Topicsai-in-mathematics

AI in Mathematics: A Dispute in Real Time

A sourced timeline of the September 2026 Navier–Stokes announcement and the attribution dispute around it — built entirely from what the parties published themselves, with the prize's own rules for what counts as solved, and nothing adjudicated.

Assertions
14
Sources consulted
7
Read in full
6/7
Cited as evidence
6
Disputed
1

7 sources sit behind this page — including any that arrive with a concept this page shares with another collection. 6 were retrieved and read in full, and only those can back an assertion. 1 were surfaced and deliberately set aside. Every one of them is named in the register below, with the reason in view. How we source this.

Background

Our own synthesis, written to orient you — not evidence. Every factual statement here is asserted and sourced further down this page.

In September 2026 an AI company said it had resolved one of the seven Millennium Prize Problems, and within three days twenty-five Fields Medallists had signed a declaration saying the way such results are being produced and announced is damaging mathematics. This collection is a timeline of that week, assembled from the documents the participants published under their own names.

What OpenAI states it did: trained a new internal model from 28 August; heard rumours on 1 September that two Millennium Prize problems had been resolved; launched coordinating groups of agents against all the open problems; resolved the Euler regularity question with nearly 100 agents over about 50 hours; then reached a Navier–Stokes result on 5 September with a group on the order of 10,000 concurrent agents, about 88 hours after launch, followed by 17 hours of Lean formalisation. Across everything attempted, 4.9 million messages and about 300 billion output tokens. The company states it does not intend to claim the prize.

What Tristan Buckmaster states, in a signed statement published about twelve hours earlier: that he and Levent Alpöge had spent most of a year on a program that was not theirs and was not proposed by a model — the credit for the idea belongs to Diego Córdoba and Luis Martínez-Zoroa — obtaining blowup under smooth forcing on 15 August and verifying it in Lean on 22 August. He describes calls on 6 September in which, he says, he was first told the model had simply been given the problem statement, and it then emerged that a team had been working on it and that even the prompt he was shown had been written by prompting Codex. He states he was told 'Why would you ruin your career?' when he said he would go public. He is also explicit about what he is not claiming: he has not seen the proof, does not know what the model did, and is not accusing anyone of anything.

The two accounts are both on this page and neither is resolved here. This index has read two documents; it has not read the proofs, the prompts, the chat logs or the calls, and it has no way to determine which account of a private conversation is accurate. Recording the conflict precisely, with each side's words attributed to the document that contains them, is the most this format can honestly do — and it is more than a summary that picks a side would do.

On whether the problem is 'solved': the Clay Mathematics Institute's own rules, adopted in 2018, require publication in a qualifying outlet, at least two years elapsed since publication, and general acceptance in the global mathematics community, before the Institute will even consider a proposed solution. By that standard nothing announced in September 2026 can be settled before September 2028, whatever the mathematics turns out to say. That is a procedural fact, not an opinion about the proof.

One more thing belongs here, five weeks earlier and involving different people: a group theorist, Andreas Thom, describing finding his own joint work inside a crucial proposition of an OpenAI paper and the public announcement describing a decade of 'no progress'. Two episodes do not make a pattern, but they make the declaration's complaint about citation legible as something other than an abstraction.

Figures

Every number below is asserted and sourced elsewhere on this page.

Eleven days

Each date is stated by one of the two parties in its own published account. The two accounts agree on this calendar; they disagree about what was said on 6 September.

Day of September 2026

1 Sep

OpenAI hears rumours, launches agents

5 Sep

Agents reach a Navier–Stokes resolution

6 Sep

Lean verification; OpenAI makes contact

8 Sep

Buckmaster statement, then OpenAI's post

10 Sep

OpenAI publishes investigation findings

11 Sep

25 Fields Medallists sign a declaration

Dates from OpenAI's post of 8 September 2026 and from Tristan Buckmaster's signed statement of the same day. Positions on the axis are the day of the month in September 2026, with the two August dates omitted from the plot for scale and given in the text.

Concepts

The vocabulary this subject is built from, and what we can show about each.

Running a Proof Attempt as a Compute Deployment

process

OpenAI states it ran coordinating agent groups — on the order of 10,000 concurrent agents for the group that produced the Navier–Stokes result, nearly 100 over approximately 50 hours for the Euler disproof — reaching a resolution on 5 September about 88 hours after launch plus 17 hours of Lean formalisation, using 4.9 million messages and about 300 billion output tokens across all problems and 2.7 million messages and approximately 130 billion output tokens on Navier–Stokes.

