[
  {
    "systems": [
      "GPT-6 Astra"
    ],
    "humans": [],
    "date": "2026-09-09",
    "outcome": "full",
    "outcome_text": "Formalised the Navier–Stokes and Euler results in Lean, checked with Comparator against the Formal Conjectures statements of alternatives (C) and (D).",
    "contribution": "secondary",
    "autonomy": "A1",
    "autonomy_reason": "Formalisation of a proof found by another system; the project's formalization.yaml records the method as an agent in Codex.",
    "lean_claimed": true,
    "sources": [
      {
        "label": "Lean formalisation",
        "url": "https://github.com/openai/NavierStokesAndEuler"
      },
      {
        "label": "formalization.yaml",
        "url": "https://github.com/openai/NavierStokesAndEuler/blob/main/formalization.yaml"
      }
    ],
    "details": {
      "scale": "2,659 Lean files, about 33 MB of source"
    },
    "id": "89ef9abfa51b",
    "statement": "millennium:NavierStokes",
    "date_text": "2026-09-09",
    "outcome_scope": "task",
    "section": "curated",
    "section_title": "Recorded from its sources",
    "machine_checked": false,
    "checked_by": [],
    "verification": "V2",
    "fidelity": "F3",
    "node": "statement",
    "statement_summary": {
      "id": "millennium:NavierStokes",
      "key": "NavierStokes",
      "label": "Existence And Smoothness Of The Navier–Stokes Equation",
      "title": "Existence And Smoothness Of The Navier–Stokes Equation",
      "collection": "Millennium",
      "url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Millennium/NavierStokes.lean",
      "page": "s/millennium-navierstokes",
      "prize": null,
      "prize_value": 0,
      "tags": [],
      "formal_status": "mixed",
      "has_formal_statement": true,
      "artifact_state": "unreachable",
      "verified": false,
      "checked_by": [],
      "fidelity": "F3",
      "declared": true
    }
  },
  {
    "systems": [
      "OpenAI internal model"
    ],
    "humans": [],
    "date": "2026-09-08",
    "outcome": "full",
    "outcome_text": "Finite-time blowup for the 3D incompressible Navier–Stokes equations with smooth forcing, on Euclidean space and on the torus: Fefferman's alternatives (C) and (D), a negative answer to the Millennium problem. Unforced Euler blowup is proved alongside.",
    "contribution": "primary",
    "autonomy": "A2",
    "autonomy_reason": "By OpenAI's account some 10,000 agents worked for 88 hours with little human mathematical input. The run was aimed at a problem the Córdoba–Martínez-Zoroa programme had already brought within reach, and began after reports of the Buckmaster–Alpöge work reached OpenAI. A2 records a result the system produced on framing it was given.",
    "lean_claimed": true,
    "sources": [
      {
        "label": "OpenAI, On the Navier–Stokes Millennium Prize Problem",
        "url": "https://openai.com/index/navier-stokes-solution/"
      },
      {
        "label": "Lean formalisation",
        "url": "https://github.com/openai/NavierStokesAndEuler"
      },
      {
        "label": "Quanta, AI has solved one of math's $1 million Millennium Prize Problems",
        "url": "https://www.quantamagazine.org/ai-has-solved-one-of-maths-1-million-millennium-prize-problems-20260908/"
      },
      {
        "label": "Wikipedia, Navier–Stokes priority controversy",
        "url": "https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_priority_controversy"
      }
    ],
    "details": {
      "compute": "At least 10,000 agents over an 88-hour run from 1 September; 2.7 million messages and 130 billion output tokens in the run that produced the proof",
      "paper": "166 pages, revised to cite Córdoba and Martínez-Zoroa"
    },
    "provenance": {
      "precursors": [
        {
          "who": "Thomas Hou and Guo Luo",
          "date": "2013",
          "what": "Numerical evidence that solutions of the 3D Euler equations can blow up in finite time.",
          "source": "https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_priority_controversy"
        },
        {
          "who": "Diego Córdoba and Luis Martínez-Zoroa",
          "date": "2023",
          "what": "An analytic programme for constructing finite-time blowup, which with Fan Zheng gave unforced blowup for 3D Euler and forced blowup for hypodissipative Navier–Stokes. Both teams that reached the Millennium setting built on it.",
          "source": "https://arxiv.org/abs/2309.08495"
        }
      ],
      "concurrent": [
        {
          "who": "Tristan Buckmaster and Levent Alpöge",
          "date": "2026-08-22",
          "announced": "2026-09-07",
