outcome_text": "Best known construction at publication", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/30.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:30", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "7899428c56dc", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:31", "key": "31", "label": "Furstenberg-Sárközy Problem", "collection": "AlphaEvolve problems", "kind": "target", "name": "Furstenberg-Sárközy Problem", "description": "If $k, m \\\\geq 2$ and $N \\\\geq 1$, let $C(k,N)$ (resp. $C(k,\\\\Z/M\\\\Z)$) denote the size of the largest subset of $\\{1,\\\\dots,N\\}$ that does not contain any two elements that differ by a perfect $k^{\\\\mathrm{th}}$ power. Establish upper and lower bounds for $C(k,N)$ and $C(k,\\\\Z/M\\\\Z)$ that are as strong as possible.", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/31.html", "page": "t/alphaevolve-31.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/31.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "unstated", "outcome_text": "", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/31.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:31", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "c901e1db393d", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:32", "key": "32", "label": "Spherical Designs", "collection": "AlphaEvolve problems", "kind": "target", "name": "Spherical Designs", "description": "A spherical $t$-design on the $d$-dimensional sphere $S^d \\\\subset \\\\R^{d+1}$ is a finite set of points $X \\\\subset S^d$ such that for any polynomial $P$ of degree at most $t$, the average value of $P$ over $X$ is equal to the average value of $P$ over the entire sphere $S^d$. For each $t \\\\in \\\\mathbb{N}$, let $C(d,t)$ be the minimal number of points in a spherical $t$-design. Establish upper and", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/32.html", "page": "t/alphaevolve-32.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/32.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "record", "outcome_text": "Best known construction at publication", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/32.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:32", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "d2e83790fbfb", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:33", "key": "33", "label": "The Thomson Problem", "collection": "AlphaEvolve problems", "kind": "target", "name": "The Thomson Problem", "description": "For any $N>1$, let $C(N)$ denote the infimum of the Coulomb energy $$ E(z_1,\\\\ldots,z_N) \\\\coloneqq \\\\sum_{1 \\\\leq i", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/33.html", "page": "t/alphaevolve-33.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/33.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "unstated", "outcome_text": "", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/33.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:33", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "bacf0192af2e", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:34", "key": "34", "label": "Tammes problem", "collection": "AlphaEvolve problems", "kind": "target", "name": "Tammes problem", "description": "For $N \\\\geq 2$, let $C(N)$ denote the maximal value of the energy $$E(z_1,\\\\dots,z_N) \\\\coloneqq \\\\min_{1 \\\\leq i", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/34.html", "page": "t/alphaevolve-34.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/34.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "unstated", "outcome_text": "", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/34.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:34", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "b528fe1de82b", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:35", "key": "35", "label": "Packing in a dilate", "collection": "AlphaEvolve problems", "kind": "target", "name": "Packing in a dilate", "description": "For any $n \\\\geq 1$ and a geometric shape $P$ (e.g. a polygon, a polytope or a sphere), let $C(n, P)$ denote the smallest scale $s$ such that one can place $n$ identical copies of $P$ with disjoint interiors inside another copy of $P$ scaled up by a factor of $s$. Establish lower and upper bounds for $C(n, P)$ that are as strong as possible.", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/35.html", "page": "t/alphaevolve-35.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/35.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "record", "outcome_text": "Best known construction at publication", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/35.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:35", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "fa003bd930cd", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:36", "key": "36", "label": "Circle packing in a square / rectangle", "collection": "AlphaEvolve problems", "kind": "target", "name": "Circle packing in a square / rectangle", "description": "For any $n \\\\geq 1$, let $C(n)$ denote the largest sum $\\\\sum_{i=1}^n r_i$ of radii such that one can place $n$ disjoint open disks of radius $r_1,\\\\dots,r_n$ inside the unit square, and let $C'(n)$ denote the largest sum $\\\\sum_{i=1}^n r_i$ of radii such that one can place $n$ disjoint open disks of radius $r_1,\\\\dots,r_n$ inside a rectangle of perimeter $4$. Establish upper and lower bounds for ", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/36.html", "page": "t/alphaevolve-36.