“I think the game is up”: what the Navier-Stokes fight is actually about
The mathematician at the centre of it is not mainly claiming he was robbed. He is saying that racing to publish is now pointless, and the damage that follows from that is bigger than one proof.
Tristan Buckmaster shared the 2019 Clay Research Award for work on exactly these equations, and spent about a year with Levent Alpöge on the route that OpenAI's agents took in four days. Asked by the ABC on 11 September what he would do next, he did not say he had been robbed. He said: “I think it's pointless. Like, I think the game is up.”
The allegation underneath the headlines is narrower and more interesting than theft. He says OpenAI began only after word of their progress reached it, that it took the same unusual route almost nobody else was working on, and that he cannot rule out his own private Codex sessions having played some part. OpenAI denies any access to their work and says an investigation found his prompts could not have influenced the system.
One piece of the story went almost unreported outside a few outlets: Sébastien Bubeck, the OpenAI researcher at the centre of the authorship row, denied asking for Alpöge's removal, published the messages in which he offered the pair priority, and apologised for the remark about ruining a career. The accounts still do not reconcile on the central point.
Who said what, what is agreed, and what the fight is really costing
1. The claim is not that a proof was copied
Buckmaster's public account is that OpenAI launched its effort after information about their progress reached it, ran an enormous amount of compute, and arrived at a proof by a very similar technical route. His words: “It turned out that an entire team had been working on the problem and an insane amount of compute had been used.” That is an allegation about timing, provenance and scale, not about a stolen manuscript, and it is worth stating precisely because the stronger version is the one that travelled.
2. The evidence he points at is the route, not the result
“The route to the Clay problem through a smooth force is the route Luis and Diego opened and the one Levent and I had quietly chosen to attack. Almost nobody else I know of was working on it.” Luis and Diego are Luis Martínez-Zoroa and Diego Córdoba, whose programme both teams built on. His point is that this was not an obvious path a model would find in days from the problem statement alone. It is a circumstantial argument, he presents it as one, and it is the strongest thing he has.
3. What OpenAI denies, and the one thing it has not been able to rule out
The announcement states that neither the researchers nor the agents saw any of their work before it was public, and that no specific user data was accessed. A dated footnote records that the section was rewritten on 10 September with findings from an investigation, which now says Buckmaster's Codex prompts in the two preceding months could not have influenced the system in any way, including through training. Earlier in the week the company's formulation was softer, acknowledging that de-identified data derived from product use helps improve its models. Both statements can be true at once, and the distance between them is why the question stayed alive for a week.
4. Bubeck denied the authorship claim, and also apologised
He called the allegations false and inflammatory on 8 September, then posted a fuller account: “I never ever asked to remove Alpöge from authorship of his own work.” He confirmed a circulating screenshot of his message to Alpöge as genuine, in which he offered a coordinated release and wrote that all the academic accolades for the result should go to the two of them. On the remark about ruining a career he did not deny saying it: “I deeply apologize for this extremely poor choice of words”, adding that he retracted it on the spot. Most summaries of this story, including the French write-up this article was commissioned from, record the denial and omit the apology.
5. Buckmaster did not accept the correction
His account is that Bubeck twice asserted he wanted Alpöge removed, and that the career remark was a threat rather than a poor choice of words. Sam Altman endorsed his researcher within about half an hour, writing that the team acted with integrity and generosity throughout. These were private conversations, nobody outside them can verify what was said, and we are not going to pretend otherwise.
6. What nobody disputes is the trigger
OpenAI's own page says the effort began on 1 September after hearing a rumour that two Millennium problems had been resolved, which it later realised concerned Alpöge and Buckmaster. That single sentence concedes the part of the story that matters most for everyone else: the machine was aimed at this problem because humans were known to be close to it. Every argument about what researchers should share in future runs through that sentence.
7. The forced and unforced detail that most coverage flattened
The two efforts are not the same result. Alpöge and Buckmaster proved finite-time blowup with smooth forcing for the porous medium equation, the two-dimensional Boussinesq system and three-dimensional Euler. OpenAI's agents proved the unforced Euler case and then the forced Navier-Stokes one, and its page recognises the pair's priority on forced Euler. The French account that prompted this piece describes their Euler breakthrough without the forcing distinction, which is the distinction the entire priority question turns on.
