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  {
    "id": "matmul:2x2-strassen",
    "key": "2x2-strassen",
    "label": "Multiplications needed for 2x2 matrix multiplication over F_2",
    "collection": "Algorithms",
    "kind": "target",
    "name": "Multiplications needed for 2x2 matrix multiplication over F_2",
    "description": "Strassen showed in 1969 that seven multiplications suffice, and seven is known to be optimal. Included as a control: a system that could not recover it would not be worth believing elsewhere.",
    "about": "Seven multiplications suffice, Strassen showed in 1969, and Winograd and, independently, Hopcroft and Kerr proved in 1971 that six do not. AlphaTensor recovered a seven-multiplication algorithm in 2022. The record cannot move; it is kept as a control.",
    "direction": "minimise",
    "unit": "multiplications",
    "format": "",
    "value_key": "rank",
    "url": "https://github.com/google-deepmind/alphatensor",
    "page": "t/matmul-2x2-strassen",
    "aliases": [],
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      "tensor decomposition"
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        "display": "7 multiplications",
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        "kind": "human",
        "date": "1969",
        "note": "Optimal for this case.",
        "source": "https://doi.org/10.1007/bf02165411"
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        "value": 7,
        "display": "7 multiplications",
        "holder": "AlphaTensor",
        "kind": "ai",
        "date": "2022-10-05",
        "note": "Recovered a rank-7 decomposition, matching Strassen's; seven is optimal, so the record cannot move.",
        "source": "https://www.nature.com/articles/s41586-022-05172-4"
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        "value": 7,
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        "note": "Recovered a rank-7 decomposition, matching Strassen's; seven is optimal, so the record cannot move.",
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        "outcome_text": "Recovered a rank-7 decomposition, matching the optimum.",
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        "autonomy_reason": "Reinforcement learning against a human-designed game.",
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        "section_title": "Record on a quantity",
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        "target": "matmul:2x2-strassen",
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        "artifact": "https://github.com/google-deepmind/alphatensor/blob/main/algorithms/factorizations_f2.npz",
        "method": "evaluator",
        "checked": "2026-08-22T21:21:52Z",
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          "rank": 7
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        },
        "verdict": "verified",
        "detail": "2x2x2 in 7 multiplications, exact identity mod 2",
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  {
    "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.",
    "about": "Applying Strassen's 1969 algorithm to 2×2 blocks multiplies 4×4 matrices with 49 multiplications. AlphaTensor found a 47-multiplication algorithm valid over the field with two elements in 2022. AlphaEvolve's 48-multiplication algorithm of 2025 works over the complex numbers, a different question, and is not a step here.",
    "direction": "minimise",
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    "format": "",
    "value_key": "rank",
    "url": "https://github.com/google-deepmind/alphatensor",
    "page": "t/matmul-4x4-mod2",
    "aliases": [],
    "tags": [
      "algorithms",
      "tensor decomposition"
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        "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"
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      {
        "value": 49,
        "display": "49 multiplications",
        "holder": "Strassen, applied recursively",
        "kind": "human",
        "date": "1969",
        "note": "Two levels of Strassen's 1969 algorithm.",
        "source": "https://doi.org/10.1007/bf02165411"
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        "value": 49,
        "display": "49 multiplications",
        "holder": "Strassen, applied recursively",
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        "source": "https://doi.org/10.1007/bf02165411",
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        "value": 47,
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      "note": "The first improvement on the recursive Strassen count for this case since 1969.",
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      "kind": "human",
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      "note": "Two levels of Strassen's 1969 algorithm.",
      "source": "https://doi.org/10.1007/bf02165411"
    },
    "held_by_machine": true,
    "claims": [
      {
        "systems": [
          "AlphaTensor"
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        "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,
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            "label": "Nature, Discovering faster matrix multiplication algorithms",
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        ],
        "details": {},
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        "contribution": "primary",
        "outcome_scope": "record",
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        "section_title": "Record on a quantity",
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        "claim": {
          "rank": 47
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          "mismatches": 0,
          "modulus": 2
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        "verdict": "verified",
        "detail": "4x4x4 in 47 multiplications, exact identity mod 2",
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  {
    "id": "matmul:omega",
    "key": "omega",
    "label": "Exponent of matrix multiplication",
    "collection": "Algorithms",
    "kind": "target",
    "name": "Exponent of matrix multiplication",
    "description": "The smallest ω such that two n×n matrices can be multiplied in about n^ω operations. The schoolbook method gives 3, and ω is at least 2; the record is the best proven upper bound.",
    "about": "Strassen's 1969 algorithm first brought the exponent below 3. By 1990 the methods of Coppersmith, Winograd and Strassen had taken it to 2.3755, and refinements since 2012 have moved it in the third and fourth decimal places. The 2026 step came from a new optimisation algorithm for the combination-loss analysis, refined with AlphaEvolve.",
    "direction": "minimise",
    "unit": "exponent (upper bound)",
    "format": "",
    "value_key": "",
    "url": "https://arxiv.org/abs/2608.16884",
    "page": "t/matmul-omega",
    "aliases": [],
    "tags": [
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      "matrix multiplication"
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        "holder": "Emilien Dupont and nine co-authors, with AlphaEvolve",
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        "note": "A reformulated optimisation problem and a new optimisation algorithm, refined with AlphaEvolve.",
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        "value": 2.371339,
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        "note": "More asymmetry yields faster matrix multiplication.",
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        "value": 2.371552,
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        "holder": "Virginia Vassilevska Williams, Yinzhan Xu, Zixuan Xu and Renfei Zhou",
        "kind": "human",
        "date": "2023-07",
        "note": "From alpha to omega.",
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      {
        "value": 2.371866,
        "display": "2.371866",
        "holder": "Ran Duan, Hongxun Wu and Renfei Zhou",
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        "date": "2022-11",
        "note": "Asymmetric hashing.",
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        "value": 2.3728596,
        "display": "2.3728596",
        "holder": "Josh Alman and Virginia Vassilevska Williams",
        "kind": "human",
        "date": "2020-10",
        "note": "A refined laser method.",
        "source": "https://arxiv.org/abs/2010.05846"
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        "value": 2.3728639,
        "display": "2.3728639",
        "holder": "François Le Gall",
        "kind": "human",
        "date": "2014-01",
        "note": "Powers of tensors and fast matrix multiplication.",
        "source": "https://arxiv.org/abs/1401.7714"
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        "value": 2.3729,
        "display": "2.3729",
        "holder": "Virginia Vassilevska Williams",
        "kind": "human",
        "date": "2012",
        "note": "Multiplying matrices faster than Coppersmith–Winograd.",
        "source": "https://doi.org/10.1145/2213977.2214056"
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        "value": 2.3755,
        "display": "2.3755",
        "holder": "Don Coppersmith and Shmuel Winograd",
        "kind": "human",
        "date": "1990",
        "note": "Matrix multiplication via arithmetic progressions; the record for two decades.",
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        "value": 2.479,
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        "holder": "Volker Strassen",
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        "note": "The laser method.",
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        "display": "2.496",
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        "date": "1982",
        "order": 2,
        "note": "On the asymptotic complexity of matrix multiplication.",
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        "value": 2.517,
        "display": "2.517",
        "holder": "Francesco Romani",
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        "date": "1982",
        "order": 1,
        "note": "Disjoint sums of tensors.",
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        "value": 2.522,
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        "holder": "Arnold Schönhage",
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        "date": "1981",
        "note": "Partial and total matrix multiplication, and the asymptotic sum inequality.",
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        "holder": "Dario Bini, Milvio Capovani, Grazia Lotti and Francesco Romani",
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        "date": "1979",
        "note": "Approximate algorithms and border rank.",
        "source": "https://doi.org/10.1016/0020-0190(79)90113-3"
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        "value": 2.796,
        "display": "2.796",
        "holder": "Victor Pan",
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        "date": "1978",
        "note": "Trilinear aggregating.",
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      {
        "value": 2.8074,
        "display": "2.8074",
        "holder": "Volker Strassen",
        "kind": "human",
        "date": "1969",
        "note": "Gaussian elimination is not optimal: seven multiplications for 2×2, applied recursively.",
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        "note": "Disjoint sums of tensors.",
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        "value": 2.496,
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        "order": 2,
        "note": "On the asymptotic complexity of matrix multiplication.",
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        "date": "1986",
        "note": "The laser method.",
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        "value": 2.3755,
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        "date": "1990",
        "note": "Matrix multiplication via arithmetic progressions; the record for two decades.",
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        "value": 2.3729,
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        "holder": "Virginia Vassilevska Williams",
        "kind": "human",
        "date": "2012",
        "note": "Multiplying matrices faster than Coppersmith–Winograd.",
        "source": "https://doi.org/10.1145/2213977.2214056",
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        "value": 2.3728639,
        "display": "2.3728639",
        "holder": "François Le Gall",
        "kind": "human",
        "date": "2014-01",
        "note": "Powers of tensors and fast matrix multiplication.",
        "source": "https://arxiv.org/abs/1401.7714",
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        "date": "2020-10",
        "note": "A refined laser method.",
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        "value": 2.371866,
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        "date": "2022-11",
        "note": "Asymmetric hashing.",
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        "note": "From alpha to omega.",
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        "date": "2024-04",
        "note": "More asymmetry yields faster matrix multiplication.",