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/36.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "superseded", "outcome_text": "Held the record at publication, since beaten", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/36.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:36", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "a34922952e10", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:37", "key": "37", "label": "The hypergraph Turán number of the tetrahedron", "collection": "AlphaEvolve problems", "kind": "target", "name": "The hypergraph Turán number of the tetrahedron", "description": "Let $C$ be the largest quantity such that, as $n \\\\to \\\\infty$, one can locate a $3$-uniform hypergraph on $n$ vertices and at least $(C-o(1)) \\\\binom{n}{3}$ edges that contains no copy of the tetrahedron $K^{(3)}_4$. What is $C$?", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/37.html", "page": "t/alphaevolve-37.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/37.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "unstated", "outcome_text": "", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/37.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:37", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "457e550a9ac4", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:38", "key": "38", "label": "Factoring $N!$ into $N$ numbers", "collection": "AlphaEvolve problems", "kind": "target", "name": "Factoring $N!$ into $N$ numbers", "description": "For a natural number $N$, let $C(N)$ be the largest quantity such that $N!$ can be factored into $N$ factors that are greater than or equal to $C(N)$ (see OEIS A034258 ). Establish upper and lower bounds on $C(N)$ that are as strong as possible.", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/38.html", "page": "t/alphaevolve-38.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/38.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "superseded", "outcome_text": "Held the record at publication, since beaten", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/38.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:38", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "6627cb857023", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:39", "key": "39", "label": "Beat the average game", "collection": "AlphaEvolve problems", "kind": "target", "name": "Beat the average game", "description": "Let $C$ denote the quantity $$C \\\\coloneqq \\\\sup_{\\\\mu} \\\\mathbb{P}[X_1 + X_2 + X_3", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/39.html", "page": "t/alphaevolve-39.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/39.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "below", "outcome_text": "Did not reach the best known construction", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/39.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:39", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "96cba1b4944e", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:4", "key": "4", "label": "An autocorrelation problem with functions that can take negative values", "collection": "AlphaEvolve problems", "kind": "target", "name": "An autocorrelation problem with functions that can take negative values", "description": "Let $C$ be the best constant for which one has $$ \\\\max_{-1/2 \\\\leq t \\\\leq 1/2}\\\\left |\\\\int_\\\\mathbb{R} f(t-x) f(x)\\ dx\\\\right| \\\\geq C \\\\left(\\\\int_{-1/4}^{1/4} f(x)\\ dx\\\\right)^2$$ for all $f \\\\colon [-1/4,1/4] \\\\to \\\\mathbb{R}$ (note $f$ can take negative values). What is $C$?", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/4.html", "page": "t/alphaevolve-4.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/4.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "record", "outcome_text": "Best known construction at publication", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/4.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:4", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "17ce75537446", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:40", "key": "40", "label": "Erdős discrepancy problem", "collection": "AlphaEvolve problems", "kind": "target", "name": "Erdős discrepancy problem", "description": "The discrepancy of a sign pattern $a_1,\\\\dots,a_N \\\\in \\\\{-1,+1\\\\}$ is the maximum value of $|a_d + a_{2d} + \\\\dots + a_{kd}|$ for homogeneous progressions $d,\\\\dots,kd$ in $\\\\{1,\\\\dots,N\\\\}$. For any $D \\\\geq 1$, let $C(D)$ denote the largest $N$ for which there exists a sign pattern $a_1,\\\\dots,a_N$ of discrepancy at most $C$. Establish upper and lower bounds on $C(D)$ that are as strong as poss", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/40.html", "page": "t/alphaevolve-40.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/40.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "below", "outcome_text": "Did not reach the best known construction", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/40.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:40", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "a8d493d98a9e", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:41", "key": "41", "label": "Points on sphere maximizing the volume", "collection": "AlphaEvolve problems", "kind": "target", "name": "Points on sphere maximizing the volume", "description": "For any $n \\\\geq 4$, Let $C(n)$ denote the maximum volume of a polyhedron with $n$ vertices that all lie on the unit sphere $\\\\mathbb S^2$. What is $C(n)$? Which polyhedra attain the maximum volume?", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/41.html", "page": "t/alphaevolve-41.