ReportedSupported by the sources below, not yet editor-reviewed.
1 source1 retrieved & read

The Attribution Dispute

other

Buckmaster states the program was not started by him and Alpöge and was not proposed by a language model: credit for the basic idea belongs to Diego Córdoba and Luis Martínez-Zoroa, who spent several years constructing forced blowups, and he states that Martínez-Zoroa 'deserves a Fields Medal' and that the smooth-force route 'is not the direction one arrives at in a few days by giving a model the problem statement'.

ReportedSupported by the sources below, not yet editor-reviewed.
1 source1 retrieved & read

Buckmaster states that progress was slow for most of a year before blowup results under smooth forcing for Boussinesq and Euler were obtained on 15 August and verified in Lean on 22 August, that the first model-generated proof he received 'was the most horrendous I have ever read', and that the Euler writeup 'can only be described as AI slop'.

ReportedSupported by the sources below, not yet editor-reviewed.
1 source1 retrieved & read

OpenAI states it contacted Alpöge and Buckmaster on 6 September to offer a concurrent release and recognise their priority on forced Euler, and that neither its researchers nor its agents saw their work before public release; Buckmaster states he was initially told the model had simply been given the problem statement, that it emerged on the call that a team had been working on it and that the prompt he was shown had itself been written by prompting Codex, and that he was told 'Why would you ruin your career?' when he said he would go public.

DisputedSources disagree. Both accounts are shown below.
2 sources2 retrieved & read

The Fields Medallists' Declaration

other

The declaration concedes that language models can now solve major outstanding problems and objects to using those problems as a benchmark: it warns that mass production of 'true/false' statements at speed 'could destroy fertile ground instead of breathing life into new ideas', that rushed announcements leave no time for proper writeups or for citing previous work and so raise 'severe attribution and plagiarism questions', and that the outcome will be determined by the decisions of the humans controlling the technology.

ReportedSupported by the sources below, not yet editor-reviewed.
1 source1 retrieved & read

A declaration titled 'A Severe Misalignment of AI in Mathematics' was published on 11 September 2026 with 25 initial signatories, all Fields Medallists, spanning awards from 1978 to 2026; its poster states it grew out of a week of discussions and was released without the more consultative process its authors would have preferred.

ReportedSupported by the sources below, not yet editor-reviewed.
1 source1 retrieved & read

What Counts as Solving a Millennium Prize Problem

other

The Clay Mathematics Institute's rules, adopted 26 September 2018, require all three of publication in a Qualifying Outlet, at least two years elapsed since publication, and general acceptance in the global mathematics community before it will consider a proposed solution — and OpenAI states it does not intend to claim the prize for its result.

ReportedSupported by the sources below, not yet editor-reviewed.
2 sources2 retrieved & read

Timeline

What actually happened, in order, with sources.

Coverage
  • 5 United States
  • 1 Germany
  • 1 United Kingdom

Where this topic’s events took place, as far as our sources establish it. Events with no single location — a standards publication, say — and events we have not yet attributed are both counted as unattributed rather than omitted.

  1. Sep 11, 2026

    Twenty-Five Fields Medallists Sign a Declaration

    otherUnited States

    On 11 September 2026 twenty-five Fields Medallists published 'A Severe Misalignment of AI in Mathematics', conceding that language models can now solve major open problems while arguing that using such problems as a company benchmark damages mathematics, leaves no room for proper writeups or citation of previous work, and raises severe attribution and plagiarism questions.

    ReportedSupported by the sources below, not yet editor-reviewed.
    1 source1 retrieved & read
  2. Sep 10, 2026

    OpenAI Investigates Itself and Publishes the Result

    otherUnited States

    On 10 September 2026 OpenAI updated its post to state that an investigation confirmed Buckmaster's Codex prompts over the preceding two months could not have influenced the system in any way including through training, and that the two sides' Euler proofs differ in that theirs was forced and OpenAI's unforced.

    ReportedSupported by the sources below, not yet editor-reviewed.
    1 source1 retrieved & read
  3. Sep 8, 2026

    OpenAI Claims a Millennium Prize Problem

    otherUnited States

    OpenAI published on 8 September 2026 that an internal system produced an analytical proof and Lean formalisation of finite-time singularity formation in Navier–Stokes under smooth forcing with finite energy, which it says resolves the problem by establishing statements 'C' and 'D' of the official formulation, while stating it does not intend to claim the Millennium Prize.

