After 25 Fields Medalists Signed On: The Math Community's Fight Is Over Who Verifies
On September 8, 2026, OpenAI announced that an internal multi-agent system had solved the existence and smoothness problem for the Navier–Stokes equations — one of the seven Millennium Prize Problems — releasing a 166-page proof manuscript along with Lean formalization materials. Two days later, 771 mathematics researchers jointly demanded that Caltech cancel an AI math hackathon; that same evening OpenAI announced it was withdrawing sponsorship. One day after that, 25 Fields Medalists published a manifesto declaring that "the goals of AI companies are severely misaligned with the goals of the mathematical community." Within four days, mathematics went from being AI's scoreboard to the site of a boundary negotiation between the AI industry and an entire discipline.
September 8: How That Proof Was Produced
According to OpenAI, the work began on September 1, after researchers heard rumors that "two other Millennium Problems had been cracked," prompting them to launch this project.
The process had three stages. First, about 100 agents spent roughly 50 hours ruling out regularity for the simpler Euler equations. Then about 10,000 agents ran concurrently for 88 hours attacking the full Navier–Stokes system. Finally, a separate model spent 17 hours translating the result into Lean and performing machine verification. Agents exchanged millions of messages over the project; executives cited a compute bill in the millions of dollars.
The conclusion points to Propositions C and D: under certain smooth forcing conditions, 3D fluids blow up in finite time — meaning no global smooth solutions exist. Nature's paraphrase: initially smooth 3D fluids develop singularities, some with unbounded acceleration.
OpenAI simultaneously stated it would not claim the Clay Mathematics Institute's million-dollar prize. The weight of this step depends on Clay's rules: a Millennium Problem solution is recognized only after publication in a qualifying journal, remaining in print for at least two years, and gaining general acceptance by the mathematical community. Clay still lists the problem as unsolved; president Martin Bridson told Nature it was "certainly an exciting day for human understanding of mathematics."
The Priority and Data-Source Dispute
On September 7, Tristan Buckmaster of NYU's Courant Institute issued a statement first. He and mathematician Levent Alpöge (Anthropic) had been advancing a route pioneered by Diego Córdoba (ICMAT Madrid) and Luis Martínez-Zoroa (CUNEF) using multiple models from both companies, with progress accelerating from mid-August. Buckmaster said he learned on September 3 that news of this progress had reached OpenAI, and spoke by phone with OpenAI researchers on September 6.
Buckmaster's two objections, recorded in Fortune's reporting: first, whether OpenAI had read his unpublished drafts and his model conversation logs; second, his claim that OpenAI researcher Sébastien Bubeck pushed to remove Alpöge from the author list for working with a competitor, and responded "Why would you ruin your career?" when Buckmaster threatened to make correspondence public. Bubeck denies these claims as false and inflammatory; OpenAI says the two proofs differ significantly.
On data sources, OpenAI's position: they had not seen these two researchers' work before the announcement and had not accessed user data, while acknowledging they cannot fully rule out that de-identified tool-use traces aided model optimization.
The European Mathematical Society (EMS) statement in September offered a third-party view: the solution closely follows the strategy of Córdoba, Martínez-Zoroa, and Fan Zheng (references 6–8 in OpenAI's paper), and builds on recent advances by Alpöge and Buckmaster — that OpenAI stands on the shoulders of giants is beyond doubt. The statement also flagged another issue: the models used are internal OpenAI models unavailable to outsiders, which — in the spirit of open science and equal opportunity — is something mathematics must address. It closes with "admiration and celebration."
September 10: The 771-Signer Letter Targets the Event, Not the Proof
Caltech's Mathathon was scheduled to begin October 30 in two rounds. Round one: 40 hours, 100 teams using LLM prompts to attack open research problems, with $20,000 in tokens per team. Teams judged "promising and clearly explained" would enter a six-month verification round with more compute. OpenAI and Anthropic together provided roughly $2 million in AI compute credits. Organizers were mostly undergraduates; over a thousand applications were received.
The open letter, led by current and former Caltech mathematicians and signed by 771 people at release (signatories had to be Caltech community members or PhD-holding researchers), raised five points:
| Concern | Specific reasoning in the letter | |---|---| | Manufacturing low-quality mathematics | Undergraduates may lack verification ability; rules don't require arXiv posting; verification is left to the community | | Corporate interests intruding | Such events give AI companies more financial and social leverage; corporate and research goals may diverge | | Time conditions | Forty hours is insufficient to deeply understand, solve, and communicate a problem | | Misleading narrative | The event site asked "when AI can solve conjectures faster, what is the mathematician's role" — a claim the letter calls both motivated and empirically unsupported | | Young researchers' reputations | Close association with the two companies could affect participants' future reputations |
The letter's hardest line: "Bluntly, AI companies are engaging in research misconduct," citing a Scientific American report on OpenAI's batch of ten AI math results in August, in which experts alleged inadequate citation of prior work. It closes by demanding organizers "abandon this harmful endeavor outright."
The $20,000 figure served as a fairness argument: no reasonable future model of mathematical research exists in which every research mathematician gets $20,000 in AI credits to prove a result.