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          "Josh Alman",
          "Virginia Vassilevska Williams",
          "Matej Balog"
        ],
        "date": "2026-08-17",
        "outcome": "record",
        "outcome_text": "An upper bound of 2.371177 on the exponent of matrix multiplication, improving 2.371339.",
        "autonomy": "A1",
        "autonomy_reason": "The authors reformulated the optimisation problem and designed a new algorithm for it; AlphaEvolve played a supporting role, refining that algorithm, as the abstract says.",
        "lean_claimed": false,
        "sources": [
          {
            "label": "Dupont et al., Improving the matrix multiplication exponent with modern optimization and AlphaEvolve",
            "url": "https://arxiv.org/abs/2608.16884"
          }
        ],
        "details": {},
        "statement": "matmul:omega",
        "node": "target",
        "contribution": "primary",
        "outcome_scope": "record",
        "section": "record",
        "section_title": "Record on a quantity",
        "date_text": "2026-08-17",
        "machine_checked": false,
        "checked_by": [],
        "id": "e304d0e0c739",
        "verification": "V1",
        "fidelity": "F0"
      }
    ],
    "checks": [],
    "declarations": [],
    "fidelity": {
      "grade": "F0",
      "reason": "Absent. No formal statement is attached to the result.",
      "anchor": "",
      "flags": []
    },
    "mentions": [],
    "relations": [],
    "verified": false,
    "checked_by": []
  },
  {
    "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 $\\\\mathbb{F}_q^d$ respectively.",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/1.html",
    "page": "t/alphaevolve-1",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/1.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. 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 $$\\\\begin{align*} \\&\\\\quad (x_{j,k},y_{j,k},0), \\\\left(x_{j,k}+\\\\frac{1}{n},y_{j,k},0\\\\right), \\\\left(x_{j,k},y_{j,k}+\\\\frac{1}{n},0\\\\right), \\\\left(x_{j,k}+\\\\frac{1}{n},y_{j,k}+\\\\frac{1}{n},0\\\\right),\\\\ &\\\\left(x_{j,k}+\\\\frac{j}{n},y_{j,k}+\\\\frac{k}{n},1\\\\right), \\\\left(x_{j,k}+\\\\frac{j+1}{n},y_{j,k}+\\\\frac{k}{n},1\\\\right), \\\\left(x_{j,k}+\\\\frac{j}{n},y_{j,k}+\\\\frac{k+1}{n},0\\\\right), \\\\left(x_{j,k}+\\\\frac{j+1}{n},y_{j,k}+\\\\frac{k+1}{n},1\\\\right) \\\\end{align*}$$ for some real numbers $x_{j,k}, y_{j,k}$. Establish upper and lower bounds for $C(n)$ that are as strong as possible.",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/10.html",
    "page": "t/alphaevolve-10",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/10.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. 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 $f(0), \\\\hat f(0) < 0$. Establish upper and lower bounds for $C$ that are as strong as possible.",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/11.html",
    "page": "t/alphaevolve-11",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/11.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. 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.",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/12.html",
    "page": "t/alphaevolve-12",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/12.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. 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 for $C(n)$ that are as strong as possible.",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/13.html",
    "page": "t/alphaevolve-13",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/13.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. 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 $C(p)$?",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/14.html",
    "page": "t/alphaevolve-14",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/14.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. 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 $j < m$. Furthermore, let $1 \\\\leq r \\\\leq \\\\infty, p \\\\geq 1$ be real and $\\\\theta \\\\in [0, 1]$ such that the following relations hold: $$\\\\frac{1}{p} = j + \\\\theta \\\\left( \\\\frac{1}{r} - m \\\\right) + \\\\frac{1 - \\\\theta}{q}, \\\\quad \\\\frac{j}{m} \\\\leq \\\\theta < 1.$$ Let $C(j,p,q,r,m)$ be the best constant such that $$\\\\|D^j u\\\\|_{L^p(\\\\mathbb{R})} \\\\leq C(j,p,q,r,m) \\\\|D^m u\\\\|_{L^r(\\\\mathbb{R})}^\\\\theta \\\\|u\\\\|_{L^q(\\\\mathbb{R})}^{1-\\\\theta}$$ for all test functions $u$, where $D$ denotes the derivative operator $\\\\frac{d}{dx}$. Establish upper and lower bounds for $C(j,p,q,r,m)$ that are as strong as possible.",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/15.html",
    "page": "t/alphaevolve-15",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/15.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. 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 $2 < p < \\\\infty$. Let $C(p)$ denote the supremum of the quantities $$ Q(f) \\\\coloneqq \\\\frac{\\\\|f\\\\|_{L^p(\\\\mathbb{R})}^{4p}}{\\\\|f\\\\|_{L^2(\\\\mathbb{R})}^{2(p+2)} \\|f'\\|_{L^2(\\\\mathbb{R})}^{2(p-2)}}$$ for all smooth rapidly decaying $f$, not identically zero. Establish upper and lower bounds for $C(p)$ that are as strong as possible.",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/16.html",
    "page": "t/alphaevolve-16",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/16.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. 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 $C(p,q,r)$?",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/17.html",
    "page": "t/alphaevolve-17",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/17.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. 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 $C$?",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/18.html",
    "page": "t/alphaevolve-18",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/18.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. 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.",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/19.html",
    "page": "t/alphaevolve-19",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/19.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. 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 $C$?",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/2.html",
    "page": "t/alphaevolve-2",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/2.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. 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 $C(n)=1$.",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/20.html",
    "page": "t/alphaevolve-20",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/20.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. 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 $$ \\\\min_{1 \\\\leq j \\\\leq n-1} \\\\left| \\\\sum_{k=1}^n l_k z_k - w_j \\\\right| \\\\leq C(n). $$ It was conjectured by Schmeisser that $C(n)=1$.",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/21.html",
    "page": "t/alphaevolve-21",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/21.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. 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 $C(p,n)$?",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/22.html",
    "page": "t/alphaevolve-22",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/22.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. 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.",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/23.html",
    "page": "t/alphaevolve-23",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/23.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. 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 $\\\\Omega(n)$?",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/24.html",
    "page": "t/alphaevolve-24",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/24.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. 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 the bound with equality?",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/25.html",
    "page": "t/alphaevolve-25",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/25.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. 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 $$t(H, W) \\\\geq t(K_2, W)^{C(H)}$$ holds for all graphons $W$, where $K_2$ is the complete graph on two vertices. What is $C(H)$?",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/26.html",
    "page": "t/alphaevolve-26",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/26.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. 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 $C^+$?",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/27.html",
    "page": "t/alphaevolve-27",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/27.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. 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 $$ C^-(n) \\\\coloneqq \\\\max_{p \\\\in \\\\mathbb{U}_{n}} \\\\left(\\min_{|z|=1}\\\\frac{|p(z)|}{\\\\sqrt{n+1}}\\\\right) $$ $$ C^+(n) \\\\coloneqq \\\\min_{p \\\\in \\\\mathbb{U}_{n}}\\\\left(\\\\max_{|z|=1}\\\\frac{|p(z)|}{\\\\sqrt{n+1}}\\\\right) $$ $$ C^w(n) \\\\coloneqq \\\\min_{p \\\\in \\\\mathbb{U}_{n}}\\\\left(\\\\max_{|z|=1}\\\\frac{|p(z)|}{\\\\sqrt{n+1}} - \\\\min_{|z|=1}\\\\frac{|p(z)|}{\\\\sqrt{n+1}}\\\\right) $$ $$ C^4(n) \\\\coloneqq \\\\min_{p \\\\in \\\\mathbb{U}_{n}} \\\\frac{(n+1)^2}{\\\\int_0^1 |p(e^{2\\\\pi \\\\theta})|^4\\ d\\\\theta - (n+1)^2} $$ (The quantity being minimized for $C^4(n)$ is known as Golay's merit factor for $p$.) What is the behavior of $C^-(n)$, $C^+(n)$, $C^w(n)$, $C^4(n)$ as $n \\\\to \\\\infty$?",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/28.html",
    "page": "t/alphaevolve-28",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/28.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. 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 $$ \\\\frac{x_{i+1}+\\\\dots+x_n}{n-i} < x_i + \\\\frac{1}{2}$$ for all $0 \\\\leq i < n$. What is $C(n)$?",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/29.html",
    "page": "t/alphaevolve-29",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/29.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. 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 $C$?",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/3.html",
    "page": "t/alphaevolve-3",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/3.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. 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 necessarily independent) taking values in a finite set of reals, then $$ {\\\\mathbf H}(\\\\pi_{r_\\\\infty}(X,Y)) \\\\leq C(\\\\{r_1,\\\\dots,r_k\\\\}; r_\\\\infty) \\\\max_{i=1,\\\\dots,k} {\\\\mathbf H}(\\\\pi_{r_i}(X,Y)),$$ where $ {\\\\mathbf H}(X) = -\\\\sum_{x} P(X = x) \\\\log(P(X=x))$ is the entropy of a random variable and $x$ ranges over the values taken by $X$. The arithmetic Kakeya conjecture asserts that $C(\\\\{r_1,\\\\dots,r_k\\\\}; r_\\\\infty)$ can be made arbitrarily close to $1$.",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/30.html",
    "page": "t/alphaevolve-30",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/30.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. 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.",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/31.html",
    "page": "t/alphaevolve-31",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/31.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. 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 lower bounds on $C(d,t)$ that are as strong as possible.",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/32.html",
    "page": "t/alphaevolve-32",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/32.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. 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 $$ E(z_1,\\\\ldots,z_N) \\\\coloneqq \\\\sum_{1 \\\\leq i < j \\\\leq N} \\\\frac{1}{\\\\|z_i - z_j\\\\|} $$ where $z_1, \\\\dots, z_N$ ranges over the unit sphere $\\\\mathbb{S}^2$. Establish upper and lower bounds on $C(N)$ that are as strong as possible. What type of configurations $z_1,\\\\dots,z_N$ come close to achieving the infimal (ground state) energy?",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/33.html",
    "page": "t/alphaevolve-33",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/33.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. 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 $$E(z_1,\\\\dots,z_N) \\\\coloneqq \\\\min_{1 \\\\leq i < j \\\\leq N} \\\\|z_i-z_j\\\\|$$ where $z_1,\\\\dots,z_N$ range over points in $\\\\mathbb{S}^2$. Establish upper and lower bounds on $C(N)$ that are as strong as possible. What type of configurations $z_1,\\\\dots,z_N$ come close to achieving the maximal energy?",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/34.html",
    "page": "t/alphaevolve-34",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/34.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. 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.",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/35.html",
