{
 "schema": 1,
 "title": "The Proof Watch",
 "blurb": "Claimed solutions to famous open mathematics problems, tracked against the field's own acceptance ladder. A claim enters when it is announced and moves only when something verifiable happens to it. The desk does not referee mathematics; it records where each claim sits on the ladder its own institutions defined, and what is still unmet.",
 "note": "Every quote here is verbatim from the document at the URL beside it, read on the date in last_checked. Nothing on this page is a judgement about whether a proof is correct. A rung left unmet is not an accusation — most of these rungs take years by design.",
 "ladder": [
  {
   "key": "announced",
   "label": "Announced",
   "what": "A claim exists in public."
  },
  {
   "key": "artifact_public",
   "label": "Artifact public",
   "what": "The argument itself is readable by outsiders, not just described."
  },
  {
   "key": "machine_checked",
   "label": "Machine-checked",
   "what": "A proof assistant (Lean, Coq, Isabelle) compiles the formalisation. Checks the steps, not the statement."
  },
  {
   "key": "preprint",
   "label": "Preprint posted",
   "what": "Filed where the field archives and timestamps work."
  },
  {
   "key": "refereed",
   "label": "Refereed publication",
   "what": "Published in a refereed journal of world-wide repute."
  },
  {
   "key": "accepted",
   "label": "General acceptance",
   "what": "The relevant community treats it as settled."
  },
  {
   "key": "prize_eligible",
   "label": "Prize-eligible",
   "what": "For Millennium Problems only: two years elapsed since qualifying publication."
  }
 ],
 "authorities": [
  {
   "id": "clay-rules",
   "title": "Rules for the Millennium Prize Problems",
   "url": "https://www.claymath.org/millennium-problems/rules/",
   "last_checked": "2026-09-08",
   "spans": [
    "the proposed solution must be published in a Qualifying Outlet",
    "at least two years must have passed since publication",
    "the proposed solution must have received general acceptance in the global mathematics community"
   ]
  },
  {
   "id": "fefferman",
   "title": "Existence and Smoothness of the Navier-Stokes Equation (Charles L. Fefferman, official problem description)",
   "url": "https://www.claymath.org/wp-content/uploads/2022/06/navierstokes.pdf",
   "last_checked": "2026-09-08",
   "spans": [
    "we ask for a proof of one of the following four statements.",
    "Then there exist a smooth, divergence-free vector field u◦ (x) on R3 and a smooth f (x, t) on R3 × [0, ∞), satisfying (4), (5), for which there exist no solutions",
    "These problems are also open and very important for the Euler equations (ν = 0), although the Euler equation is not on the Clay Institute's list of prize problems."
   ]
  }
 ],
 "claims": [
  {
   "id": "openai-navier-stokes-2026",
   "problem": "Navier-Stokes existence and smoothness",
   "problem_class": "Clay Millennium Problem",
   "claimant": "OpenAI",
   "claimed_on": "2026-09-08",
   "claimed_precisely": "Finite-time blowup for the forced 3D Navier-Stokes equations - the breakdown direction, with a smooth forcing term.",
   "why_it_matters": "Fefferman's official statement asks for a proof of one of four statements. (C) and (D) are the breakdown cases and both explicitly admit a smooth forcing f, so a forced-blowup result is one of the four accepted routes and not a lesser substitute for it. What it is not is statement (A) or (B), the unforced existence questions most readers have in mind when they hear the problem described.",
   "artifacts": [
    {
     "label": "Announcement",
     "url": "https://openai.com/index/navier-stokes-solution/"
    },
    {
     "label": "Proof (PDF)",
     "url": "https://cdn.openai.com/pdf/32d9f210-8b73-45e0-91bc-82a30aef8a9a/navier-stokes.pdf"
    }
   ],
   "machine_involvement": "An unreleased internal model; OpenAI describes roughly 10,000 concurrent agents over about 88 hours from 1 September, with a further period spent formalising in Lean.",
   "ladder_status": {
    "announced": {
     "met": true,
     "on": "2026-09-08"
    },
    "artifact_public": {
     "met": true,
     "on": "2026-09-08",
     "note": "Manuscript and Lean formalisation released with the announcement."
    },
    "machine_checked": {
     "met": true,
     "note": "OpenAI states the proof is formalised in Lean. A Lean certificate checks that the steps follow, never that the statement formalised is the one the prize asks for."
    },
    "preprint": {
     "met": false,
     "note": "Posted on OpenAI's own domain rather than an archive of record."
    },
    "refereed": {
     "met": false
    },
    "accepted": {
     "met": false
    },
    "prize_eligible": {
     "met": false,
     "note": "Cannot be met before two years have passed since a qualifying publication, and the clock has not started."