          "what": "Blowup with smooth forcing for Euler, the incompressible porous media equation and Boussinesq, obtained on 15 August with AI assistance and verified in Lean a week later. Announced about twelve hours before OpenAI.",
          "source": "https://cims.nyu.edu/~tristanb/statement.pdf"
        }
      ],
      "disputes": [
        {
          "topic": "Priority and customer data",
          "summary": "Buckmaster says news of the pair's progress reached OpenAI before its run, and asks whether their use of Codex informed the model. OpenAI's account changed between 8 and 13 September, ending with the statement that no user input after 3 July could have influenced the system.",
          "source": "https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_priority_controversy"
        },
        {
          "topic": "Attribution",
          "summary": "The first version of OpenAI's paper did not cite Córdoba and Martínez-Zoroa. The references were added in the 166-page revision.",
          "source": "https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_priority_controversy"
        }
      ],
      "prize": {
        "name": "Clay Millennium Prize",
        "status": "OpenAI does not intend to claim it, and the Clay Mathematics Institute has acknowledged no party."
      }
    },
    "id": "ed88695692e8",
    "statement": "millennium:NavierStokes",
    "date_text": "2026-09-08",
    "outcome_scope": "statement",
    "section": "curated",
    "section_title": "Recorded from its sources",
    "machine_checked": false,
    "checked_by": [],
    "verification": "V2",
    "fidelity": "F3",
    "node": "statement",
    "statement_summary": {
      "id": "millennium:NavierStokes",
      "key": "NavierStokes",
      "label": "Existence And Smoothness Of The Navier–Stokes Equation",
      "title": "Existence And Smoothness Of The Navier–Stokes Equation",
      "collection": "Millennium",
      "url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Millennium/NavierStokes.lean",
      "page": "s/millennium-navierstokes",
      "prize": null,
      "prize_value": 0,
      "tags": [],
      "formal_status": "mixed",
      "has_formal_statement": true,
      "artifact_state": "unreachable",
      "verified": false,
      "checked_by": [],
      "fidelity": "F3",
      "declared": true
    }
  },
  {
    "systems": [
      "Claude",
      "GPT-5.6 Sol",
      "GPT-6 Astra"
    ],
    "humans": [
      "Levent Alpöge",
      "Tristan Buckmaster",
      "Matei P. Coiculescu"
    ],
    "date": "2026-09-07",
    "outcome": "full",
    "outcome_text": "Finite-time blowup for the incompressible porous media equation on the two-dimensional torus with a force smooth in space and time, extending Córdoba and Martínez-Zoroa's result with a spatially smooth force.",
    "contribution": "primary",
    "autonomy": "A1",
    "autonomy_reason": "By the authors' account, mathematicians pushed the Córdoba–Martínez-Zoroa programme to smooth forcing with a great deal of help from language models; the programme was neither started nor proposed by a model.",
    "lean_claimed": false,
    "sources": [
      {
        "label": "Extending the Córdoba–Martínez-Zoroa IPM blow-up to uniformly space-time smooth forcing",
        "url": "https://cims.nyu.edu/~tristanb/ipm.pdf"
      },
      {
        "label": "Tristan Buckmaster, statement of 7 September 2026",
        "url": "https://cims.nyu.edu/~tristanb/statement.pdf"
      },
      {
        "label": "Terence Tao, on the three blowup results",
        "url": "https://terrytao.wordpress.com/2026/09/07/finite-time-blowup-with-smooth-forcing-term-for-the-incompressible-porous-medium-boussinesq-and-incompressible-euler-equations/"
      }
    ],
    "details": {
      "found": "Before the Boussinesq and Euler results, by the authors' account",
      "lean": "The authors say their results were verified in Lean; no public repository is cited"
    },
    "provenance": {
      "precursors": [
        {
          "who": "Diego Córdoba and Luis Martínez-Zoroa",
          "date": "2023",
          "what": "Blowup for the porous media equation with a spatially smooth force, which this result extends to a force smooth in time as well.",
          "source": "https://arxiv.org/abs/2309.08495"
        }
      ]
    },
    "id": "0ccd14772447",
    "statement": "pde:PorousMediaSmoothForcingBlowup",
    "date_text": "2026-09-07",
    "outcome_scope": "statement",
    "section": "curated",
    "section_title": "Recorded from its sources",
    "machine_checked": false,
    "checked_by": [],
    "verification": "V1",
    "fidelity": "F0",
    "node": "statement",
    "statement_summary": {
      "id": "pde:PorousMediaSmoothForcingBlowup",
