On September 8, 2026, OpenAI announced that an internal multi-agent system had resolved the existence and smoothness problem for the Navier-Stokes equations—one of the seven Millennium Prize Problems—and released a 166-page proof manuscript along with Lean formalization material. Two days later, 771 mathematics researchers jointly demanded that Caltech cancel an AI math hackathon; that evening, OpenAI announced its withdrawal as a sponsor. One day after that, 25 Fields Medalists published a manifesto whose wording was that "the goals of AI companies are seriously misaligned with the goals of the mathematical community." Within four days, mathematics went from being AI's scoreboard to the subject of a boundary negotiation between the AI industry and an entire discipline.
September 8: How the Proof Was Produced
According to OpenAI, the work began on September 1, when researchers heard rumors that "two more Millennium Problems had been cracked," and launched a dedicated effort.
The process ran in three stages. First, about 100 agents spent roughly 50 hours ruling out regularity for the simpler Euler equations. Then around 10,000 agents ran concurrently for 88 hours on the full Navier-Stokes system. Finally, a separate model spent 17 hours translating the result into Lean and machine-verifying it. Agents exchanged messages numbering in the millions, and executives quoted a compute bill in the millions of dollars.
The conclusion targets Propositions C and D: under certain smooth external forcing conditions, three-dimensional 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 declared it would not claim the million-dollar prize set by the Clay Mathematics Institute. The weight of that step depends on Clay's rules: a Millennium Problem solution must be published in a qualifying journal, remain in print for at least two years, and achieve general acceptance by the mathematical community. The Clay Institute still lists the problem as unsolved; president Martin Bridson told Nature it was "undoubtedly an exciting day for human understanding of mathematics."
Priority and Data-Provenance Disputes
On September 7, Tristan Buckmaster of NYU's Courant Institute published 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 the progress had reached OpenAI, and spoke with OpenAI researchers on September 6.
Buckmaster's two challenges, recorded in Fortune's reporting: whether OpenAI had read his unpublished drafts and conversations with the models; and his claim that OpenAI researcher Sébastien Bubeck pushed to remove Alpöge from the author list because he worked for a competitor, replying "Why would you ruin your career?" when Buckmaster threatened to publish correspondence. Bubeck denied these accounts as false and inflammatory; OpenAI said the two proofs differ substantially.
On data provenance, OpenAI's position: it had not seen the two researchers' work before the announcement and had not accessed user data, while noting it could not fully rule out that de-identified tool-use traces helped model optimization.
The European Mathematical Society (EMS) statement in September offered a third-party view: the solution closely follows the strategy proposed by Córdoba, Martínez-Zoroa, and Fan Zheng (references 6–8 in OpenAI's paper), and builds on recent progress by Alpöge and Buckmaster—OpenAI is unquestionably standing on the shoulders of giants. The statement also flagged another issue: the models used are OpenAI-internal and inaccessible to outsiders, which under the spirit of open science and equal opportunity is something mathematics must address. It nonetheless closed with "admiration and celebration."
September 10: The 771-Signer Letter Targeted the Event, Not the Proof
Caltech's Mathathon was scheduled to begin October 30, in two rounds. Round one: 40 hours, 100 teams using large-model prompts to attack open research problems, each team with $20,000 in tokens. Teams judged "promising and clearly explained" would enter a six-month verification round with more compute. OpenAI and Anthropic together contributed roughly $2 million in AI compute credits. Most organizers were undergraduates; over a thousand applications were received.