8. The objection from mathematicians is about the write-up, not the Lean
Javier Gómez-Serrano of Brown, a Navier-Stokes specialist, called the write-up incomprehensible in its current form, and put the real point plainly: “What I don't think the mathematical community will accept is the write-up as it is.” That is a complaint about 166 pages of machine-written exposition, not about the formal proof, which is the one artifact designed to be checked mechanically. The two are being reported as one objection and they are opposites: the Lean file exists precisely so that nobody has to be able to read the paper.
9. Tao's objection, in his own words rather than the summary
“In most cases in pure mathematics, the problems are posed not because we desperately want the solution to these problems in and of themselves, but because we have seen from past experience that human-directed efforts to solve these problems tend to spur further development of the field.” And the consequence: “Prematurely solving the problem by purely AI-powered methods, particularly without full transparency into the solution process, can contaminate this process to the point where it actually becomes a net negative for the progress of mathematics as a whole.” Note the condition he attaches. The complaint is about opacity as much as speed.
10. The money, with the arithmetic corrected
The figures in circulation are unreliable and mostly derive from one mistake: the 300 billion output tokens OpenAI reports were spent across every problem it attempted, while Navier-Stokes itself took about 130 billion. Cost estimates built on the larger number, such as the 22.5 million dollars widely quoted, inherit the error. Mark Chen, OpenAI's chief research officer, described the effort as around a thousand times a typical project and in the ballpark of millions of dollars. Karthik Duraisamy of the University of Michigan estimated roughly 6 million dollars at retail rates and about 1 million in internal inference cost. The shape is what matters: several million dollars of compute aimed at a one million dollar prize the company then declined to claim.
11. The second-order cost is the one Buckmaster is actually talking about
His verdict on racing is not sour grapes, it is a resource argument: “very few mathematicians will have resources of that scale”, as Gómez-Serrano put it. If a year of quiet work can be overtaken in four days by a lab that hears you are close, the rational response is to stop telling anyone what you are working on. Tao has warned that this breaks with centuries of open practice. Buckmaster's own questions are more concrete: “What do we do about credit? What do we do about hiring? What do we do about PhDs?” Hiring committees and doctoral programmes run on priority, and priority is what just became purchasable.
12. The practical lesson for anyone doing original work in a chatbot
Whatever happened here, the episode makes a governance point that does not depend on who is telling the truth: unpublished work sitting in a vendor's product is sitting on someone else's infrastructure, under terms you did not negotiate, at a company that may be racing you. Providers do distinguish between consumer tiers and business or API tiers in how they handle content, and the settings differ by product and change over time, so the only safe assumption is the one you can verify in writing for the specific product you are using. A private session is not a lab notebook.
There are two disputes here and they are being argued as one. The narrow dispute is whether OpenAI's agents benefited, directly or indirectly, from two mathematicians' private work, and it will probably never be settled, because the evidence is private conversations and a training pipeline nobody outside the company can inspect. The broad dispute is whether it now makes sense for a mathematician to spend a year on a famous problem, and on that one Buckmaster has already conceded, which is a more serious event than either side winning the first argument.
The reassuring reading is that the mathematics survives regardless. The proof is public, the Lean file is public, and whether it holds is a question a machine can answer without anybody trusting anybody. The unreassuring reading is that the machinery of credit is not mechanised at all and just took visible damage, and that the response most likely to spread is the least useful one: keep your drafts off other people's systems, tell nobody what you are close to, and publish only when it is finished. That is how you protect a career. It is also how a field stops talking to itself.
Where we track this
- What OpenAI's proof actually claims →
Our read of the 166-page paper, the Lean repository and the Clay statement: why the forcing objection fails, and what has and has not been verified.
- The AGI Clock →
The announcement is logged there at minus 30 days, with the reasoning in the open. This dispute changes nothing on the clock, because credit is not capability.
Questions people ask
- Did OpenAI steal Tristan Buckmaster's work?