    "page": "t/alphaevolve-35",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/35.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. 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 $C(n)$ and $C'(n)$ that are as strong as possible.",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/36.html",
    "page": "t/alphaevolve-36",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/36.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. 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 $C$?",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/37.html",
    "page": "t/alphaevolve-37",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/37.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. 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.",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/38.html",
    "page": "t/alphaevolve-38",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/38.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. 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 $$C \\\\coloneqq \\\\sup_{\\\\mu} \\\\mathbb{P}[X_1 + X_2 + X_3 < 2X_4]$$ where $\\\\mu$ ranges over probability measures on $[0, \\\\infty)$ and let $X_1, \\\\ldots, X_4 \\\\sim \\\\mu$ are independent random variables with law $\\\\mu$. Establish upper and lower bounds on $C$ that are as strong as possible.",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/39.html",
    "page": "t/alphaevolve-39",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/39.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. 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 $C$?",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/4.html",
    "page": "t/alphaevolve-4",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/4.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. 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 possible.",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/40.html",
    "page": "t/alphaevolve-40",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/40.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. 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?",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/41.html",
    "page": "t/alphaevolve-41",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/41.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. 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.",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/42.html",
    "page": "t/alphaevolve-42",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/42.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. 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.",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/43.html",
    "page": "t/alphaevolve-43",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/43.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. 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.",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/44.html",
    "page": "t/alphaevolve-44",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/44.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. 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.",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/45.html",
    "page": "t/alphaevolve-45",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/45.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. 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 $C(\\\\rho)$?",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/46.html",
    "page": "t/alphaevolve-46",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/46.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. 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 $$ C(n,m,d) \\\\coloneqq \\\\inf \\\\frac{ \\\\frac{1}{n^m} \\\\sum_{j_1, j_2, \\\\ldots, j_m = 1}^{n} |||A_{j_1}A_{j_2}\\\\ldots A_{j_m}|||}{ \\\\frac{(n-m)!}{n!} \\\\sum_{\\\\substack{j_1, j_2, \\\\ldots, j_m = 1 \\\\\\\\ \\\\text{all distinct}}}^{n} |||A_{j_1}A_{j_2}\\\\ldots A_{j_m}|||} $$ where the infimum is taken over all matrices $A_1,\\\\dots,A_n$ and invariant norms $|||\\\\cdot|||$. What is $C(n,m,d)$?",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/47.html",
    "page": "t/alphaevolve-47",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/47.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. 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 $C(n,K)$.",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/48.html",
    "page": "t/alphaevolve-48",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/48.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. 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 $C(n)$.",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/49.html",
    "page": "t/alphaevolve-49",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/49.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. 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 $C$?",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/5.html",
    "page": "t/alphaevolve-5",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/5.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. 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 $\\\\max_{1 \\\\leq i < j \\\\leq n} \\\\|x_i-x_j\\\\|$ between the points is at least $C(d,n)$ times the minimum distance $\\\\min_{1 \\\\leq i < j \\\\leq n} \\\\|x_i-x_j\\\\|$. Establish upper and lower bounds for $C(d,n)$. What are the configurations that attain the minimal ratio between the two distances?",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/50.html",
    "page": "t/alphaevolve-50",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/50.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. 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 $k \\\\geq 2$?",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/51.html",
    "page": "t/alphaevolve-51",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/51.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. 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.",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/52.html",
    "page": "t/alphaevolve-52",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/52.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. 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?",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/53.html",
    "page": "t/alphaevolve-53",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/53.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. 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?",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/54.html",
    "page": "t/alphaevolve-54",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/54.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. 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.",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/55.html",
    "page": "t/alphaevolve-55",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/55.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. 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.",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/56.html",
    "page": "t/alphaevolve-56",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/56.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. 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 possible.",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/57.html",
    "page": "t/alphaevolve-57",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/57.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. 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.",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/58.html",
    "page": "t/alphaevolve-58",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/58.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. 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.",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/59.html",
    "page": "t/alphaevolve-59",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/59.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. 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 $C$?",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/6.html",
    "page": "t/alphaevolve-6",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/6.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. 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.",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/60.html",
    "page": "t/alphaevolve-60",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/60.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. 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 lower bounds on $C$ that are as strong as possible.",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/61.html",
    "page": "t/alphaevolve-61",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/61.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. 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 $C$?",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/62.html",
    "page": "t/alphaevolve-62",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/62.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. 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 $C$?",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/63.html",
    "page": "t/alphaevolve-63",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/63.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. 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 $C$?",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/64.html",
    "page": "t/alphaevolve-64",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/64.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. 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 grid has exactly one unit square that is not covered by any tile.",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/65.html",
    "page": "t/alphaevolve-65",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/65.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. 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 the function at any point for $1$ currency unit, (2) to ask a simple yes/no question from an Oracle who knows the hidden function for $10$ currency units, and (3) to ask any question from a different LLM that does not know the hidden function for $10$ currency units, and optionally execute any code returned by it.",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/66.html",
    "page": "t/alphaevolve-66",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/66.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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": "Absent. No formal statement is attached to the result.",
      "anchor": "",
      "flags": []
    },
    "mentions": [],
    "relations": [],
    "verified": false,
    "checked_by": []
  },
  {
    "id": "alphaevolve:67",
    "key": "67",
    "label": "Logic puzzles (problem 67)",
    "collection": "AlphaEvolve problems",
    "kind": "target",
    "name": "Logic puzzles (problem 67)",
    "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. The second question can depend on the answer you get to the first question.",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/67.html",
    "page": "t/alphaevolve-67",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/67.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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/67.html"
          },
          {
            "label": "Mathematical exploration and discovery at scale",
            "url": "https://arxiv.org/abs/2511.02864"
          }
        ],
        "details": {},
        "statement": "alphaevolve:67",
        "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": "0dd7fad9bc28",
        "verification": "V1",
        "fidelity": "F0"
      }
    ],
    "checks": [],
    "declarations": [],
    "fidelity": {
      "grade": "F0",
      "reason": "Absent. No formal statement is attached to the result.",
      "anchor": "",
      "flags": []
    },
    "mentions": [],
    "relations": [],
    "verified": false,
    "checked_by": []
  },
  {
    "id": "alphaevolve:68",
    "key": "68",
    "label": "Logic puzzles (problem 68)",
    "collection": "AlphaEvolve problems",
    "kind": "target",
    "name": "Logic puzzles (problem 68)",
    "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. The second question can depend on the answer you get to the first question.",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/68.html",
    "page": "t/alphaevolve-68",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/68.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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/68.html"
          },
          {
            "label": "Mathematical exploration and discovery at scale",
            "url": "https://arxiv.org/abs/2511.02864"
          }
        ],
        "details": {},
        "statement": "alphaevolve:68",
        "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": "6b83641a728c",
        "verification": "V1",
        "fidelity": "F0"
      }
    ],
    "checks": [],
    "declarations": [],
    "fidelity": {
      "grade": "F0",
      "reason": "Absent. No formal statement is attached to the result.",
      "anchor": "",
      "flags": []
    },
    "mentions": [],
    "relations": [],
    "verified": false,
    "checked_by": []
  },
  {
    "id": "alphaevolve:7",