      "key": "PorousMediaSmoothForcingBlowup",
      "label": "Finite-time blowup for the incompressible porous media equation with smooth forcing",
      "title": "Finite-time blowup for the incompressible porous media equation with smooth forcing",
      "collection": "Fluid dynamics",
      "url": "https://cims.nyu.edu/~tristanb/ipm.pdf",
      "page": "s/pde-porousmediasmoothforcingblowup",
      "prize": null,
      "prize_value": 0,
      "tags": [
        "partial differential equations",
        "fluid dynamics",
        "finite-time blowup"
      ],
      "formal_status": "unclassified",
      "has_formal_statement": false,
      "artifact_state": null,
      "verified": false,
      "checked_by": [],
      "fidelity": "F0",
      "declared": false
    }
  },
  {
    "systems": [
      "Claude",
      "GPT-5.6 Sol",
      "GPT-6 Astra"
    ],
    "humans": [
      "Tristan Buckmaster",
      "Levent Alpöge"
    ],
    "date": "2026-09-07",
    "outcome": "full",
    "outcome_text": "A finite-time singularity for the incompressible Euler equations on ℝ³ with a force smooth in space and time up to and including the blowup time.",
    "contribution": "primary",
    "autonomy": "A1",
    "autonomy_reason": "By the authors' account, two mathematicians pushed the Córdoba–Martínez-Zoroa programme to smooth forcing with a great deal of help from language models; the programme was neither started nor proposed by a model, and GPT-6 Astra was used only for write-ups and auditing.",
    "lean_claimed": false,
    "sources": [
      {
        "label": "Blowup for the Euler equations with smooth forcing",
        "url": "https://cims.nyu.edu/~tristanb/euler.pdf"
      },
      {
        "label": "Tristan Buckmaster, statement of 7 September 2026",
        "url": "https://cims.nyu.edu/~tristanb/statement.pdf"
      },
      {
        "label": "Terence Tao, on the three blowup results",
        "url": "https://terrytao.wordpress.com/2026/09/07/finite-time-blowup-with-smooth-forcing-term-for-the-incompressible-porous-medium-boussinesq-and-incompressible-euler-equations/"
      }
    ],
    "details": {
      "found": "15 August 2026, by the authors' account",
      "lean": "Verified in Lean on 22 August 2026, by the authors' account; no public repository is cited",
      "write-up": "Buckmaster says the Euler write-up is much closer to what models produce under human direction than to a paper written by a person"
    },
    "provenance": {
      "precursors": [
        {
          "who": "Diego Córdoba and Luis Martínez-Zoroa",
          "date": "2023",
          "what": "The programme of forced blowup constructions, with rough forcing, that this work pushes to smooth forcing.",
          "source": "https://arxiv.org/abs/2309.08495"
        }
      ],
      "disputes": [
        {
          "topic": "Priority",
          "summary": "Announced about twelve hours before OpenAI's Navier–Stokes claim; Buckmaster's statement sets out his contacts with OpenAI in the days before it.",
          "source": "https://cims.nyu.edu/~tristanb/statement.pdf"
        }
      ]
    },
    "id": "58935e97737d",
    "statement": "pde:EulerSmoothForcingBlowup",
    "date_text": "2026-09-07",
    "outcome_scope": "statement",
    "section": "curated",
    "section_title": "Recorded from its sources",
    "machine_checked": false,
    "checked_by": [],
    "verification": "V1",
    "fidelity": "F0",
    "node": "statement",
    "statement_summary": {
      "id": "pde:EulerSmoothForcingBlowup",
      "key": "EulerSmoothForcingBlowup",
      "label": "Finite-time blowup for the 3D incompressible Euler equations with smooth forcing",
      "title": "Finite-time blowup for the 3D incompressible Euler equations with smooth forcing",
      "collection": "Fluid dynamics",
      "url": "https://cims.nyu.edu/~tristanb/euler.pdf",
      "page": "s/pde-eulersmoothforcingblowup",
      "prize": null,
      "prize_value": 0,
      "tags": [
        "partial differential equations",
        "fluid dynamics",
        "finite-time blowup"
      ],
      "formal_status": "unclassified",
      "has_formal_statement": false,
      "artifact_state": null,
      "verified": false,
      "checked_by": [],
      "fidelity": "F0",
      "declared": false
    }
  },
  {
    "systems": [
      "Claude",
      "GPT-5.6 Sol",
      "GPT-6 Astra"
    ],
    "humans": [
      "Levent Alpöge",
      "Tristan Buckmaster"
    ],
    "date": "2026-09-07",
    "outcome": "full",
    "outcome_text": "Finite-time blowup for the inviscid Boussinesq system on ℝ² with smooth forcing in both equations: the temperature stays bounded while its gradient and the vorticity blow up.",