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/41.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "unstated", "outcome_text": "", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/41.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:41", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "468c16f8c15d", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:42", "key": "42", "label": "Sum-difference problem I", "collection": "AlphaEvolve problems", "kind": "target", "name": "Sum-difference problem I", "description": "Let $C$ be the least constant such that $$ |A+A|/|A| \\\\leq (|A-A|/|A|)^{C}$$ for any non-empty finite set $A$ of integers. Establish upper and lower bounds for $C$ that are as strong as possible.", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/42.html", "page": "t/alphaevolve-42.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/42.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "record", "outcome_text": "Best known construction at publication", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/42.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:42", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "9ee02c34cde6", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:43", "key": "43", "label": "Sum-difference problem II", "collection": "AlphaEvolve problems", "kind": "target", "name": "Sum-difference problem II", "description": "Let $C$ be the least constant such that $$ |A-A| \\\\leq |A+A|^{C}$$ for any non-empty finite set $A$ of integers. Establish upper and lower bounds for $C$ that are as strong as possible.", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/43.html", "page": "t/alphaevolve-43.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/43.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "below", "outcome_text": "Did not reach the best known construction", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/43.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:43", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "0e1bb0227eaa", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:44", "key": "44", "label": "Sum-difference problem III", "collection": "AlphaEvolve problems", "kind": "target", "name": "Sum-difference problem III", "description": "Let $C$ be the supremum of all constants such that there exist arbitrarily large finite sets of integers $A,B$ with $|A+B| \\\\lesssim |A|$ and $|A-B| \\\\gtrsim |A|^{C}$. Establish upper and lower bounds for $C$ that are as strong as possible.", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/44.html", "page": "t/alphaevolve-44.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/44.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "superseded", "outcome_text": "Held the record at publication, since beaten", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/44.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:44", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "0ea013e800c7", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:45", "key": "45", "label": "Sum-product problems", "collection": "AlphaEvolve problems", "kind": "target", "name": "Sum-product problems", "description": "Given a natural number $N$ and a ring $R$ of size at least $N$, let $C(R, N)$ denote the least possible value of $\\\\max(|A+A|, |A \\\\cdot A|)$ where $A$ ranges over subsets of $R$ of cardinality $N$. Establish upper and lower bounds for $C(R, N)$ that are as strong as possible.", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/45.html", "page": "t/alphaevolve-45.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/45.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "unstated", "outcome_text": "", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/45.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:45", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "cef074151222", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:46", "key": "46", "label": "Minimal triangle density in graphs", "collection": "AlphaEvolve problems", "kind": "target", "name": "Minimal triangle density in graphs", "description": "For $0 \\\\leq \\\\rho \\\\leq 1$, let $C(\\\\rho)$ denote the largest quantity such that any graph on $n$ vertices and $(\\\\rho+o(1)) \\\\binom{n}{2}$ edges will have at least $(C(\\\\rho)-o(1)) \\\\binom{n}{3}$ triangles. What is $C(\\\\rho)$?", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/46.html", "page": "t/alphaevolve-46.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/46.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "matched", "outcome_text": "Matched the best known construction", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/46.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:46", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "37956aa6e859", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:47", "key": "47", "label": "Matrix multiplications and AM-GM inequalities", "collection": "AlphaEvolve problems", "kind": "target", "name": "Matrix multiplications and AM-GM inequalities", "description": "For positive-semidefinite $d \\\\times d$ matrices $A_1, \\\\ldots, A_n$ and any unitarily invariant norm $|||\\\\cdot|||$ (including the operator norm and Schatten $p$-norms) and $m \\\\leq n$, define $$ C(n,m,d) \\\\coloneqq \\\\inf \\\\frac{ \\\\frac{1}{n^m} \\\\sum_{j_1, j_2, \\\\ldots, j_m = 1}^{n} |||A_{j_1}A_{j_2}\\\\ldots A_{j_m}|||}{ \\\\frac{(n-m)!