    "key": "7",
    "label": "Difference Bases",
    "collection": "AlphaEvolve problems",
    "kind": "target",
    "name": "Difference Bases",
    "description": "For any natural number $n$, let $\\\\Delta(n)$ be the size of the smallest set $B$ of integers such that every natural number from $1$ to $n$ is expressible as a difference of two elements of $B$ (such sets are known as difference bases for the interval $\\\\{1,\\\\dots,n\\\\}$). Write $C(n) \\\\coloneqq \\\\Delta^2(n)/n$, and $C \\\\coloneqq \\\\inf_{n \\\\geq 1} C(n)$. Establish upper and lower bounds on $C$ that are as strong as possible.",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/7.html",
    "page": "t/alphaevolve-7",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/7.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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/7.html"
          },
          {
            "label": "Mathematical exploration and discovery at scale",
            "url": "https://arxiv.org/abs/2511.02864"
          }
        ],
        "details": {},
        "statement": "alphaevolve:7",
        "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": "2c41a906b171",
        "verification": "V1",
        "fidelity": "F0"
      }
    ],
    "checks": [],
    "declarations": [],
    "fidelity": {
      "grade": "F0",
      "reason": "Absent. No formal statement is attached to the result.",
      "anchor": "",
      "flags": []
    },
    "mentions": [],
    "relations": [],
    "verified": false,
    "checked_by": []
  },
  {
    "id": "alphaevolve:8",
    "key": "8",
    "label": "Kissing numbers",
    "collection": "AlphaEvolve problems",
    "kind": "target",
    "name": "Kissing numbers",
    "description": "For a dimension $n \\\\geq 1$, define the kissing number $C(n)$ to be the maximum number of non-overlapping unit spheres that can be arranged to simultaneously touch a central unit sphere in $n$-dimensional space. Establish upper and lower bounds on $C(n)$ that are as strong as possible.",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/8.html",
    "page": "t/alphaevolve-8",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/8.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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/8.html"
          },
          {
            "label": "Mathematical exploration and discovery at scale",
            "url": "https://arxiv.org/abs/2511.02864"
          }
        ],
        "details": {},
        "statement": "alphaevolve:8",
        "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": "fb0d7eed86f3",
        "verification": "V1",
        "fidelity": "F0"
      }
    ],
    "checks": [],
    "declarations": [],
    "fidelity": {
      "grade": "F0",
      "reason": "Absent. No formal statement is attached to the result.",
      "anchor": "",
      "flags": []
    },
    "mentions": [],
    "relations": [],
    "verified": false,
    "checked_by": []
  },
  {
    "id": "alphaevolve:9",
    "key": "9",
    "label": "Kakeya needle problem",
    "collection": "AlphaEvolve problems",
    "kind": "target",
    "name": "Kakeya needle problem",
    "description": "Let $n \\\\geq 2$. Let $C^T(n)$ denote the minimal area $|\\\\bigcup_{j=1}^n T_j|$ of a union of triangles $T_j$ with vertices $(x_j,0)$, $(x_j + 1/n, 0)$, $(x_j + j/n, 1)$ for some real numbers $x_1,\\\\dots,x_n$, and similarly define $C^P(n)$ denote the minimal area $|\\\\bigcup_{j=1}^n P_j|$ of a union of parallelograms $P_j$ with vertices $(x_j,0), (x_j+1/n,0), (x_j+j/n,1), (x_j+(j+1)/n,0)$ for some real numbers $x_1,\\\\dots,x_n$. Finally, define $S^T(n)$ to be the maximal \"score\" $$ \\\\frac{\\\\sum_{i=1}^n |T_i|}{\\\\left(\\\\sum_{i=1}^n \\\\sum_{j=1}^n |T_i \\\\cap T_j|\\\\right)^{1/2} |\\\\bigcup_{i=1}^n T_i|^{1/2}}$$ over triangles $T_i$ as above, and define $S^P(n)$ similarly. Establish upper and lower bounds for $C^T(n)$, $C^P(n)$, $S^T(n)$, $S^P(n)$ that are as strong as possible.",
    "about": "",
    "direction": "unstated",
    "unit": "",
    "format": "",
    "value_key": "",
    "url": "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/9.html",
    "page": "t/alphaevolve-9",
    "aliases": [
      "https://github.com/google-deepmind/alphaevolve_repository_of_problems/blob/main/problems/9.html"
    ],
    "tags": [
      "construction"
    ],
    "records": [],
    "steps": [],
    "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/9.html"
          },
          {
            "label": "Mathematical exploration and discovery at scale",
            "url": "https://arxiv.org/abs/2511.02864"
          }
        ],
        "details": {},
        "statement": "alphaevolve:9",
        "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": "166d3eadcb21",
        "verification": "V1",
        "fidelity": "F0"
      }
    ],
    "checks": [],
    "declarations": [],
    "fidelity": {
      "grade": "F0",
      "reason": "Absent. No formal statement is attached to the result.",
      "anchor": "",
      "flags": []
    },
    "mentions": [],
    "relations": [],
    "verified": false,
    "checked_by": []
  },
  {
    "id": "primes:bounded-gaps",
    "key": "bounded-gaps",
    "label": "Bounded gaps between primes",
    "collection": "Analytic number theory",
    "kind": "target",
    "name": "Bounded gaps between primes",
    "description": "The smallest H for which infinitely many pairs of consecutive primes are at most H apart. The twin prime conjecture says H is 2; before 2013 no finite bound was known.",
    "about": "Zhang gave the first finite bound in 2013, and within a year the Polymath projects and Maynard's multidimensional sieve brought it to 246, where it stayed for twelve years. On 31 August 2026 Julia Stadlmann posted 240. Within three days Shiva Kintali, directing several AI models, reached 236; Axiom Math reached 212, with a Lean certificate of the deduction produced by AxiomProver; and OpenAI released a report reaching 186, which it says is due to GPT-6 Astra. Science News reports that OpenAI's release came within two hours of Axiom's announcement, on 3 September, and the two steps are ordered that way here.",
    "direction": "minimise",
    "unit": "gap between consecutive primes",
    "format": "",
    "value_key": "",
    "url": "https://arxiv.org/abs/2608.31126",
    "page": "t/primes-bounded-gaps",
    "aliases": [],
    "tags": [
      "number theory",
      "prime gaps"
    ],
    "records": [
      {
        "value": 186,
        "display": "186",
        "holder": "GPT-6 Astra (OpenAI)",
        "kind": "ai",
        "date": "2026-09-03",
        "order": 2,
        "note": "The report, dated 30 August, says the proof is due to GPT 6 Astra; its Lean formalisation is conditional on stated numerical bounds and exponential-sum estimates. Science News reports it was released with the model on 3 September, within two hours of Axiom's announcement.",
        "source": "https://cdn.openai.com/pdf/51126fac-1b68-4128-9666-c908bcc16033/short_gaps.pdf",
        "artifact": "https://github.com/openai/PrimeGaps186"
      },
      {
        "value": 212,
        "display": "212",
        "holder": "François Charton, Letong Hong, Kenny Lau, Ken Ono, Guillaume Remy, Ho Chung Siu, Ashvin Swaminathan, Jesse Thorner and Yunzhou Xie (Axiom Math), with AxiomProver",
        "kind": "ai",
        "date": "2026-09-03",
        "order": 1,
        "note": "Builds on Stadlmann's work. AxiomProver generated a Lean certificate of the deduction from natural-language specifications, taking the equidistribution estimates as hypotheses, as Appendix A of the preliminary draft says.",
        "source": "https://primegaps.axiommath.ai/bgp212.pdf"
      },
      {
        "value": 236,
        "display": "236",
        "holder": "Shiva Kintali, with Claude, ChatGPT, Codex and open-weight models",
        "kind": "ai",
        "date": "2026-09-01",
        "note": "Extends Stadlmann's framework with exactly checked computer certificates. The paper's statement on AI: a harness the author built directed several models, and the author is responsible for the proofs and the code.",
        "source": "https://shivakintali.github.io/papers/Prime-Gaps.pdf",
        "artifact": "https://github.com/shivakintali/Prime-Gaps"
      },
      {
        "value": 240,
        "display": "240",
        "holder": "Julia Stadlmann",
        "kind": "human",
        "date": "2026-08-31",
        "note": "Combines the Bombieri–Vinogradov theorem with newer equidistribution estimates for smooth moduli.",
        "source": "https://arxiv.org/abs/2608.31126"
      },
      {
        "value": 246,
        "display": "246",
        "holder": "Polymath8b",
        "kind": "human",
        "date": "2014",
        "note": "Published in Research in the Mathematical Sciences in 2014; the record for twelve years.",
        "source": "https://arxiv.org/abs/1407.4897"
      },
      {
        "value": 600,
        "display": "600",
        "holder": "James Maynard",
        "kind": "human",
        "date": "2013-11-19",
        "note": "The multidimensional sieve, needing only the Bombieri–Vinogradov theorem; published in the Annals of Mathematics in 2015.",
        "source": "https://arxiv.org/abs/1311.4600"
      },
      {
        "value": 4680,
        "display": "4,680",
        "holder": "Polymath8a",
        "kind": "human",
        "date": "2013-07",
        "note": "Reached by the Polymath8a project in 2013, sharpening Zhang's method; published in Algebra & Number Theory in 2014.",
        "source": "https://doi.org/10.2140/ant.2014.8.2067"
      },
      {
        "value": 70000000,
        "display": "70,000,000",
        "holder": "Yitang Zhang",
        "kind": "human",
        "date": "2013-05",
        "note": "The first finite bound; published in the Annals of Mathematics in 2014.",
        "source": "https://doi.org/10.4007/annals.2014.179.3.7"
      }
    ],
    "steps": [
      {
        "value": 70000000,
        "display": "70,000,000",
        "holder": "Yitang Zhang",
        "kind": "human",
        "date": "2013-05",
        "note": "The first finite bound; published in the Annals of Mathematics in 2014.",
        "source": "https://doi.org/10.4007/annals.2014.179.3.7",
        "moved": true,
        "candidate": false,
        "best_after": 70000000
      },
      {
        "value": 4680,
        "display": "4,680",
        "holder": "Polymath8a",
        "kind": "human",
        "date": "2013-07",
        "note": "Reached by the Polymath8a project in 2013, sharpening Zhang's method; published in Algebra & Number Theory in 2014.",
        "source": "https://doi.org/10.2140/ant.2014.8.2067",
        "moved": true,
        "candidate": false,
        "best_after": 4680
      },
      {
        "value": 600,
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        "note": "The multidimensional sieve, needing only the Bombieri–Vinogradov theorem; published in the Annals of Mathematics in 2015.",
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        "value": 246,
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        "date": "2014",
        "note": "Published in Research in the Mathematical Sciences in 2014; the record for twelve years.",
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        "moved": true,
        "candidate": false,
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        "value": 240,
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        "note": "Extends Stadlmann's framework with exactly checked computer certificates. The paper's statement on AI: a harness the author built directed several models, and the author is responsible for the proofs and the code.",
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        "date": "2026-09-03",
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      "date": "2026-09-03",
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            "label": "Kintali, Improved bounded gaps between primes",
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            "url": "https://github.com/shivakintali/Prime-Gaps"
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          "Ho Chung Siu",
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            "label": "Axiom Math, A new bound for small gaps between primes (preliminary draft)",
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    "id": "zeta:critical-line-proportion",
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    "label": "Proportion of zeta zeros proved to lie on the critical line",
    "collection": "Analytic number theory",
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    "name": "Proportion of zeta zeros proved to lie on the critical line",
    "description": "The Riemann hypothesis asserts that every nontrivial zero of the zeta function lies on the critical line. Short of proving it, mathematicians bound the proportion of zeros that provably do.",
    "about": "Selberg proved in 1942 that a positive proportion of the zeros lie on the line, without an explicit constant. Levinson's method gave more than a third in 1974, and refinements of it carried the bound past two fifths and, by 2018, to just over five twelfths. The 2026 step proves a stronger statement, that the zeros are simple as well as on the line; its paper says the proof was discovered by Claude and verified and communicated by Levent Alpöge and Ralph Furman.",