    "contribution": "primary",
    "autonomy": "A1",
    "autonomy_reason": "By the authors' account, two mathematicians pushed the Córdoba–Martínez-Zoroa programme to smooth forcing with a great deal of help from language models; the programme was neither started nor proposed by a model, and GPT-6 Astra was used only for write-ups and auditing.",
    "lean_claimed": false,
    "sources": [
      {
        "label": "Blowup for the Boussinesq equations with smooth forcing",
        "url": "https://cims.nyu.edu/~tristanb/boussinesq.pdf"
      },
      {
        "label": "Tristan Buckmaster, statement of 7 September 2026",
        "url": "https://cims.nyu.edu/~tristanb/statement.pdf"
      },
      {
        "label": "Terence Tao, on the three blowup results",
        "url": "https://terrytao.wordpress.com/2026/09/07/finite-time-blowup-with-smooth-forcing-term-for-the-incompressible-porous-medium-boussinesq-and-incompressible-euler-equations/"
      }
    ],
    "details": {
      "found": "15 August 2026, by the authors' account",
      "lean": "Verified in Lean on 22 August 2026, by the authors' account; no public repository is cited",
      "ai statement": "The paper carries a section setting out how AI was used"
    },
    "provenance": {
      "precursors": [
        {
          "who": "Diego Córdoba and Luis Martínez-Zoroa",
          "date": "2023",
          "what": "The multiscale programme of forced blowup constructions that this work follows.",
          "source": "https://arxiv.org/abs/2309.08495"
        }
      ]
    },
    "id": "3edb0739c066",
    "statement": "pde:BoussinesqSmoothForcingBlowup",
    "date_text": "2026-09-07",
    "outcome_scope": "statement",
    "section": "curated",
    "section_title": "Recorded from its sources",
    "machine_checked": false,
    "checked_by": [],
    "verification": "V1",
    "fidelity": "F0",
    "node": "statement",
    "statement_summary": {
      "id": "pde:BoussinesqSmoothForcingBlowup",
      "key": "BoussinesqSmoothForcingBlowup",
      "label": "Finite-time blowup for the inviscid Boussinesq equations with smooth forcing",
      "title": "Finite-time blowup for the inviscid Boussinesq equations with smooth forcing",
      "collection": "Fluid dynamics",
      "url": "https://cims.nyu.edu/~tristanb/boussinesq.pdf",
      "page": "s/pde-boussinesqsmoothforcingblowup",
      "prize": null,
      "prize_value": 0,
      "tags": [
        "partial differential equations",
        "fluid dynamics",
        "finite-time blowup"
      ],
      "formal_status": "unclassified",
      "has_formal_statement": false,
      "artifact_state": null,
      "verified": false,
      "checked_by": [],
      "fidelity": "F0",
      "declared": false
    }
  },
  {
    "systems": [
      "Claude"
    ],
    "humans": [],
    "date": "2026-09-04",
    "outcome": "full",
    "outcome_text": "A complete Lean 4 proof of Fermat's Last Theorem on the standard axioms, deriving Mathlib's own statement, with the classical inputs proved in the strength the argument needs rather than assumed.",
    "contribution": "secondary",
    "autonomy": "A1",
    "autonomy_reason": "The mathematics is Wiles and Taylor's. The machine's contribution is the formalisation, which Anthropic describes as largely autonomous and the project's formalization.yaml records as agent work.",
    "lean_claimed": true,
    "sources": [
      {
        "label": "Anthropic, Formalizing Fermat's Last Theorem",
        "url": "https://www.anthropic.com/research/formalizing-fermats-last-theorem"
      },
      {
        "label": "Lean formalisation",
        "url": "https://github.com/anthropics/fermats-last-theorem"
      }
    ],
    "details": {
      "scale": "About 13 million lines of Lean and some 30,000 intermediate theorems, in 11 days",
      "builds on": "The Imperial College London FLT project led by Kevin Buzzard, and flt-regular"
    },
    "id": "2cba2d1d7d11",
    "statement": "mathlib:FermatLastTheorem",
    "date_text": "2026-09-04",
    "outcome_scope": "task",
    "section": "curated",
    "section_title": "Recorded from its sources",
    "machine_checked": false,
    "checked_by": [],
    "verification": "V2",
    "fidelity": "F2",
    "node": "statement",
    "statement_summary": {
      "id": "mathlib:FermatLastTheorem",
      "key": "FermatLastTheorem",
      "label": "Fermat's Last Theorem",
      "title": "Fermat's Last Theorem",
      "collection": "Mathlib",
      "url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/FLT/Basic.html",
      "page": "s/mathlib-fermatlasttheorem",
      "prize": null,
      "prize_value": 0,
      "tags": [
        "number theory",