}{n!} \\\\sum_{\\\\substack{j_1, j_2, \\\\ldots, j_m = 1 \\\\\\\\ \\\\text{a", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/47.html", "page": "t/alphaevolve-47.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/47.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "unstated", "outcome_text": "", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/47.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:47", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "5026ac6f5f86", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:48", "key": "48", "label": "Heilbronn problem in a fixed bounding box", "collection": "AlphaEvolve problems", "kind": "target", "name": "Heilbronn problem in a fixed bounding box", "description": "For any $n \\\\geq 3$ and any convex body $K$ in the plane, let $C(n,K)$ be the largest quantity such that in every configuration of $n$ points in $K$, there exists a triple of points determining a triangle of area at most $C(n,K)$ times the area of $K$. Establish upper and lower bounds on $C(n,K)$.", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/48.html", "page": "t/alphaevolve-48.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/48.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "record", "outcome_text": "Best known construction at publication", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/48.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:48", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "2430a7b51f5a", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:49", "key": "49", "label": "Heilbronn problem in an arbitrary convex bounding box", "collection": "AlphaEvolve problems", "kind": "target", "name": "Heilbronn problem in an arbitrary convex bounding box", "description": "For any $n \\\\geq 3$ let $C(n)$ be the largest quantity such that in every configuration of $n$ points in the plane, there exists a triple of points determining a triangle of area at most $C(n)$ times the area of their convex hull. Establish upper and lower bounds on $C(n)$.", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/49.html", "page": "t/alphaevolve-49.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/49.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "record", "outcome_text": "Best known construction at publication", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/49.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:49", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "381462bb3537", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:5", "key": "5", "label": "Erdős minimum overlap problem", "collection": "AlphaEvolve problems", "kind": "target", "name": "Erdős minimum overlap problem", "description": "Let $C$ be the largest constant for which $$ \\\\sup_{x \\\\in [-2,2]} \\\\int_{-1}^1 f(t) g(x+t)\\ dt\\\\geq C$$ for all non-negative $f,g: [-1,1] \\\\to [0,1]$ with $f+g=1$ on $[-1,1]$ and $\\\\int_\\\\mathbb{R} f = 1$, where we extend $f,g$ by zero outside of $[-1,1]$. What is $C$?", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/5.html", "page": "t/alphaevolve-5.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/5.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "record", "outcome_text": "Best known construction at publication", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/5.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:5", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "6a083de5192b", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:50", "key": "50", "label": "Max to min ratios", "collection": "AlphaEvolve problems", "kind": "target", "name": "Max to min ratios", "description": "Let $n,d \\\\geq 2$. Let $C(d,n)$ denote the largest quantity such that, given any $n$ distinct points $x_1,\\\\dots,x_n$ in $\\\\R^d$, the maximum distance $\\\\max_{1 \\\\leq i", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/50.html", "page": "t/alphaevolve-50.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/50.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "record", "outcome_text": "Best known construction at publication", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/50.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:50", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "6b440520f388", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:51", "key": "51", "label": "Erdős–Gyárfás conjecture", "collection": "AlphaEvolve problems", "kind": "target", "name": "Erdős–Gyárfás conjecture", "description": "Let $G$ be a finite graph with minimum degree at least $3$. Must $G$ contain a cycle of length $2^k$ for some $k \\\\geq 2$?", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/51.html", "page": "t/alphaevolve-51.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/51.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "unstated", "outcome_text": "", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/51.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:51", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "b66f0cb0cb21", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:52", "key": "52", "label": "Erdős squarefree problem", "collection": "AlphaEvolve problems", "kind": "target", "name": "Erdős squarefree problem", "description": "For any natural number $N$, let $C(N)$ denote the largest cardinality of a subset $A$ of $\\\\{1,\\\\dots,N\\\\}$ with the property that $ab+1$ is square-free for all $a,b \\\\in A$. Establish upper and lower bounds for $C(N)$ that are as strong as possible.", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/52.html", "page": "t/alphaevolve-52.