[
  {
    "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.html",
      "prize": null,
      "prize_value": 0,
      "tags": [],
      "formal_status": "mixed",
      "has_formal_statement": true,
      "artifact_state": "unreachable",
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      "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."
      }
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    "statement": "millennium:NavierStokes",
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      "id": "millennium:NavierStokes",
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      "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",
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    "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.",
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        "label": "Anthropic, Formalizing Fermat's Last Theorem",
        "url": "https://www.anthropic.com/research/formalizing-fermats-last-theorem"
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        "url": "https://github.com/anthropics/fermats-last-theorem"
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      "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"
    },
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    "statement": "mathlib:FermatLastTheorem",
    "date_text": "2026-09-04",
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    "checked_by": [],
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    "fidelity": "F2",
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      "id": "mathlib:FermatLastTheorem",
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      "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.html",
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      "formal_status": "unclassified",
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  },
  {
    "systems": [
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    ],
    "humans": [],
    "date": "2026-08-10",
    "outcome": "record",
    "outcome_text": "Raised the proven proportion from 41.6% to 67.2%, and 5/6 of zeros shown distinct.",
    "autonomy": "A3",
    "autonomy_reason": "The prompt was to attempt the Riemann hypothesis; the mathematical choices were the model's, and the accompanying formalization.yaml records the work as autonomous. Humans validated the result rather than contributing to it.",
    "lean_claimed": true,
    "sources": [
      {
        "label": "Anthropic, Learning more about Claude's mathematical capabilities",
        "url": "https://www.anthropic.com/research/riemann-zeta"
      },
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        "label": "Lean formalisation",
        "url": "https://github.com/anthropics/formal-math/tree/main/zeta23"
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      "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",
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    "date_text": "2026-08-10",
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    "autonomy_source": "declared autonomous in the project's formalization.yaml",
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      "id": "zeta:critical-line-proportion",
      "key": "critical-line-proportion",
      "label": "Proportion of zeta zeros proved simple and on the critical line",
      "title": "Proportion of zeta zeros proved simple and on the critical line",
      "collection": "Analytic number theory",
      "url": "https://www.anthropic.com/research/riemann-zeta",
      "page": "t/zeta-critical-line-proportion.html",
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        "Riemann zeta function"
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        "kind": "ai",
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        "note": "Found while attempting the Riemann hypothesis itself, by combining work of Aryan and of Baluyot, Goldston, Suriajaya and Turnage-Butterbaugh with a 2000 paper of Bombieri.",
        "source": "https://arxiv.org/abs/2608.13637",
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