The open letter was led by current and former Caltech mathematicians, signed by 771 people at publication; signatories had to be Caltech community members or PhD-level researchers. It made five points:
| Concern | Specific reasoning in the letter | |---|---| | Manufacturing low-quality mathematics | Undergraduates may lack verification ability; rules do not require arXiv posting; verification is left to the community | | Corporate interests intruding | Such events give AI companies more financial and social leverage; company goals may not align with research goals | | Time conditions | Forty hours is insufficient to deeply understand, solve, and articulate a problem | | Misleading narrative | The event website asked "when AI can solve conjectures faster, what is the mathematician's role"—a claim the letter called both motivated and lacking evidence | | Young researchers' reputations | Close ties to the two companies could affect participants' future reputations |
The letter's hardest line: "bluntly, AI companies are engaging in research misconduct," citing Scientific American's report on OpenAI's August batch of ten AI math results, in which experts alleged prior work was not properly cited. It closed by demanding the organizers "abandon this harmful endeavor entirely."
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 "the concerns raised by members of the mathematics community," and expressed a desire for more dialogue. At press time, Anthropic had not publicly responded.
The organizers replied the same day, acknowledging "several of these concerns are valid," and, after consulting faculty advisors, made arXiv posting a hard requirement of the verification round, extended it to six months with required preprints, video presentations, and automated formalizations, and added an education track. They also rebutted parts of the 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 mathematics community's standards."
A Green Checkmark Is Not Understanding
The value of proof assistants like Lean is that they check the statement as encoded: machine-readable definitions, assumptions, and derivations. They can catch notational errors and unstated conditions human readers might miss.
What they do not judge is equally clear: whether the statement captures the mathematical idea people actually meant, how the result connects to the rest of the field, and whether the proof has been written in a form students can learn from. These three points are exactly the dividing line between a result that becomes a tool and one that becomes 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, why that construction was chosen, and where the same trick can be used next. The manifesto's line about "no time to isolate new methods and ideas" points precisely at this layer.
September 11: The Manifesto of 25 Fields Medalists
The next day, mathandai.org posted 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 noted 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 have always been proxy metrics; the real goals are 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 how results are released. The manifesto states that these solutions are often announced hastily, with no time for full write-up, no time to isolate new methods and ideas, and no time to cite others' relevant prior work; as in all creative professions, this raises serious attribution and plagiarism problems.
The manifesto's only call to action is that these issues "must be urgently addressed by the mathematical community, by these companies, and by society at large."
Who Pays the Verification Bill
The documents from these two weeks point to one structural problem: the speed of producing proofs and the speed of verifying them no longer match.
A result can be announced over a weekend; verifying a new proof takes experts weeks or months. The 771-signer letter put the gap bluntly: AI companies treat frontier mathematical research as an advertising vehicle, competing for the publicity title of "first to prove a major conjecture," while rigorous verification, dissemination, and even refutation are left entirely to human mathematicians—labor with "no pay, no attribution, no recognition."
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 a non-renewable resource.
The Leiden Manifesto on AI and Mathematics, released in June and endorsed by the International Mathematical Union, already called for safeguards on openness, attribution, and independence from commercial interests. This conflict did not begin with the Mathathon; the event simply entered a sponsor category already under value contestation.
What Remains Unverified
OpenAI's Navier-Stokes result has not passed any step of independent verification. Clay's recognition process—publication, two-year wait, general acceptance—has not begun. Buckmaster and Alpöge's work is likewise under verification.
Other threads are developing in parallel. Per AI Weekly, group theorist Andreas Thom of TU Dresden publicly questioned whether his private ChatGPT conversations had fed into an OpenAI result on non-sofic groups.
Three indicators to watch: whether Anthropic follows OpenAI out of 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 executable attribution-and-verification standard for "machine-produced + human-endorsed" results. The EMS has already framed inaccessible internal models as an open-science problem—the most actionable item in the short term: by what standard should academia accept results produced by private models?
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 for 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 the Córdoba, Martínez-Zoroa, and Fan Zheng strategy; the open-science problem of inaccessible internal models. 4. "Open Letter about the Mathathon" published by Proofs and Prompts (2026-09-10, 771 signatures): five concerns, the $20,000 credit, demand to cancel. 5. mathandai.org manifesto (2026-09-11) and Tao's blog repost: 25 Fields Medalist initial signatories, 800+ verified endorsements, manifesto text.