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/52.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "unstated", "outcome_text": "", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/52.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:52", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "5ea245772fad", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:53", "key": "53", "label": "Equidistant points in convex polygons", "collection": "AlphaEvolve problems", "kind": "target", "name": "Equidistant points in convex polygons", "description": "Is it true that every convex polygon has a vertex with no other 4 vertices equidistant from it?", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/53.html", "page": "t/alphaevolve-53.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/53.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "unstated", "outcome_text": "", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/53.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:53", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "5bcfaf1993e2", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:54", "key": "54", "label": "Pairwise touching cylinders", "collection": "AlphaEvolve problems", "kind": "target", "name": "Pairwise touching cylinders", "description": "Is it possible for seven infinite circular cylinders $C_1,\\\\dots,C_7$ of unit radius to touch all the others?", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/54.html", "page": "t/alphaevolve-54.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/54.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "matched", "outcome_text": "Matched the best known construction", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/54.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:54", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "a801b33f0d42", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:55", "key": "55", "label": "Erdős squares in a square problem", "collection": "AlphaEvolve problems", "kind": "target", "name": "Erdős squares in a square problem", "description": "For any natural $n$, let $C(n)$ denote the maximum possible sum of side lengths of $n$ squares with disjoint interiors contained inside a unit square. Obtain upper and lower bounds for $C(n)$ that are as strong as possible.", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/55.html", "page": "t/alphaevolve-55.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/55.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "unstated", "outcome_text": "", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/55.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:55", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "886405aa2cbf", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:56", "key": "56", "label": "Good asymptotic constructions of Szemerédi--Trotter", "collection": "AlphaEvolve problems", "kind": "target", "name": "Good asymptotic constructions of Szemerédi--Trotter", "description": "If $n,m$ are natural numbers, let $C(n,m)$ denote the maximum number of incidences that are possible between $n$ points and $m$ lines in the plane. Establish upper and lower bounds on $C(n,m)$ that are as strong as possible.", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/56.html", "page": "t/alphaevolve-56.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/56.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "unstated", "outcome_text": "", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/56.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:56", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "d2e3aeee8843", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:57", "key": "57", "label": "Rudin problem for polynomials", "collection": "AlphaEvolve problems", "kind": "target", "name": "Rudin problem for polynomials", "description": "Let $d \\\\geq 2$ and $D \\\\geq 1$. For $p \\\\in \\\\{4,\\\\infty\\\\}$, let $C^p(d,D)$ be the maximum of the ratio $$ \\\\frac{\\\\|u\\\\|_{L^p({\\\\mathbb S}^d)}}{\\\\|u\\\\|_{L^2({\\\\mathbb S}^d)}}$$ where $u$ ranges over (real) spherical harmonics of degree $D$ on the $d$-dimensional sphere $\\\\mathbb S^d$, which we normalize to have unit measure. Establish upper and lower bounds on $C^p(d,D)$ that are as strong as p", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/57.html", "page": "t/alphaevolve-57.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/57.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "unstated", "outcome_text": "", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/57.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:57", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "feb1d8679254", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:58", "key": "58", "label": "Erdős--Szekeres Happy Ending problem", "collection": "AlphaEvolve problems", "kind": "target", "name": "Erdős--Szekeres Happy Ending problem", "description": "For $k \\\\geq 3$, let $C(k)$ be the smallest integer such that every set of $C(k)$ points in the plane in general position contains a convex $k$-gon. Obtain upper and lower bounds for $C(k)$ that are as strong as possible.", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/58.html", "page": "t/alphaevolve-58.