        "Diophantine equations"
      ],
      "formal_status": "unclassified",
      "has_formal_statement": false,
      "artifact_state": null,
      "verified": false,
      "checked_by": [],
      "fidelity": "F2",
      "declared": true
    }
  },
  {
    "systems": [
      "AxiomProver"
    ],
    "humans": [
      "François Charton",
      "Letong Hong",
      "Kenny Lau",
      "Ken Ono",
      "Guillaume Remy",
      "Ho Chung Siu",
      "Ashvin A. Swaminathan",
      "Jesse Thorner",
      "Yunzhou Xie"
    ],
    "date": "2026-09-03",
    "outcome": "superseded",
    "outcome_text": "Proved a bound of 212, building on Stadlmann's work, with a Lean certificate of the deduction.",
    "autonomy": "A1",
    "autonomy_reason": "The mathematics is the authors'; AxiomProver played a supporting role, producing the Lean certificate of the deduction from their natural-language specifications (Appendix A).",
    "lean_claimed": false,
    "sources": [
      {
        "label": "Axiom Math, A new bound for small gaps between primes (preliminary draft)",
        "url": "https://primegaps.axiommath.ai/bgp212.pdf"
      }
    ],
    "details": {
      "formalisation": "A Lean certificate of the deduction, taking the equidistribution estimates as hypotheses"
    },
    "statement": "primes:bounded-gaps",
    "node": "target",
    "contribution": "primary",
    "outcome_scope": "record",
    "section": "record",
    "section_title": "Record on a quantity",
    "date_text": "2026-09-03",
    "machine_checked": false,
    "checked_by": [],
    "id": "1e75b762c9b1",
    "verification": "V2",
    "fidelity": "F2",
    "statement_summary": {
      "id": "primes:bounded-gaps",
      "key": "bounded-gaps",
      "label": "Bounded gaps between primes",
      "title": "Bounded gaps between primes",
      "collection": "Analytic number theory",
      "url": "https://arxiv.org/abs/2608.31126",
      "page": "t/primes-bounded-gaps",
      "prize": null,
      "prize_value": 0,
      "tags": [
        "number theory",
        "prime gaps"
      ],
      "formal_status": "record",
      "has_formal_statement": false,
      "artifact_state": "clean",
      "verified": false,
      "checked_by": [],
      "fidelity": "F2",
      "declared": true,
      "node": "target",
      "best": {
        "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"
      },
      "direction": "minimise"
    }
  },
  {
    "systems": [
      "GPT-6 Astra"
    ],
    "humans": [],
    "date": "2026-09-03",
    "outcome": "record",
    "outcome_text": "Proved that infinitely many pairs of consecutive primes are at most 186 apart.",
    "autonomy": "A3",
    "autonomy_reason": "OpenAI's report says the proof is due to GPT 6 Astra and describes no human contribution to it.",
    "lean_claimed": true,
    "sources": [
      {
        "label": "OpenAI, Improved short gaps between primes",
        "url": "https://cdn.openai.com/pdf/51126fac-1b68-4128-9666-c908bcc16033/short_gaps.pdf"
      },
      {
        "label": "Conditional Lean formalisation",
        "url": "https://github.com/openai/PrimeGaps186"
      }
    ],
    "details": {
      "formalisation": "Lean 4, conditional on stated numerical integral and cap bounds and on exponential-sum estimates"
    },
    "statement": "primes:bounded-gaps",
    "node": "target",
    "contribution": "primary",
    "outcome_scope": "record",
    "section": "record",
    "section_title": "Record on a quantity",
    "date_text": "2026-09-03",
    "machine_checked": false,
    "checked_by": [],
    "id": "e36c3b5ee17d",
    "verification": "V2",
    "fidelity": "F2",
    "statement_summary": {
      "id": "primes:bounded-gaps",
      "key": "bounded-gaps",
      "label": "Bounded gaps between primes",
      "title": "Bounded gaps between primes",
      "collection": "Analytic number theory",
      "url": "https://arxiv.org/abs/2608.31126",
      "page": "t/primes-bounded-gaps",
      "prize": null,
      "prize_value": 0,
      "tags": [
        "number theory",
        "prime gaps"
      ],
      "formal_status": "record",
      "has_formal_statement": false,
      "artifact_state": "clean",
      "verified": false,
      "checked_by": [],
      "fidelity": "F2",
      "declared": true,
      "node": "target",
      "best": {
        "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"
      },
      "direction": "minimise"
    }
  },
  {
    "systems": [
      "Claude",
      "ChatGPT",
      "Codex"
    ],
    "humans": [
      "Shiva Kintali"
    ],
    "date": "2026-09-01",
    "outcome": "superseded",
    "outcome_text": "Proved that infinitely many pairs of consecutive primes are at most 236 apart, improving Stadlmann's 240.",