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    "url": "https://arxiv.org/abs/2608.13637",
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        "value": 0.6725,
        "display": "67.25%",
        "holder": "Claude, with Levent Alpöge and Ralph Furman",
        "kind": "ai",
        "date": "2026-08-10",
        "note": "At least two thirds of the zeros, counted with multiplicity, are simple and on the line, and 0.6725 with the Montgomery–Taylor window; at least five sixths are distinct. Announced on 10 August; the paper, with its Lean formalisation, followed on arXiv on 13 August.",
        "source": "https://arxiv.org/abs/2608.13637",
        "artifact": "https://github.com/anthropics/formal-math/tree/main/zeta23"
      },
      {
        "value": 0.4167,
        "display": "more than 5/12",
        "holder": "Kyle Pratt, Nicolas Robles, Alexandru Zaharescu and Dirk Zeindler",
        "kind": "human",
        "date": "2018-02-28",
        "note": "Slightly over five twelfths; published in Research in the Mathematical Sciences in 2020.",
        "source": "https://arxiv.org/abs/1802.10521"
      },
      {
        "value": 0.41,
        "display": "more than 41%",
        "holder": "Hung Bui, Brian Conrey and Matthew Young",
        "kind": "human",
        "date": "2010-02-22",
        "note": "Published in Acta Arithmetica in 2011.",
        "source": "https://arxiv.org/abs/1002.4127"
      },
      {
        "value": 0.4,
        "display": "more than 2/5",
        "holder": "J. Brian Conrey",
        "kind": "human",
        "date": "1989",
        "note": "More than two fifths of the zeros lie on the critical line.",
        "source": "https://doi.org/10.1515/crll.1989.399.1"
      },
      {
        "value": 0.3333,
        "display": "more than 1/3",
        "holder": "Norman Levinson",
        "kind": "human",
        "date": "1974",
        "note": "More than one third of the zeros lie on the critical line.",
        "source": "https://doi.org/10.1016/0001-8708(74)90074-7"
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        "value": 0.3333,
        "display": "more than 1/3",
        "holder": "Norman Levinson",
        "kind": "human",
        "date": "1974",
        "note": "More than one third of the zeros lie on the critical line.",
        "source": "https://doi.org/10.1016/0001-8708(74)90074-7",
        "moved": true,
        "candidate": false,
        "best_after": 0.3333
      },
      {
        "value": 0.4,
        "display": "more than 2/5",
        "holder": "J. Brian Conrey",
        "kind": "human",
        "date": "1989",
        "note": "More than two fifths of the zeros lie on the critical line.",
        "source": "https://doi.org/10.1515/crll.1989.399.1",
        "moved": true,
        "candidate": false,
        "best_after": 0.4
      },
      {
        "value": 0.41,
        "display": "more than 41%",
        "holder": "Hung Bui, Brian Conrey and Matthew Young",
        "kind": "human",
        "date": "2010-02-22",
        "note": "Published in Acta Arithmetica in 2011.",
        "source": "https://arxiv.org/abs/1002.4127",
        "moved": true,
        "candidate": false,
        "best_after": 0.41
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      {
        "value": 0.4167,
        "display": "more than 5/12",
        "holder": "Kyle Pratt, Nicolas Robles, Alexandru Zaharescu and Dirk Zeindler",
        "kind": "human",
        "date": "2018-02-28",
        "note": "Slightly over five twelfths; published in Research in the Mathematical Sciences in 2020.",
        "source": "https://arxiv.org/abs/1802.10521",
        "moved": true,
        "candidate": false,
        "best_after": 0.4167
      },
      {
        "value": 0.6725,
        "display": "67.25%",
        "holder": "Claude, with Levent Alpöge and Ralph Furman",
        "kind": "ai",
        "date": "2026-08-10",
        "note": "At least two thirds of the zeros, counted with multiplicity, are simple and on the line, and 0.6725 with the Montgomery–Taylor window; at least five sixths are distinct. Announced on 10 August; the paper, with its Lean formalisation, followed on arXiv on 13 August.",
        "source": "https://arxiv.org/abs/2608.13637",
        "artifact": "https://github.com/anthropics/formal-math/tree/main/zeta23",
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      "note": "At least two thirds of the zeros, counted with multiplicity, are simple and on the line, and 0.6725 with the Montgomery–Taylor window; at least five sixths are distinct. Announced on 10 August; the paper, with its Lean formalisation, followed on arXiv on 13 August.",
      "source": "https://arxiv.org/abs/2608.13637",
      "artifact": "https://github.com/anthropics/formal-math/tree/main/zeta23"
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      "holder": "Kyle Pratt, Nicolas Robles, Alexandru Zaharescu and Dirk Zeindler",
      "kind": "human",
      "date": "2018-02-28",
      "note": "Slightly over five twelfths; published in Research in the Mathematical Sciences in 2020.",
      "source": "https://arxiv.org/abs/1802.10521"
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    "claims": [
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        ],
        "humans": [],
        "date": "2026-08-10",
        "outcome": "record",
        "outcome_text": "Proved that at least two thirds of the zeta zeros are simple and on the critical line, 67.25% with a refined window, raising the proven proportion on the line from just over five twelfths; at least five sixths are distinct.",
        "autonomy": "A3",
        "autonomy_reason": "The prompt was to attempt the Riemann hypothesis; the mathematical choices were the model's, the paper states the proof was discovered autonomously by Claude, and the accompanying formalization.yaml records the work as autonomous. Humans validated the result rather than contributing to it.",
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            "url": "https://www.anthropic.com/research/riemann-zeta"
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            "url": "https://github.com/anthropics/formal-math/tree/main/zeta23"
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          "review": "Examined by Brian Conrey and Dan Goldston; formalisation author-verified by Ralph Furman"
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        "node": "target",
        "contribution": "primary",
        "outcome_scope": "record",
        "section": "record",
        "section_title": "Record on a quantity",
        "date_text": "2026-08-10",
        "machine_checked": true,
        "checked_by": [
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        "autonomy_source": "declared autonomous in the project's formalization.yaml",
        "id": "d89f5a143623",
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        "checked": "2026-08-22T21:18:52Z",
        "built": true,
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        "declarations": {
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        "results": 5,
        "comparator": true,
        "alignment": true,
        "divergences": "liminf bounds are rendered as: for all ε > 0 there is T₀ such that for all T ≥ T₀, (c − ε)·N ≤ X. \"Nontrivial zero\" is rendered as a zero with 0 < Re ρ < 1. Windows are T₁ < Im ρ ≤ T₂ (positive ordinates). Left sides count with multiplicity; N₀*, N₀ˢ, N_d count distinct points (the strong direction). Theorem B of the paper is formalized for primitive characters of modulus q > 1. The repository states the theorems at the paper's constants; the weaker Cauchy–Schwarz-form variants that earlier revisions also certified are implied by these and are no longer separately stated. The 5/6 constant is obtained from the rank–trace inequality with parameter c = 3 where the paper's text uses Proposition 4.5(iii) with c = 2. (In the non-submitted ξ′ configuration the proportion statements carry fixed decimal constants rather than ε-forms.) See README.md, \"Reading notes for the statements\".",
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        "role": "",
        "substantive": "",
        "sources": [
          {
            "title": "More than two thirds of the zeta zeros are simple and on the critical line",
            "relationship": "formalizes",
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          }
        ],
        "related": [
          {
            "id": "https://github.com/AlexKontorovich/PrimeNumberTheoremAnd",
            "relationship": "adapts",
            "note": "Files under Zeta23/FromPNTPlus/ are derived from the PrimeNumberTheoremAnd project with local modifications; each carries its upstream notice (file, commit, copyright, licence). See NOTICE."
          }
        ],
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        "checked_by": [
          "qed.bot"
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        "join": "artifact",
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      "grade": "F2",
      "reason": "The correspondence is declared through a Comparator challenge, an alignment table and written divergences.",
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      "flags": [
        "declares divergences from its source"
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    },
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    "checked_by": [
      "qed.bot"
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  {
    "id": "borsuk:dimension",
    "key": "dimension",
    "label": "Smallest dimension in which Borsuk's conjecture is known to fail",
    "collection": "Discrete geometry",
    "kind": "target",
    "name": "Smallest dimension in which Borsuk's conjecture is known to fail",
    "description": "Borsuk asked in 1933 whether every bounded set in n-dimensional space can be split into n + 1 parts of smaller diameter. It is true in dimensions 2 and 3 and false in high dimensions; the record is the smallest dimension with a known counterexample.",
    "about": "Kahn and Kalai's 1993 counterexample works in dimension 1325; constructions from codes brought the dimension to 298 by 2003, and Bondarenko's two-distance set from the G2(4) graph led to 65 and then 64 in 2013. Several improvements between 1994 and 2002 are not yet drawn here. In May 2026 Max Grinsztajn posted a counterexample in dimension 63, obtained with assistance from GPT-5.5 Pro. In August an arXiv note reported the same dimension from a construction generated by GPT 5.6 Sol, and was withdrawn when its author found the earlier posting.",
    "direction": "minimise",
    "unit": "dimension",
    "format": "",
    "value_key": "",
    "url": "https://github.com/maaxgrin/borsuk-63-counterexample",
    "page": "t/borsuk-dimension",
    "aliases": [],
    "tags": [
      "discrete geometry",
      "Borsuk's problem"
    ],
    "records": [
      {
        "value": 63,
        "display": "63",
        "holder": "Max Grinsztajn, with GPT-5.5 Pro",
        "kind": "ai",
        "date": "2026-05-26",
        "note": "A 321-point set in which every subset of smaller diameter has at most five points, so at least 65 parts are needed. The repository says the construction and proof were obtained with assistance from GPT-5.5 Pro.",
        "source": "https://github.com/maaxgrin/borsuk-63-counterexample",
        "artifact": "https://github.com/maaxgrin/borsuk-63-counterexample"
      },
      {
        "value": 63,
        "display": "63",
        "holder": "Yibo Ji, with GPT 5.6 Sol",
        "kind": "ai",
        "date": "2026-08-12",
        "note": "The note said the example and proof were generated entirely by ChatGPT using GPT 5.6 Sol; its author withdrew it on 14 August on finding the earlier posting.",
        "source": "https://arxiv.org/abs/2608.12561"
      },
      {
        "value": 64,
        "display": "64",
        "holder": "Thomas Jenrich",
        "kind": "human",
        "date": "2013-08",
        "note": "A 352-point subset of Bondarenko's set; published with Andries Brouwer in the Electronic Journal of Combinatorics in 2014.",
        "source": "https://arxiv.org/abs/1308.0206"
      },
      {
        "value": 65,
        "display": "65",
        "holder": "Andriy Bondarenko",
        "kind": "human",
        "date": "2013-05",
        "note": "A 416-point two-distance set from the G2(4) graph; published in Discrete & Computational Geometry in 2014.",
        "source": "https://arxiv.org/abs/1305.2584"
      },
      {
        "value": 298,
        "display": "298",
        "holder": "Aicke Hinrichs and Christian Richter",
        "kind": "human",
        "date": "2003",
        "note": "New sets with large Borsuk numbers.",
        "source": "https://doi.org/10.1016/s0012-365x(02)00833-6"
      },
      {
        "value": 1325,
        "display": "1325",
        "holder": "Jeff Kahn and Gil Kalai",
        "kind": "human",
        "date": "1993",
        "note": "The first counterexample; Weißbach later showed the dimension-1325 case needs a corrected argument, which holds.",
        "source": "https://doi.org/10.1090/s0273-0979-1993-00398-7"
      }
    ],
    "steps": [
      {
        "value": 1325,
        "display": "1325",
        "holder": "Jeff Kahn and Gil Kalai",
        "kind": "human",
        "date": "1993",