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/58.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "unstated", "outcome_text": "", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/58.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:58", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "32f501b894a1", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:59", "key": "59", "label": "Subsets of the grid with no isosceles triangles", "collection": "AlphaEvolve problems", "kind": "target", "name": "Subsets of the grid with no isosceles triangles", "description": "For $n$ a natural number, let $C(n)$ denote the size of the largest subset of $[n]^2 = \\\\{1,\\\\dots,n\\\\}^2$ that does not contain a (possibly flat) isosceles triangle. In other words, $$C(n) \\\\coloneqq \\\\max_{S\\\\subset [n]^2}\\\\{|S|: a,b,c\\\\in S \\\\text{ distinct} \\\\implies \\\\|a-b\\\\| \\\\neq \\\\|b-c\\\\|\\\\}.$$ Obtain upper and lower bounds for $C(n)$ that are as strong as possible.", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/59.html", "page": "t/alphaevolve-59.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/59.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "record", "outcome_text": "Best known construction at publication", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/59.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:59", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "8c5fe1bea9bc", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:6", "key": "6", "label": "An autocorrelation problem related to difference bases", "collection": "AlphaEvolve problems", "kind": "target", "name": "An autocorrelation problem related to difference bases", "description": "Let $C$ be the smallest constant such that $$ \\\\min_{0 \\\\leq t \\\\leq 1} \\\\int_\\\\mathbb{R} f(x) f(x+t)\\ dx \\\\leq C \\\\|f\\\\|_{L^1(\\\\mathbb{R})}^2 $$ for $f \\\\in L^1(\\\\mathbb{R})$. What is $C$?", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/6.html", "page": "t/alphaevolve-6.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/6.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "below", "outcome_text": "Did not reach the best known construction", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/6.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:6", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "5572bbce5704", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:60", "key": "60", "label": "The no 5 on a sphere problem", "collection": "AlphaEvolve problems", "kind": "target", "name": "The no 5 on a sphere problem", "description": "For $n$ a natural number, let $C(n)$ denote the size of the largest subset of $[n]^3 = \\\\{1,\\\\dots,n\\\\}^3$ such that no 5 points lie on a sphere or a plane. Obtain upper and lower bounds for $C(n)$ that are as strong as possible.", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/60.html", "page": "t/alphaevolve-60.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/60.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "record", "outcome_text": "Best known construction at publication", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/60.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:60", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "586e181c0de0", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:61", "key": "61", "label": "The Ring Loading Problem", "collection": "AlphaEvolve problems", "kind": "target", "name": "The Ring Loading Problem", "description": "Let $C$ be the infimum of all reals $\\\\alpha$ for which the following statement holds: for all positive integers $m$ and nonnegative reals $u_1, \\\\ldots, u_m$ and $v_1, \\\\ldots, v_m$ with $u_i + v_i \\\\leq 1$, there exist $z_1, \\\\ldots, z_m$ such that for every $k$, we have $z_k \\\\in \\\\{v_k, -u_k\\\\}$, and $$\\\\left|\\\\sum_{i=1}^k z_i - \\\\sum_{i=k+1}^m z_i\\\\right|\\\\leq \\\\alpha.$$ Obtain upper and lowe", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/61.html", "page": "t/alphaevolve-61.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/61.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "record", "outcome_text": "Best known construction at publication", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/61.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:61", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "56193ef2fbd4", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:62", "key": "62", "label": "The moving sofa problem", "collection": "AlphaEvolve problems", "kind": "target", "name": "The moving sofa problem", "description": "Define $C$ to be the largest area of a connected bounded subset $S$ of $\\\\R^2$ that can continuously pass through an $L$-shaped corner of unit width (e.g., $[0,1] \\\\times [0,+\\\\infty) \\\\cup [0,+\\\\infty) \\\\times [0,1]$). What is $C$?", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/62.html", "page": "t/alphaevolve-62.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/62.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "matched", "outcome_text": "Matched the best known construction", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/62.