    "autonomy": "A1",
    "autonomy_reason": "The author ran a harness directing several models, corrected and redirected them, and takes responsibility for the proofs and the code, as the paper's statement on AI use says.",
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    "sources": [
      {
        "label": "Kintali, Improved bounded gaps between primes",
        "url": "https://shivakintali.github.io/papers/Prime-Gaps.pdf"
      },
      {
        "label": "Code and certificates",
        "url": "https://github.com/shivakintali/Prime-Gaps"
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    ],
    "details": {},
    "statement": "primes:bounded-gaps",
    "node": "target",
    "contribution": "primary",
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      "id": "primes:bounded-gaps",
      "key": "bounded-gaps",
      "label": "Bounded gaps between primes",
      "title": "Bounded gaps between primes",
      "collection": "Analytic number theory",
      "url": "https://arxiv.org/abs/2608.31126",
      "page": "t/primes-bounded-gaps",
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        "number theory",
        "prime gaps"
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        "value": 186,
        "display": "186",
        "holder": "GPT-6 Astra (OpenAI)",
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        "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"
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  {
    "systems": [
      "Claude"
    ],
    "humans": [
      "Levent Alpöge",
      "Ava Howell"
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    "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.",
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        "label": "ICARM Elliptic Curve Rank Leaderboard, curve 302",
        "url": "https://elliptic-rank.icarm.cloud/curve/302"
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    ],
    "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",
    "statement_summary": {
      "id": "elliptic:rank-record",
      "key": "rank-record",
      "label": "Largest known rank of an elliptic curve over the rationals",
      "title": "Largest known rank of an elliptic curve over the rationals",
      "collection": "Number theory",
      "url": "https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html",
      "page": "t/elliptic-rank-record",
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      "tags": [
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        "elliptic curves"
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      "formal_status": "record",
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      "checked_by": [],
      "fidelity": "F0",
      "declared": false,
      "node": "target",
      "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"
      },
      "direction": "maximise"
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  },
  {
    "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",
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    "contribution": "primary",
    "outcome_scope": "record",
    "section": "record",
    "section_title": "Record on a quantity",
    "date_text": "2026-08-20",
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    "checked_by": [],
    "id": "5aba75d14892",
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    "statement_summary": {
      "id": "elliptic:rank-record",
      "key": "rank-record",
      "label": "Largest known rank of an elliptic curve over the rationals",
      "title": "Largest known rank of an elliptic curve over the rationals",
      "collection": "Number theory",
      "url": "https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html",
      "page": "t/elliptic-rank-record",
      "prize": null,
      "prize_value": 0,
      "tags": [
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        "elliptic curves"
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      "checked_by": [],
      "fidelity": "F0",
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      "node": "target",
      "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"
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      "direction": "maximise"
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  },
  {
    "systems": [
      "AlphaEvolve"
    ],
    "humans": [
      "Emilien Dupont",
      "Marvin Eisenberger",
      "Borislav Kozlovskii",
      "Abbas Mehrabian",
      "Francisco J. R. Ruiz",
      "Abigail See",
      "Renfei Zhou",
      "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.",
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      {