That evening, OpenAI research lead Dan Roberts announced withdrawal from sponsorship, citing "concerns raised by members of the mathematics community" and expressing hope for more dialogue. Anthropic had not publicly responded as of writing.
The organizers replied the same day, acknowledging "several of these concerns are valid." After discussion with faculty advisors, they made arXiv posting a hard requirement of the verification round, extended it to six months, required preprints, video presentations, and automated formalized solutions, and added an education track. They also pushed back on some framing: the main organizers are undergraduates mostly planning careers in theoretical math, and most applicants are PhD students with publication records. Their most candid line: "We cannot guarantee that all work produced by this event will meet the mathematical community's standards."
A Green Checkmark Is Not Understanding
The value of proof assistants like Lean lies in checking the encoded statement: a machine-readable string of definitions, assumptions, and derivations. It blocks notational errors and unstated conditions that human readers might miss.
What it does not judge is equally clear: whether the statement captures the mathematical idea people actually care about, how the result connects to the rest of the field, and whether the proof is written in a form students can learn from. These three things mark the divide between "a result becoming a tool" and "a result becoming a sealed box."
A 40,000-line formalized proof can be entirely correct. The next mathematician still needs to know which lemma is the key one, why that construction was chosen, and where the same trick applies next. The manifesto's line about having "no time to isolate new methods and ideas" points at exactly this layer.
September 11: The Manifesto of 25 Fields Medalists
The next day, mathandai.org published a manifesto with 25 Fields Medalists as initial signatories, including Terence Tao, Peter Scholze, Maryna Viazovska, James Maynard, and Maxim Kontsevich. By the evening of September 11, over 800 verified endorsements (via ORCID or academic email) had accumulated. Tao reposted it on his blog, saying he was "proud to be one of the 25 initial signatories" and noting the manifesto grew out of discussions over the past week — he was not an author.
The manifesto names no company, model, or specific result. Its argument has two layers.
The first is about goals. Famous problems are only proxy indicators; the real goal is conceptual understanding and insight. Rapidly mass-producing true-or-false results "may destroy fertile soil rather than breathe life into new ideas."
The second is about publication practices. The manifesto states that these solutions are often announced in haste, without time for full write-up, without time to isolate new methods and ideas, without time to credit others' related prior work — and as in all creative professions, this creates serious attribution and plagiarism problems.
The manifesto's single call to action: these issues "must be urgently addressed by the mathematical community, by these companies, and by society at large."
Who Pays the Bill for Verification Labor
The documents from these two weeks point to the same structural problem: the speed of producing proofs no longer matches the speed of verifying them.
A result can be announced in a weekend; verifying a new proof takes experts weeks or months. The 771-signer letter put the gap precisely: AI companies treat frontier mathematical research as an advertising vehicle, competing for the publicity title of "first to prove a major conjecture," while the work of rigorous verification, dissemination, and even refutation is left entirely to human mathematicians — labor that is "unpaid, uncredited, and unacknowledged."
Terence Tao told The New York Times that these problems are "lighthouses" that concentrate human effort; letting AI solve them may hollow out mathematicians' understanding of their own field. Elsewhere he described good problems as non-renewable resources.
The Leiden Manifesto on AI and Mathematics, released in June and already endorsed by the International Mathematical Union, calls for safeguards on openness, attribution, and independence from commercial interests. This conflict didn't start with the Mathathon; the event merely entered a sponsorship category already mired in a values debate.
What Remains Unverified
OpenAI's Navier–Stokes result has not completed any step of independent verification. Clay's recognition process — publication, a two-year wait, and general acceptance — hasn't even begun. Buckmaster and Alpöge's results are likewise under verification.
Other threads are developing concurrently. Per AI Weekly, group theorist Andreas Thom of TU Dresden has publicly questioned whether his private ChatGPT conversations fed into an OpenAI result on non-sofic groups.
Three indicators to watch: whether Anthropic follows OpenAI in withdrawing sponsorship; whether the Mathathon actually implements mandatory arXiv posting and the six-month verification round before October 30; and whether the mathematical community produces an enforceable "machine-generated + human-endorsed" attribution and verification standard. The EMS has already framed "inaccessible models" as an open-science problem — the most actionable item in the short term: by what standard should academia accept results produced by internal models no one else can access?
Sources
1. VnExpress International, "AI agent swarm may have cracked million-dollar math problem open since 1934 in 88 hours" (2026-09-09): 100 agents/50 hours, 10,000 agents/88 hours, 17 hours Lean; Clay rules and Bridson's remarks. 2. Fortune, "OpenAI says it cracked Navier-Stokes..." (2026-09-08): Buckmaster and Alpöge's technical route, the September 3 and 6 timestamps, both sides' accounts. 3. European Mathematical Society, "EMS statement on recent Navier–Stokes announcement": citations of Córdoba, Martínez-Zoroa, Fan Zheng strategies; the open-science issue of inaccessible internal models. 4. Proofs and Prompts, "Open Letter about the Mathathon" (2026-09-10, 771 signatures): five concerns, the $20,000 credit, demand to cancel. 5. mathandai.org manifesto (2026-09-11) and Terence Tao's blog repost: 25 Fields Medalist initial signatories, 800+ verified endorsements, manifesto text.