        "note": "The first counterexample; Weißbach later showed the dimension-1325 case needs a corrected argument, which holds.",
        "source": "https://doi.org/10.1090/s0273-0979-1993-00398-7",
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        "candidate": false,
        "best_after": 1325
      },
      {
        "value": 298,
        "display": "298",
        "holder": "Aicke Hinrichs and Christian Richter",
        "kind": "human",
        "date": "2003",
        "note": "New sets with large Borsuk numbers.",
        "source": "https://doi.org/10.1016/s0012-365x(02)00833-6",
        "moved": true,
        "candidate": false,
        "best_after": 298
      },
      {
        "value": 65,
        "display": "65",
        "holder": "Andriy Bondarenko",
        "kind": "human",
        "date": "2013-05",
        "note": "A 416-point two-distance set from the G2(4) graph; published in Discrete & Computational Geometry in 2014.",
        "source": "https://arxiv.org/abs/1305.2584",
        "moved": true,
        "candidate": false,
        "best_after": 65
      },
      {
        "value": 64,
        "display": "64",
        "holder": "Thomas Jenrich",
        "kind": "human",
        "date": "2013-08",
        "note": "A 352-point subset of Bondarenko's set; published with Andries Brouwer in the Electronic Journal of Combinatorics in 2014.",
        "source": "https://arxiv.org/abs/1308.0206",
        "moved": true,
        "candidate": false,
        "best_after": 64
      },
      {
        "value": 63,
        "display": "63",
        "holder": "Max Grinsztajn, with GPT-5.5 Pro",
        "kind": "ai",
        "date": "2026-05-26",
        "note": "A 321-point set in which every subset of smaller diameter has at most five points, so at least 65 parts are needed. The repository says the construction and proof were obtained with assistance from GPT-5.5 Pro.",
        "source": "https://github.com/maaxgrin/borsuk-63-counterexample",
        "artifact": "https://github.com/maaxgrin/borsuk-63-counterexample",
        "moved": true,
        "candidate": false,
        "best_after": 63
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      {
        "value": 63,
        "display": "63",
        "holder": "Yibo Ji, with GPT 5.6 Sol",
        "kind": "ai",
        "date": "2026-08-12",
        "note": "The note said the example and proof were generated entirely by ChatGPT using GPT 5.6 Sol; its author withdrew it on 14 August on finding the earlier posting.",
        "source": "https://arxiv.org/abs/2608.12561",
        "moved": false,
        "candidate": false,
        "best_after": 63
      }
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      "value": 63,
      "display": "63",
      "holder": "Max Grinsztajn, with GPT-5.5 Pro",
      "kind": "ai",
      "date": "2026-05-26",
      "note": "A 321-point set in which every subset of smaller diameter has at most five points, so at least 65 parts are needed. The repository says the construction and proof were obtained with assistance from GPT-5.5 Pro.",
      "source": "https://github.com/maaxgrin/borsuk-63-counterexample",
      "artifact": "https://github.com/maaxgrin/borsuk-63-counterexample"
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    "previous": {
      "value": 63,
      "display": "63",
      "holder": "Yibo Ji, with GPT 5.6 Sol",
      "kind": "ai",
      "date": "2026-08-12",
      "note": "The note said the example and proof were generated entirely by ChatGPT using GPT 5.6 Sol; its author withdrew it on 14 August on finding the earlier posting.",
      "source": "https://arxiv.org/abs/2608.12561"
    },
    "held_by_machine": true,
    "claims": [
      {
        "systems": [
          "GPT-5.5 Pro"
        ],
        "humans": [
          "Max Grinsztajn"
        ],
        "date": "2026-05-26",
        "outcome": "record",
        "outcome_text": "A 321-point set in dimension 63 that cannot be split into 64 parts of smaller diameter, improving 64.",
        "autonomy": "A1",
        "autonomy_reason": "The repository states the construction and proof were obtained with assistance from GPT-5.5 Pro.",
        "lean_claimed": false,
        "sources": [
          {
            "label": "Grinsztajn, A 63-dimensional counterexample to Borsuk's conjecture",
            "url": "https://github.com/maaxgrin/borsuk-63-counterexample"
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        ],
        "details": {},
        "statement": "borsuk:dimension",
        "node": "target",
        "contribution": "primary",
        "outcome_scope": "record",
        "section": "record",
        "section_title": "Record on a quantity",
        "date_text": "2026-05-26",
        "machine_checked": false,
        "checked_by": [],
        "id": "8fe849d74fe1",
        "verification": "V2",
        "fidelity": "F0"
      },
      {
        "systems": [
          "GPT-5.6 Sol"
        ],
        "humans": [
          "Yibo Ji"
        ],
        "date": "2026-08-12",
        "outcome": "matched",
        "outcome_text": "The same dimension, 63, from a construction the author said was generated entirely by ChatGPT; withdrawn on 14 August when he found the earlier posting.",
        "autonomy": "A3",
        "autonomy_reason": "The note stated the example and proof were generated entirely by ChatGPT using GPT 5.6 Sol, with the author verifying them.",
        "lean_claimed": false,
        "sources": [
          {
            "label": "Ji, An AI generated counterexample to Borsuk problem in dimension 63 (withdrawn)",
            "url": "https://arxiv.org/abs/2608.12561"
          }
        ],
        "details": {},
        "statement": "borsuk:dimension",
        "node": "target",
        "contribution": "primary",
        "outcome_scope": "record",
        "section": "record",
        "section_title": "Record on a quantity",
        "date_text": "2026-08-12",
        "machine_checked": false,
        "checked_by": [],
        "id": "2a720f16ef9e",
        "verification": "V2",
        "fidelity": "F0"
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    ],
    "checks": [],
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    "fidelity": {
      "grade": "F0",
      "reason": "Absent. No formal statement is attached to the result.",
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  },
  {
    "id": "kissing:dimension-11",
    "key": "dimension-11",
    "label": "Kissing number in dimension 11",
    "collection": "Discrete geometry",
    "kind": "target",
    "name": "Kissing number in dimension 11",
    "description": "The largest number of non-overlapping unit spheres that can touch a central unit sphere in 11 dimensions. The exact value is unknown; the record is the largest known configuration, and the best upper bound is 868.",
    "about": "Leech and Sloane's constructions from codes gave 566 in 1971, and Zinoviev and Ericson reached 582 in 1999. Ganzhinov found 592 in 2022 from highly symmetric line systems. AlphaEvolve reported 593 in 2025, and agents working on the EinsteinArena platform moved the bound to 604 by May 2026 through a sequence of submissions and public discussion. Henry Cohn's table of kissing numbers, whose references the older steps follow, lists 604 against an upper bound of 868.",
    "direction": "maximise",
    "unit": "spheres",
    "format": "",
    "value_key": "",
    "url": "https://arxiv.org/abs/2606.10402",
    "page": "t/kissing-dimension-11",
    "aliases": [],
    "tags": [
      "discrete geometry",
      "sphere packing"
    ],
    "records": [
      {
        "value": 604,
        "display": "604",
        "holder": "Agents on EinsteinArena",
        "kind": "ai",
        "date": "2026-06-09",
        "note": "Reached by May 2026 through a sequence of agent submissions, public discussion and verifier refinement, as the paper describes.",
        "source": "https://arxiv.org/abs/2606.10402"
      },
      {
        "value": 593,
        "display": "593",
        "holder": "AlphaEvolve",
        "kind": "ai",
        "date": "2025-06-16",
        "note": "Reported in the AlphaEvolve white paper; the EinsteinArena paper states 593 as the bound it improved.",
        "source": "https://arxiv.org/abs/2506.13131"
      },
      {
        "value": 592,
        "display": "592",
        "holder": "Mikhail Ganzhinov",
        "kind": "human",
        "date": "2022-07-17",
        "note": "Highly symmetric line systems; published in Linear Algebra and its Applications in 2025.",
        "source": "https://arxiv.org/abs/2207.08266"
      },
      {
        "value": 582,
        "display": "582",
        "holder": "Victor Zinoviev and Thomas Ericson",
        "kind": "human",
        "date": "1999",
        "note": "New lower bounds for contact numbers in small dimensions, as referenced in Cohn's table.",
        "source": "https://cohn.mit.edu/kissing-numbers/"
      },
      {
        "value": 566,
        "display": "566",
        "holder": "John Leech and Neil Sloane",
        "kind": "human",
        "date": "1971",
        "note": "Sphere packings and error-correcting codes, as referenced in Cohn's table.",
        "source": "https://doi.org/10.4153/cjm-1971-081-3"
      }
    ],
    "steps": [
      {
        "value": 566,
        "display": "566",
        "holder": "John Leech and Neil Sloane",
        "kind": "human",
        "date": "1971",
        "note": "Sphere packings and error-correcting codes, as referenced in Cohn's table.",
        "source": "https://doi.org/10.4153/cjm-1971-081-3",
        "moved": true,
        "candidate": false,
        "best_after": 566
      },
      {
        "value": 582,
        "display": "582",
        "holder": "Victor Zinoviev and Thomas Ericson",
        "kind": "human",
        "date": "1999",
        "note": "New lower bounds for contact numbers in small dimensions, as referenced in Cohn's table.",
        "source": "https://cohn.mit.edu/kissing-numbers/",
        "moved": true,
        "candidate": false,
        "best_after": 582
      },
      {
        "value": 592,
        "display": "592",
        "holder": "Mikhail Ganzhinov",
        "kind": "human",
        "date": "2022-07-17",
        "note": "Highly symmetric line systems; published in Linear Algebra and its Applications in 2025.",
        "source": "https://arxiv.org/abs/2207.08266",
        "moved": true,
        "candidate": false,
        "best_after": 592
      },
      {
        "value": 593,
        "display": "593",
        "holder": "AlphaEvolve",
        "kind": "ai",
        "date": "2025-06-16",
        "note": "Reported in the AlphaEvolve white paper; the EinsteinArena paper states 593 as the bound it improved.",
        "source": "https://arxiv.org/abs/2506.13131",
        "moved": true,
        "candidate": false,
        "best_after": 593
      },
      {
        "value": 604,
        "display": "604",
        "holder": "Agents on EinsteinArena",
        "kind": "ai",
        "date": "2026-06-09",
        "note": "Reached by May 2026 through a sequence of agent submissions, public discussion and verifier refinement, as the paper describes.",
        "source": "https://arxiv.org/abs/2606.10402",
        "moved": true,
        "candidate": false,
        "best_after": 604
      }
    ],
    "best": {
      "value": 604,
      "display": "604",
      "holder": "Agents on EinsteinArena",
      "kind": "ai",
      "date": "2026-06-09",
      "note": "Reached by May 2026 through a sequence of agent submissions, public discussion and verifier refinement, as the paper describes.",
      "source": "https://arxiv.org/abs/2606.10402"
    },
    "previous": {
      "value": 593,
      "display": "593",
      "holder": "AlphaEvolve",
      "kind": "ai",
      "date": "2025-06-16",
      "note": "Reported in the AlphaEvolve white paper; the EinsteinArena paper states 593 as the bound it improved.",
      "source": "https://arxiv.org/abs/2506.13131"
    },
    "held_by_machine": true,
    "claims": [
      {
        "systems": [
          "AlphaEvolve"
        ],
        "humans": [],
        "date": "2025-06-16",
        "outcome": "superseded",
        "outcome_text": "A configuration of 593 spheres in dimension 11, improving 592.",
        "autonomy": "A2",
        "autonomy_reason": "An evolutionary coding agent searching against an evaluator written by people, who also chose the problem.",
        "lean_claimed": false,
        "sources": [
          {
            "label": "Novikov et al., AlphaEvolve: A coding agent for scientific and algorithmic discovery",
            "url": "https://arxiv.org/abs/2506.13131"
          }
        ],
        "details": {},
        "statement": "kissing:dimension-11",
        "node": "target",
        "contribution": "primary",
        "outcome_scope": "record",
        "section": "record",
        "section_title": "Record on a quantity",
        "date_text": "2025-06-16",
        "machine_checked": false,
        "checked_by": [],
        "id": "ea544b0fc1f4",
        "verification": "V1",
        "fidelity": "F0"
      },
      {
        "systems": [
          "Various"
        ],
        "humans": [],
        "date": "2026-06-09",
        "outcome": "record",
        "outcome_text": "A configuration of 604 spheres in dimension 11, improving 593.",
        "autonomy": "A2",