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:62", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "b58bb6712036", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:63", "key": "63", "label": "The ambidextrous moving sofa problem", "collection": "AlphaEvolve problems", "kind": "target", "name": "The ambidextrous moving sofa problem", "description": "Define $C$ to be the largest area of a connected planar shape $C$ that can continuously pass through both a left-turning and right-turning L-shaped corner of unit width (e.g., both $[0,1] \\\\times [0,+\\\\infty) \\\\cup [0,+\\\\infty) \\\\times [0,1]$ and $[0,1] \\\\times [0,+\\\\infty) \\\\cup (-\\\\infty,1] \\\\times [0,1]$). What is $C$?", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/63.html", "page": "t/alphaevolve-63.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/63.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "matched", "outcome_text": "Matched the best known construction", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/63.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:63", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "d13ab0a55eaf", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:64", "key": "64", "label": "The three-dimensional moving sofa problem with two perpendicular turns", "collection": "AlphaEvolve problems", "kind": "target", "name": "The three-dimensional moving sofa problem with two perpendicular turns", "description": "Define $C$ to be the largest volume of a connected bounded subset $S_3$ of $\\\\R^3$ that can continuously pass through a three-dimensional snake-shaped corridor with a unit square cross-section, consisting of two turns in the $x-y$ and $y-z$ planes that are far apart. What is $C$?", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/64.html", "page": "t/alphaevolve-64.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/64.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "record", "outcome_text": "Best known construction at publication", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/64.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:64", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "95ba03acb13a", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:65", "key": "65", "label": "Variant of IMO 2025, Problem 6", "collection": "AlphaEvolve problems", "kind": "target", "name": "Variant of IMO 2025, Problem 6", "description": "Consider a $2025 \\\\times 2025$ (and more generally an $n \\\\times n$) grid of unit squares. Matilda wishes to place on the grid some rectangular tiles, possibly of different sizes, such that each side of every tile lies on a grid line and every unit square is covered by at most one tile. Determine the minimum number of tiles $C(n)$) Matilda needs to place so that each row and each column of the gri", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/65.html", "page": "t/alphaevolve-65.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/65.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "matched", "outcome_text": "Matched the best known construction", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/65.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:65", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "1cd51dfe6246", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:66", "key": "66", "label": "The function guessing game", "collection": "AlphaEvolve problems", "kind": "target", "name": "The function guessing game", "description": "Write a program that is able to guess a hidden function. $f \\\\colon \\\\R\\\\rightarrow \\\\R$. Your program will receive a reward of $1000$ currency units for every function that it guessed correctly (the $L^1$ norm of the difference between the correct and the guessed functions had to be below a small threshold). To gather information about the hidden function, your program is allowed to (1) evaluate ", "direction": "unstated", "unit": "", "value_key": "", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/66.html", "page": "t/alphaevolve-66.html", "aliases": [ "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/66.html" ], "tags": [ "construction" ], "records": [], "best": null, "previous": null, "held_by_machine": false, "claims": [ { "systems": [ "AlphaEvolve" ], "humans": [], "date": "2025-11-05", "outcome": "unstated", "outcome_text": "", "autonomy": "A2", "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problems.", "lean_claimed": false, "sources": [ { "label": "AlphaEvolve repository of problems", "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/66.html" }, { "label": "Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864" } ], "details": {}, "statement": "alphaevolve:66", "node": "target", "contribution": "primary", "outcome_scope": "record", "section": "record", "section_title": "Record on a quantity", "date_text": "2025-11-05", "machine_checked": false, "checked_by": [], "id": "7b429fdb0446", "verification": "V1", "fidelity": "F0" } ], "checks": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "mentions": [], "relations": [], "verified": false, "checked_by": [] }, { "id": "alphaevolve:67", "key": "67", "label": "Logic puzzles", "collection": "AlphaEvolve problems", "kind": "target", "name": "Logic puzzles", "description": "We have three guards in front of three doors. The guards are, in some order, an angel (always tells the truth), the devil (always lies), and the gatekeeper (answers truthfully if and only if the question is about the prize behind Door A). The prizes behind the doors are €0, €100, and €110. You can ask two yes/no questions and want to maximize your expected profit. 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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.) 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