        "label": "Dupont et al., Improving the matrix multiplication exponent with modern optimization and AlphaEvolve",
        "url": "https://arxiv.org/abs/2608.16884"
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    "statement": "matmul:omega",
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    "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",
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      "label": "Exponent of matrix multiplication",
      "title": "Exponent of matrix multiplication",
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      "url": "https://arxiv.org/abs/2608.16884",
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        "matrix multiplication"
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      "checked_by": [],
      "fidelity": "F0",
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      "node": "target",
      "best": {
        "value": 2.371177,
        "display": "2.371177",
        "holder": "Emilien Dupont and nine co-authors, with AlphaEvolve",
        "kind": "ai",
        "date": "2026-08-17",
        "note": "A reformulated optimisation problem and a new optimisation algorithm, refined with AlphaEvolve.",
        "source": "https://arxiv.org/abs/2608.16884"
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  {
    "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,
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      {
        "label": "Ji, An AI generated counterexample to Borsuk problem in dimension 63 (withdrawn)",
        "url": "https://arxiv.org/abs/2608.12561"
      }
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    "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",
    "statement_summary": {
      "id": "borsuk:dimension",
      "key": "dimension",
      "label": "Smallest dimension in which Borsuk's conjecture is known to fail",
      "title": "Smallest dimension in which Borsuk's conjecture is known to fail",
      "collection": "Discrete geometry",
      "url": "https://github.com/maaxgrin/borsuk-63-counterexample",
      "page": "t/borsuk-dimension",
      "prize": null,
      "prize_value": 0,
      "tags": [
        "discrete geometry",
        "Borsuk's problem"
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      "has_formal_statement": false,
      "artifact_state": "clean",
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      "checked_by": [],
      "fidelity": "F0",
      "declared": false,
      "node": "target",
      "best": {
        "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"
      },
      "direction": "minimise"
    }
  },
  {
    "systems": [
      "Claude"
    ],
    "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.",
    "lean_claimed": true,
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      {
        "label": "Alpöge and Furman, More than two thirds of the zeta zeros are simple and on the critical line",
        "url": "https://arxiv.org/abs/2608.13637"
      },
      {
        "label": "Anthropic, Learning more about Claude's mathematical capabilities",
        "url": "https://www.anthropic.com/research/riemann-zeta"
      },
      {
        "label": "Lean formalisation",
        "url": "https://github.com/anthropics/formal-math/tree/main/zeta23"
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    ],
    "details": {
      "method": "Two Claude Code sessions, 31 million output tokens, roughly 60 subagents",
      "review": "Examined by Brian Conrey and Dan Goldston; formalisation author-verified by Ralph Furman"
    },
    "statement": "zeta:critical-line-proportion",
    "node": "target",
    "contribution": "primary",
    "outcome_scope": "record",
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    "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",
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    "fidelity": "F2",
    "statement_summary": {
      "id": "zeta:critical-line-proportion",
      "key": "critical-line-proportion",
      "label": "Proportion of zeta zeros proved to lie on the critical line",
      "title": "Proportion of zeta zeros proved to lie on the critical line",
      "collection": "Analytic number theory",
      "url": "https://arxiv.org/abs/2608.13637",
      "page": "t/zeta-critical-line-proportion",
      "prize": null,
      "prize_value": 0,
      "tags": [
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        "Riemann zeta function"
      ],
      "formal_status": "record",
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      "artifact_state": "clean",
      "verified": true,
      "checked_by": [
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      "fidelity": "F2",