        "autonomy_reason": "Agents searched and discussed against verifiers the platform's builders wrote; the paper says the advance came from a sequence of agent submissions and agent-to-agent borrowing of ideas.",
        "lean_claimed": false,
        "sources": [
          {
            "label": "Bianchi, Kwon, Pappu and Zou, Harnessing the collective intelligence of AI agents in the wild",
            "url": "https://arxiv.org/abs/2606.10402"
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        ],
        "details": {
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        "section_title": "Record on a quantity",
        "date_text": "2026-06-09",
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        "checked_by": [],
        "id": "a9ce2a06198e",
        "verification": "V1",
        "fidelity": "F0"
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  {
    "id": "capset:dimension-8",
    "key": "dimension-8",
    "label": "Largest known cap set in dimension 8",
    "collection": "Extremal combinatorics",
    "kind": "target",
    "name": "Largest known cap set in dimension 8",
    "description": "A cap set is a subset of F_3^n containing no three points in a line, equivalently no three distinct points summing to zero. Larger explicit constructions raise the known lower bound.",
    "about": "In dimension 8 the largest known cap had 496 points, from Yves Edel's product constructions, until FunSearch found one of 512 in 2023. Whether larger caps exist in dimension 8 is open.",
    "direction": "maximise",
    "unit": "points",
    "format": "",
    "value_key": "size",
    "url": "https://github.com/google-deepmind/funsearch/tree/main/cap_set",
    "page": "t/capset-dimension-8",
    "aliases": [],
    "tags": [
      "combinatorics",
      "cap set"
    ],
    "records": [
      {
        "value": 512,
        "display": "512 points",
        "holder": "FunSearch",
        "kind": "ai",
        "date": "2023-12-14",
        "note": "The largest cap set in dimension 8 found by the search, published with the Nature paper.",
        "source": "https://deepmind.google/blog/funsearch-making-new-discoveries-in-mathematical-sciences-using-large-language-models/",
        "artifact": "https://github.com/google-deepmind/funsearch/blob/main/cap_set/n8_size512.txt"
      },
      {
        "value": 496,
        "display": "496 points",
        "holder": "Yves Edel",
        "kind": "human",
        "date": "2004",
        "note": "From extensions of generalized product caps; the best known in dimension 8 before FunSearch.",
        "source": "https://doi.org/10.1023/a:1027365901231"
      }
    ],
    "steps": [
      {
        "value": 496,
        "display": "496 points",
        "holder": "Yves Edel",
        "kind": "human",
        "date": "2004",
        "note": "From extensions of generalized product caps; the best known in dimension 8 before FunSearch.",
        "source": "https://doi.org/10.1023/a:1027365901231",
        "moved": true,
        "candidate": false,
        "best_after": 496
      },
      {
        "value": 512,
        "display": "512 points",
        "holder": "FunSearch",
        "kind": "ai",
        "date": "2023-12-14",
        "note": "The largest cap set in dimension 8 found by the search, published with the Nature paper.",
        "source": "https://deepmind.google/blog/funsearch-making-new-discoveries-in-mathematical-sciences-using-large-language-models/",
        "artifact": "https://github.com/google-deepmind/funsearch/blob/main/cap_set/n8_size512.txt",
        "moved": true,
        "candidate": false,
        "best_after": 512
      }
    ],
    "best": {
      "value": 512,
      "display": "512 points",
      "holder": "FunSearch",
      "kind": "ai",
      "date": "2023-12-14",
      "note": "The largest cap set in dimension 8 found by the search, published with the Nature paper.",
      "source": "https://deepmind.google/blog/funsearch-making-new-discoveries-in-mathematical-sciences-using-large-language-models/",
      "artifact": "https://github.com/google-deepmind/funsearch/blob/main/cap_set/n8_size512.txt"
    },
    "previous": {
      "value": 496,
      "display": "496 points",
      "holder": "Yves Edel",
      "kind": "human",
      "date": "2004",
      "note": "From extensions of generalized product caps; the best known in dimension 8 before FunSearch.",
      "source": "https://doi.org/10.1023/a:1027365901231"
    },
    "held_by_machine": true,
    "claims": [
      {
        "systems": [
          "FunSearch"
        ],
        "humans": [],
        "date": "2023-12-14",
        "outcome": "record",
        "outcome_text": "An explicit cap set of 512 points in dimension 8.",
        "autonomy": "A2",
        "autonomy_reason": "A search scaffold: the system proposed and evolved candidate programs, while the evaluator defining success was written by people.",
        "lean_claimed": false,
        "sources": [
          {
            "label": "DeepMind, FunSearch",
            "url": "https://deepmind.google/blog/funsearch-making-new-discoveries-in-mathematical-sciences-using-large-language-models/"
          }
        ],
        "details": {},
        "statement": "capset:dimension-8",
        "node": "target",
        "contribution": "primary",
        "outcome_scope": "record",
        "section": "record",
        "section_title": "Record on a quantity",
        "date_text": "2023-12-14",
        "machine_checked": true,
        "checked_by": [
          "qed.bot"
        ],
        "id": "2f198664e59c",
        "verification": "V3",
        "fidelity": null
      }
    ],
    "checks": [
      {
        "target": "capset:dimension-8",
        "evaluator": "cap_set",
        "artifact": "https://github.com/google-deepmind/funsearch/blob/main/cap_set/n8_size512.txt",
        "method": "evaluator",
        "checked": "2026-08-22T21:21:52Z",
        "claim": {
          "size": 512,
          "dimension": 8
        },
        "measured": {
          "size": 512,
          "dimension": 8,
          "duplicates": 0,
          "progression_free": true,
          "violations": 0,
          "example_violation": null
        },
        "verdict": "verified",
        "detail": "512 points in dimension 8, no three collinear",
        "systems": [
          "FunSearch"
        ]
      }
    ],
    "declarations": [],
    "fidelity": {
      "grade": null,
      "reason": "An object, checked against the problem's own constraints rather than a formal statement.",
      "anchor": "",
      "flags": []
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    "mentions": [],
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    "checked_by": [
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  },
  {
    "id": "elliptic:rank-record",
    "key": "rank-record",
    "label": "Largest known rank of an elliptic curve over the rationals",
    "collection": "Number theory",
    "kind": "target",
    "name": "Largest known rank of an elliptic curve over the rationals",
    "description": "The rank of an elliptic curve over the rational numbers counts its independent rational points of infinite order. Whether ranks are unbounded is open; the record is a lower bound, shown by exhibiting independent points.",
    "about": "Records rose from 3 in 1938 to 28 when Noam Elkies found his curve in 2006, and stood there for eighteen years until Elkies and Zev Klagsbrun reached 29 in 2024. In August 2026 curves of rank at least 30 and 31 were submitted to ICARM's Elliptic Curve Rank Leaderboard, which verifies each submission before admitting it; the rank-31 entry is credited to Claude, Levent Alpöge and Ava Howell. Steps before 2026 follow Andrej Dujella's table of rank records, which cites each original paper.",
    "direction": "maximise",
    "unit": "rank (lower bound)",
    "format": "",
    "value_key": "",
    "url": "https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html",
    "page": "t/elliptic-rank-record",
    "aliases": [],
    "tags": [
      "number theory",
      "elliptic curves"
    ],
    "records": [
      {
        "value": 31,
        "display": "at least 31",
        "holder": "Claude, Levent Alpöge and Ava Howell",
        "kind": "ai",
        "date": "2026-08-23",
        "note": "Curve 302. The leaderboard's entry reads: found by Claude, Levent Alpöge, and Ava Howell. Rank exactly 31 is certified under BSD and GRH; the unconditional result is the lower bound shown by 31 independent points.",
        "source": "https://elliptic-rank.icarm.cloud/curve/302",
        "artifact": "https://elliptic-rank.icarm.cloud/curve/302.json"
      },
      {
        "value": 30,
        "display": "at least 30",
        "holder": "Levent Alpöge and Ava Howell, with Claude",
        "kind": "ai",
        "date": "2026-08-20",
        "note": "Curve 273, submitted to ICARM's leaderboard by the user ranksunbounded. Dujella's table credits Alpöge and Howell; Scientific American reports that they used an internal variant of Claude.",
        "source": "https://icarm.io/news/new-record-breaking-elliptic-curve-reported/"
      },
      {
        "value": 29,
        "display": "at least 29",
        "holder": "Noam Elkies and Zev Klagsbrun",
        "kind": "human",
        "date": "2024",
        "note": "From Dujella's table of rank records.",
        "source": "https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html"
      },
      {
        "value": 28,
        "display": "at least 28",
        "holder": "Noam Elkies",
        "kind": "human",
        "date": "2006",
        "note": "From Dujella's table of rank records; the record for eighteen years.",
        "source": "https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html"
      },
      {
        "value": 24,
        "display": "at least 24",
        "holder": "Robert Martin and William McMillen",
        "kind": "human",
        "date": "2000",
        "note": "From Dujella's table of rank records.",
        "source": "https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html"
      },
      {
        "value": 23,
        "display": "at least 23",
        "holder": "Robert Martin and William McMillen",
        "kind": "human",
        "date": "1998",
        "note": "From Dujella's table of rank records.",
        "source": "https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html"
      },
      {
        "value": 22,
        "display": "at least 22",
        "holder": "Stéphane Fermigier",
        "kind": "human",
        "date": "1997",
        "note": "From Dujella's table of rank records.",
        "source": "https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html"
      },
      {
        "value": 21,
        "display": "at least 21",
        "holder": "Kazuhiko Nagao and Tomonori Kouya",
        "kind": "human",
        "date": "1994",
        "note": "From Dujella's table of rank records.",
        "source": "https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html"
      },
      {
        "value": 20,
        "display": "at least 20",
        "holder": "Kazuhiko Nagao",
        "kind": "human",
        "date": "1993",
        "note": "From Dujella's table of rank records.",
        "source": "https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html"
      },
      {
        "value": 19,
        "display": "at least 19",
        "holder": "Stéphane Fermigier",
        "kind": "human",
        "date": "1992",
        "order": 3,
        "note": "From Dujella's table of rank records.",
        "source": "https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html"
      },
      {
        "value": 17,
        "display": "at least 17",
        "holder": "Kazuhiko Nagao",
        "kind": "human",
        "date": "1992",
        "order": 2,
        "note": "From Dujella's table of rank records.",
        "source": "https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html"
      },
      {
        "value": 15,
        "display": "at least 15",
        "holder": "Jean-François Mestre",
        "kind": "human",
        "date": "1992",
        "order": 1,
        "note": "From Dujella's table of rank records.",
        "source": "https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html"
      },
      {
        "value": 14,
        "display": "at least 14",
        "holder": "Jean-François Mestre",
        "kind": "human",
        "date": "1986",
        "note": "From Dujella's table of rank records.",
        "source": "https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html"
      },
      {
        "value": 12,
        "display": "at least 12",
        "holder": "Jean-François Mestre",
        "kind": "human",
        "date": "1982",
        "note": "From Dujella's table of rank records.",
        "source": "https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html"
      },
      {
        "value": 9,
        "display": "at least 9",
        "holder": "Armand Brumer and Kenneth Kramer",
        "kind": "human",
        "date": "1977",
        "order": 2,
        "note": "From Dujella's table of rank records.",
        "source": "https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html"
      },
      {
        "value": 8,
        "display": "at least 8",
        "holder": "Fritz Grunewald and Rainer Zimmert",
        "kind": "human",
        "date": "1977",
        "order": 1,
        "note": "From Dujella's table of rank records.",