    ReportedSupported by the sources below, not yet editor-reviewed.
    3 sources3 retrieved & read
  4. Aug 22, 2026

    Two Mathematicians Verify a Machine Proof

    otherUnited States

    Buckmaster states that he and Levent Alpöge obtained blowup results with smooth forcing for Boussinesq and incompressible Euler on 15 August and verified the proof in Lean on 22 August, using Anthropic's Claude and OpenAI's Codex in a personal collaboration funded from his own research funds.

    ReportedSupported by the sources below, not yet editor-reviewed.
    1 source1 retrieved & read
  5. Aug 1, 2026

    A Group Theorist Finds His Own Work Inside the Proof

    otherGermany

    Andreas Thom states that a solution to the existence of a non-sofic group circulated on 31 July 2026, that his joint work with Gábor Kun played a decisive role in a crucial proposition of the OpenAI paper, and that colleagues found the public announcement's framing misleading in describing 'no progress' in the last decade while relying on their 2019 paper.

    ReportedSupported by the sources below, not yet editor-reviewed.
    1 source1 retrieved & read
  6. Jul 25, 2024

    A Machine Reaches Silver-Medal Standard at the Olympiad

    otherUnited Kingdom

    Google DeepMind stated that AlphaProof and AlphaGeometry 2 solved four of six problems at the 2024 International Mathematical Olympiad for 28 of 42 points — silver-medal standard against a gold threshold of 29 reached by 58 of 609 contestants — with the problems manually translated into formal language first, and with the systems taking up to three days where contestants have two sessions of 4.5 hours.

    ReportedSupported by the sources below, not yet editor-reviewed.
    1 source1 retrieved & read
  7. Sep 26, 2018

    The Prize Fixes Its Own Standard of Proof

    otherUnited States

    On 26 September 2018 the Clay Mathematics Institute's Board adopted revised Millennium Prize rules requiring publication in a Qualifying Outlet, at least two years since publication, and general acceptance in the global mathematics community before a proposed solution is considered.

    ReportedSupported by the sources below, not yet editor-reviewed.
    1 source1 retrieved & read

Source register

All 7 sources behind this page — what we read, what we tried to read and could not, and what we looked at and set aside, with the reason in view for each. A concept shared with another collection brings its own references with it, so some entries here were surfaced for a neighbouring topic rather than this one.

Cited as evidence
6
Tried, could not read
0
Surfaced, set aside
1
Cited sources 6 distinct links

Original publisher links. Files open on the publisher’s site; we do not host copies. A linked document is not an additional source or an independent verification.

Surfaced, set aside1

These came up while researching and were deliberately not used. We do not claim to have read them — each is listed with why it was passed over, so the shape of the survey is visible and not just its conclusions.

Coverage & limits

What this page does and does not claim.

Twenty-eighth packet through the generic ingestion pipeline, researched 12 September 2026, four days after the events at its centre. It is deliberately built from party-authored documents: OpenAI's own post, Buckmaster's own signed statement, the declaration on the page its signatories published it on, and the Clay Institute's own rules. No assertion rests on a news report of what someone said, though news coverage was used to locate those documents. The central dispute is recorded and not adjudicated: where OpenAI and Buckmaster describe the same calls differently, both accounts appear with the speaker named, and this collection states plainly that it has read two documents rather than the proofs, prompts, logs or calls. Three known gaps. Fefferman's official problem statement, which defines the 'C' and 'D' labels both parties use, was not retrieved, so those labels are reported only as the parties describe them. Neither the OpenAI proof nor the Alpöge–Buckmaster papers were read, and no assertion here characterises whether any proof is correct. And no compute cost appears: third-party estimates circulated widely but OpenAI publishes agent, message and token counts without a cost, and no retrievable source establishes one. The Andreas Thom guest post is cited at the blog front page on which it was read rather than at its permanent URL, which will drift. This subject also sits outside the index's three lanes and was built at the founder's request as a test of the format on a live, contested story. Not yet editor-reviewed; every assertion reads as reported, and every quotation is attributed to the document that contains it.

Source check, 2026-09-17. Numeric-presence checks passed for 14 assertions using available source text, which may be cached. This is not verification of their meaning. What this check does and does not prove →

  • Not editor-reviewed unless labelled. Assertions marked Reported are assembled from the sources shown and have not yet been checked by an editor. Only Primary source and Corroborated mean a human verified them.
  • Disagreements are preserved, not resolved. Where sources conflict, both accounts appear and the assertion is marked Disputed.
  • Retrieval status is disclosed per source. A source we could not open is never counted as evidence for an assertion.

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