      "declared": true,
      "node": "target",
      "best": {
        "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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  },
  {
    "systems": [
      "Claude Fable 5"
    ],
    "humans": [
      "Levent Alpöge"
    ],
    "date": "2026-07-19",
    "outcome": "full",
    "outcome_text": "An explicit polynomial map in three variables with constant Jacobian determinant −2 that sends distinct points to the same image, disproving the Jacobian conjecture in every dimension above two. The two-variable case remains open.",
    "contribution": "primary",
    "autonomy": "A2",
    "autonomy_reason": "Alpöge presented the counterexample and credited it to Claude Fable 5; how it was found has not been disclosed. A2 records a result credited to the system on a problem a person chose, with no public account of how much of the search was the system's own.",
    "lean_claimed": true,
    "sources": [
      {
        "label": "Wikipedia, Jacobian conjecture",
        "url": "https://en.wikipedia.org/wiki/Jacobian_conjecture"
      },
      {
        "label": "Formal Conjectures, the counterexample and the disproof in Lean",
        "url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Wikipedia/JacobianConjecture.lean"
      },
      {
        "label": "Terence Tao, A digestion of the Jacobian conjecture counterexample",
        "url": "https://terrytao.wordpress.com/2026/07/21/a-digestion-of-the-jacobian-conjecture-counterexample/"
      },
      {
        "label": "The Conversation, why a tiny social media post has mathematicians rethinking AI",
        "url": "https://theconversation.com/hello-there-the-jacobian-conjecture-is-false-thanx-why-a-tiny-social-media-post-has-mathematicians-rethinking-ai-283883"
      }
    ],
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      "formal record": "Formal Conjectures proves in Lean that the map's Jacobian determinant is −2 and that it identifies two distinct points, and from a variant with determinant 1 that the conjecture fails in general; the two-variable statement is still stated with sorry",
      "by hand": "The counterexample can be verified with any computer algebra system"
    },
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          "what": "A geometric reformulation of the counterexample, credited to GPT, announced the next day; Terence Tao developed it into an explanation through the factorisation of binary cubics.",
          "source": "https://en.wikipedia.org/wiki/Jacobian_conjecture"
        }
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          "topic": "Attribution",
          "summary": "Adrian Vasiu said the AI-generated paper describing the counterexample borrowed notation and concepts from a draft by Alexander Borisov, Ofer Gabber and Vasiu released in January 2025. Alpöge later said the three should receive credit for their foundational work.",
          "source": "https://en.wikipedia.org/wiki/Jacobian_conjecture"
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    "systems": [
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    "humans": [],
    "details": {},
    "date": "2026-06-29",
    "date_text": "29 Jun, 2026",
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    "autonomy": "A3",
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      "collection": "Erdős",
      "url": "https://www.erdosproblems.com/36",
      "page": "s/erdos-36",
      "prize": "no",
      "prize_value": 0,
      "tags": [
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        "additive combinatorics"
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  {
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      "GPT-5.5"
    ],
    "humans": [
      "Ákos Dúcz",
      "Dániel Varga"
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    ],
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    "fidelity": "F0",
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      "id": "erdos:1070",
      "key": "1070",
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      "collection": "Erdős",
      "url": "https://www.erdosproblems.com/1070",
      "page": "s/erdos-1070",
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      "tags": [
        "geometry"
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      "formal_status": "unclassified",
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  {
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    "statement": "erdos:731",
    "systems": [
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    ],
    "humans": [
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