    }
   },
   "disputes": [
    {
     "by": "Tristan Buckmaster (NYU)",
     "on": "2026-09-08",
     "url": "https://cims.nyu.edu/~tristanb/statement.pdf",
     "last_checked": "2026-09-08",
     "summary": "Buckmaster states he was told the result before it was announced, describes what he was offered, and raises a question about training data. He explicitly does not claim the proof is wrong.",
     "spans": [
      "the forcing function is smooth option c and d in Fefferman.",
      "I was told the proof is about 100 pages. I have not seen it.",
      "I asked whether the model had been trained on, or had access to, our sessions in Codex, into which we had been putting all our drafts for the whole of this project. I was told the model did not look up user data. I asked again, about training, and I did not get an answer.",
      "The reply was, \"Why would you ruin your career?\"",
      "I have not seen OpenAI's proof. I do not know what their model did, or how. I do not know whether our data was used. I am not accusing anyone of anything."
     ]
    }
   ],
   "last_checked": "2026-09-08"
  },
  {
   "id": "buckmaster-alpoge-euler-2026",
   "problem": "Finite-time blowup with smooth forcing: incompressible porous media, Boussinesq, 3D incompressible Euler",
   "problem_class": "Open problem, not a Clay prize problem",
   "claimant": "Tristan Buckmaster (NYU) and Levent Alpoge",
   "claimed_on": "2026-09-08",
   "claimed_precisely": "Three results made public together: finite-time blowup with smooth forcing for incompressible porous media, for Boussinesq, and for 3D incompressible Euler.",
   "why_it_matters": "Euler is not on the Clay list, by the official Navier-Stokes description's own words, so however strong these results are they are not a Millennium Prize solution. Buckmaster also assigns the underlying idea to others rather than to a machine.",
   "artifacts": [
    {
     "label": "Statement",
     "url": "https://cims.nyu.edu/~tristanb/statement.pdf"
    }
   ],
   "machine_involvement": "Buckmaster states several LLMs were used throughout - Anthropic's Claude and OpenAI's Codex - on a personal collaboration with no institutional agreement, and that a Lean verification was completed on 22 August.",
   "ladder_status": {
    "announced": {
     "met": true,
     "on": "2026-09-08"
    },
    "artifact_public": {
     "met": true,
     "on": "2026-09-08",
     "note": "Three papers released with the statement."
    },
    "machine_checked": {
     "met": true,
     "note": "Buckmaster states the proof was verified in Lean on 22 August."
    },
    "preprint": {
     "met": false
    },
    "refereed": {
     "met": false
    },
    "accepted": {
     "met": false
    },
    "prize_eligible": {
     "met": false,
     "note": "Not applicable - Euler is not a Clay prize problem."
    }
   },
   "disputes": [],
   "notes_spans": [
    "Today, Levent Alpöge and I have made public three results: finite-time blowup with smooth forcing for incompressible porous media, for Boussinesq, and for 3d incompressible Euler.",
    "The program this fits into was not started by us nor was it proposed by a Large Language Model.",
    "My work with Levent has been a purely personal collaboration, free of any institutional agreements or official involvement by either of our employers.",
    "The Euler writeup, in particular, can only be described as AI slop."
   ],
   "notes_url": "https://cims.nyu.edu/~tristanb/statement.pdf",
   "last_checked": "2026-09-08"
  },
  {
   "id": "perelman-poincare",
   "problem": "Poincare conjecture",
   "problem_class": "Clay Millennium Problem",
   "claimant": "Grigori Perelman",
   "claimed_on": "2002-11-11",
   "claimed_precisely": "Proof of the geometrisation conjecture, which implies the Poincare conjecture, posted in three preprints between November 2002 and July 2003.",
   "why_it_matters": "The only Millennium Problem resolved to the Clay Institute's satisfaction, and the calibration entry for this page: seven and a half years from preprint to prize, no machine involved, and the prize declined.",
   "artifacts": [
    {
     "label": "Preprint (2002)",
     "url": "https://arxiv.org/abs/math/0211159"
    },
    {
     "label": "Clay problem page",
     "url": "https://www.claymath.org/millennium/poincare-conjecture/"
    }
   ],
   "machine_involvement": "None.",
   "ladder_status": {
    "announced": {
     "met": true,
     "on": "2002-11-11"
    },
    "artifact_public": {
     "met": true,
     "on": "2002-11-11"
    },
    "machine_checked": {
     "met": false,
     "note": "Verified by human referees over several years, not by a proof assistant."
    },
    "preprint": {
     "met": true,
     "on": "2002-11-11"
    },
    "refereed": {
     "met": true,
     "note": "Expository verifications published by independent teams, 2006-2008."
    },
    "accepted": {
     "met": true,
     "on": "2006"
    },
    "prize_eligible": {
     "met": true,
     "on": "2010-03-18",
     "note": "Prize awarded 2010 and declined by Perelman."
    }
   },
   "disputes": [],
   "last_checked": "2026-09-08"
  }
 ],
 "url": "https://thestochasticparrot.com/proof-watch/",
 "license": "https://creativecommons.org/licenses/by/4.0/"
}