        "source": "https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html"
      },
      {
        "value": 7,
        "display": "at least 7",
        "holder": "David Penney and Carl Pomerance",
        "kind": "human",
        "date": "1975",
        "note": "From Dujella's table of rank records.",
        "source": "https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html"
      },
      {
        "value": 6,
        "display": "at least 6",
        "holder": "David Penney and Carl Pomerance",
        "kind": "human",
        "date": "1974",
        "note": "From Dujella's table of rank records.",
        "source": "https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html"
      },
      {
        "value": 4,
        "display": "at least 4",
        "holder": "A. Wiman",
        "kind": "human",
        "date": "1945",
        "note": "From Dujella's table of rank records.",
        "source": "https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html"
      },
      {
        "value": 3,
        "display": "at least 3",
        "holder": "G. Billing",
        "kind": "human",
        "date": "1938",
        "note": "From Dujella's table of rank records.",
        "source": "https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html"
      }
    ],
    "steps": [
      {
        "value": 3,
        "display": "at least 3",
        "holder": "G. Billing",
        "kind": "human",
        "date": "1938",
        "note": "From Dujella's table of rank records.",
        "source": "https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html",
        "moved": true,
        "candidate": false,
        "best_after": 3
      },
      {
        "value": 4,
        "display": "at least 4",
        "holder": "A. Wiman",
        "kind": "human",
        "date": "1945",
        "note": "From Dujella's table of rank records.",
        "source": "https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html",
        "moved": true,
        "candidate": false,
        "best_after": 4
      },
      {
        "value": 6,
        "display": "at least 6",
        "holder": "David Penney and Carl Pomerance",
        "kind": "human",
        "date": "1974",
        "note": "From Dujella's table of rank records.",
        "source": "https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html",
        "moved": true,
        "candidate": false,
        "best_after": 6
      },
      {
        "value": 7,
        "display": "at least 7",
        "holder": "David Penney and Carl Pomerance",
        "kind": "human",
        "date": "1975",
        "note": "From Dujella's table of rank records.",
        "source": "https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html",
        "moved": true,
        "candidate": false,
        "best_after": 7
      },
      {
        "value": 8,
        "display": "at least 8",
        "holder": "Fritz Grunewald and Rainer Zimmert",
        "kind": "human",
        "date": "1977",
        "order": 1,
        "note": "From Dujella's table of rank records.",
        "source": "https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html",
        "moved": true,
        "candidate": false,
        "best_after": 8
      },
      {
        "value": 9,
        "display": "at least 9",
        "holder": "Armand Brumer and Kenneth Kramer",
        "kind": "human",
        "date": "1977",
        "order": 2,
        "note": "From Dujella's table of rank records.",
        "source": "https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html",
        "moved": true,
        "candidate": false,
        "best_after": 9
      },
      {
        "value": 12,
        "display": "at least 12",
        "holder": "Jean-François Mestre",
        "kind": "human",
        "date": "1982",
        "note": "From Dujella's table of rank records.",
        "source": "https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html",
        "moved": true,
        "candidate": false,
        "best_after": 12
      },
      {
        "value": 14,
        "display": "at least 14",
        "holder": "Jean-François Mestre",
        "kind": "human",
        "date": "1986",
        "note": "From Dujella's table of rank records.",
        "source": "https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html",
        "moved": true,
        "candidate": false,
        "best_after": 14
      },
      {
        "value": 15,
        "display": "at least 15",
        "holder": "Jean-François Mestre",
        "kind": "human",
        "date": "1992",
        "order": 1,
        "note": "From Dujella's table of rank records.",
        "source": "https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html",
        "moved": true,
        "candidate": false,
        "best_after": 15
      },
      {
        "value": 17,
        "display": "at least 17",
        "holder": "Kazuhiko Nagao",
        "kind": "human",
        "date": "1992",
        "order": 2,
        "note": "From Dujella's table of rank records.",
        "source": "https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html",
        "moved": true,
        "candidate": false,
        "best_after": 17
      },
      {
        "value": 19,
        "display": "at least 19",
        "holder": "Stéphane Fermigier",
        "kind": "human",
        "date": "1992",
        "order": 3,
        "note": "From Dujella's table of rank records.",
        "source": "https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html",
        "moved": true,
        "candidate": false,
        "best_after": 19
      },
      {
        "value": 20,
        "display": "at least 20",
        "holder": "Kazuhiko Nagao",
        "kind": "human",
        "date": "1993",
        "note": "From Dujella's table of rank records.",
        "source": "https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html",
        "moved": true,
        "candidate": false,
        "best_after": 20
      },
      {
        "value": 21,
        "display": "at least 21",
        "holder": "Kazuhiko Nagao and Tomonori Kouya",
        "kind": "human",
        "date": "1994",
        "note": "From Dujella's table of rank records.",
        "source": "https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html",
        "moved": true,
        "candidate": false,
        "best_after": 21
      },
      {
        "value": 22,
        "display": "at least 22",
        "holder": "Stéphane Fermigier",
        "kind": "human",
        "date": "1997",
        "note": "From Dujella's table of rank records.",
        "source": "https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html",
        "moved": true,
        "candidate": false,
        "best_after": 22
      },
      {
        "value": 23,
        "display": "at least 23",
        "holder": "Robert Martin and William McMillen",
        "kind": "human",
        "date": "1998",
        "note": "From Dujella's table of rank records.",
        "source": "https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html",
        "moved": true,
        "candidate": false,
        "best_after": 23
      },
      {
        "value": 24,
        "display": "at least 24",
        "holder": "Robert Martin and William McMillen",
        "kind": "human",
        "date": "2000",
        "note": "From Dujella's table of rank records.",
        "source": "https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html",
        "moved": true,
        "candidate": false,
        "best_after": 24
      },
      {
        "value": 28,
        "display": "at least 28",
        "holder": "Noam Elkies",
        "kind": "human",
        "date": "2006",
        "note": "From Dujella's table of rank records; the record for eighteen years.",
        "source": "https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html",
        "moved": true,
        "candidate": false,
        "best_after": 28
      },
      {
        "value": 29,
        "display": "at least 29",
        "holder": "Noam Elkies and Zev Klagsbrun",
        "kind": "human",
        "date": "2024",
        "note": "From Dujella's table of rank records.",
        "source": "https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html",
        "moved": true,
        "candidate": false,
        "best_after": 29
      },
      {
        "value": 30,
        "display": "at least 30",
        "holder": "Levent Alpöge and Ava Howell, with Claude",
        "kind": "ai",
        "date": "2026-08-20",
        "note": "Curve 273, submitted to ICARM's leaderboard by the user ranksunbounded. Dujella's table credits Alpöge and Howell; Scientific American reports that they used an internal variant of Claude.",
        "source": "https://icarm.io/news/new-record-breaking-elliptic-curve-reported/",
        "moved": true,
        "candidate": false,
        "best_after": 30
      },
      {
        "value": 31,
        "display": "at least 31",
        "holder": "Claude, Levent Alpöge and Ava Howell",
        "kind": "ai",
        "date": "2026-08-23",
        "note": "Curve 302. The leaderboard's entry reads: found by Claude, Levent Alpöge, and Ava Howell. Rank exactly 31 is certified under BSD and GRH; the unconditional result is the lower bound shown by 31 independent points.",
        "source": "https://elliptic-rank.icarm.cloud/curve/302",
        "artifact": "https://elliptic-rank.icarm.cloud/curve/302.json",
        "moved": true,
        "candidate": false,
        "best_after": 31
      }
    ],
    "best": {
      "value": 31,
      "display": "at least 31",
      "holder": "Claude, Levent Alpöge and Ava Howell",
      "kind": "ai",
      "date": "2026-08-23",
      "note": "Curve 302. The leaderboard's entry reads: found by Claude, Levent Alpöge, and Ava Howell. Rank exactly 31 is certified under BSD and GRH; the unconditional result is the lower bound shown by 31 independent points.",
      "source": "https://elliptic-rank.icarm.cloud/curve/302",
      "artifact": "https://elliptic-rank.icarm.cloud/curve/302.json"
    },
    "previous": {
      "value": 30,
      "display": "at least 30",
      "holder": "Levent Alpöge and Ava Howell, with Claude",
      "kind": "ai",
      "date": "2026-08-20",
      "note": "Curve 273, submitted to ICARM's leaderboard by the user ranksunbounded. Dujella's table credits Alpöge and Howell; Scientific American reports that they used an internal variant of Claude.",
      "source": "https://icarm.io/news/new-record-breaking-elliptic-curve-reported/"
    },
    "held_by_machine": true,
    "claims": [
      {
        "systems": [
          "Claude"
        ],
        "humans": [
          "Levent Alpöge",
          "Ava Howell"
        ],
        "date": "2026-08-20",
        "outcome": "superseded",
        "outcome_text": "An elliptic curve over the rationals of rank at least 30, the first above Elkies and Klagsbrun's 29.",
        "autonomy": "A1",
        "autonomy_reason": "Reported as found by two mathematicians using an internal variant of Claude; the leaderboard's entry for this curve names only its submitter.",
        "lean_claimed": false,
        "sources": [
          {
            "label": "ICARM, New record breaking elliptic curve reported",
            "url": "https://icarm.io/news/new-record-breaking-elliptic-curve-reported/"
          },
          {
            "label": "Scientific American, Anthropic's AI steals mathematicians' record for most complicated curve",
            "url": "https://www.scientificamerican.com/article/anthropics-ai-steals-mathematicians-record-for-most-complicated-curve/"
          }
        ],
        "details": {},
        "statement": "elliptic:rank-record",
        "node": "target",
        "contribution": "primary",
        "outcome_scope": "record",
        "section": "record",
        "section_title": "Record on a quantity",
        "date_text": "2026-08-20",
        "machine_checked": false,
        "checked_by": [],
        "id": "5aba75d14892",
        "verification": "V2",
        "fidelity": "F0"
      },
      {
        "systems": [
          "Claude"
        ],
        "humans": [
          "Levent Alpöge",
          "Ava Howell"
        ],
        "date": "2026-08-23",
        "outcome": "record",
        "outcome_text": "An elliptic curve over the rationals of rank at least 31, verified by ICARM's leaderboard.",
        "autonomy": "A1",
        "autonomy_reason": "The leaderboard credits the curve jointly to Claude and two mathematicians.",
        "lean_claimed": false,
        "sources": [
          {
            "label": "ICARM Elliptic Curve Rank Leaderboard, curve 302",
            "url": "https://elliptic-rank.icarm.cloud/curve/302"
          }
        ],
        "details": {},
        "statement": "elliptic:rank-record",
        "node": "target",
        "contribution": "primary",
        "outcome_scope": "record",
        "section": "record",
        "section_title": "Record on a quantity",
        "date_text": "2026-08-23",
        "machine_checked": false,
        "checked_by": [],
        "id": "a1f4631eb597",
        "verification": "V2",
        "fidelity": "F0"
      }
    ],
    "checks": [],
    "declarations": [],
    "fidelity": {
      "grade": "F0",
      "reason": "Absent. No formal statement is attached to the result.",
      "anchor": "",
      "flags": []
    },
    "mentions": [],
    "relations": [],
    "verified": false,
    "checked_by": []
  }
]