- That is not quite his claim, and it has not been established. He says OpenAI began work after information about his and Levent Alpöge's progress reached it, took the same unusual technical route, and that he cannot rule out that his private Codex sessions played a part. OpenAI says neither its researchers nor its agents saw their work before publication, and that an investigation found his Codex prompts could not have influenced the system, including through training.
- What did Sébastien Bubeck actually say?
- He called the allegations false and inflammatory on 8 September 2026, denied ever asking for Levent Alpöge to be removed from authorship of his own work, and published messages in which he offered the pair a coordinated release and priority for the result. On the reported remark about ruining a career he apologised, calling it an extremely poor choice of words and saying he retracted it at the time. Buckmaster maintains that the removal was requested twice and that the remark was a threat.
- Why do mathematicians say the proof is incomprehensible?
- The criticism, from Javier Gómez-Serrano of Brown University, is about the 166-page write-up rather than the formal proof. His point is that the mathematical community will not accept the exposition in its current form. The Lean formalization is a separate artifact designed to be checked by machine, so readability and verifiability are different questions here.
- How much did the computation cost?
- Nobody outside OpenAI knows. Mark Chen, its chief research officer, described the effort as roughly a thousand times a typical project and in the ballpark of millions of dollars. An independent estimate by Karthik Duraisamy put it near 6 million dollars at retail rates and about 1 million in internal inference cost. Higher figures such as 22.5 million are generally computed from the 300 billion tokens spent across all the problems attempted, not the roughly 130 billion spent on Navier-Stokes.
- Is it now unsafe to discuss unpublished research with an AI assistant?
- The episode is a reason to check rather than to panic. Nothing here has been proven about training on those sessions, and OpenAI denies it. The durable lesson is that unpublished work held in a vendor's product sits on infrastructure you do not control, at a company that may be working on the same problem. Handling differs between consumer, business and API tiers and changes over time, so verify the terms for the specific product rather than assuming.
Written 16 September 2026 from an owner-supplied French-language article published by Developpez on 15 September 2026, and checked the same day against sources. Verified through first-tier reporting: Buckmaster's quotations, including “I think it's pointless. Like, I think the game is up.”, the remark that the companies have zero respect for the scientific community, his questions about credit, hiring and PhDs, and his comment on Australia's national interest, all from the ABC interview published 11 September 2026, which also carries OpenAI's statement that it is categorically impossible for his Codex prompts over the previous two months to have influenced the system; his account of the route through a smooth force and of Bubeck's requests, from TechCrunch, 8 September 2026; Bubeck's denial, his published messages offering priority, and his apology for the remark about ruining a career, together with Sam Altman's endorsement of his team, from The Next Web and Officechai, 8 September 2026; Terence Tao's two quoted passages and Javier Gómez-Serrano's assessment of the write-up, from MIT Technology Review, 8 September 2026, with the wording of the incomprehensible remark corroborated by further coverage; and the compute estimates from Mark Chen and from Karthik Duraisamy of the University of Michigan. Read directly at source for this piece and its companion: OpenAI's announcement page of 8 September 2026 with its dated 10 September update, the 166-page Navier-Stokes paper, the 57-page Euler paper, the Lean repository openai/NavierStokesAndEuler, Fefferman's official Clay problem statement and the Clay prize rules. Corrections to the source article: the Navier-Stokes run took 88 hours, not 90; the 300 billion output tokens were spent across every problem OpenAI attempted, while Navier-Stokes itself used about 130 billion, and cost figures built on the larger number, including the 22.5 million dollars quoted, inherit that error; the 15 million dollar figure could not be traced to a first-tier source and is omitted; the word incomprehensible is Gómez-Serrano's and applies to the write-up, and attributing it jointly to him and to Terence Tao misstates both positions, since Tao's published objection is about insight and transparency rather than legibility; Bubeck did not merely contest the account, he denied the authorship request outright and separately apologised for the career remark, which the source omits; Alpöge and Buckmaster's Euler result is the forced case while OpenAI's agents proved the unforced one, a distinction the source drops and on which the priority question turns; and OpenAI's statement about de-identified data improving its models predates the narrower finding now on its page, so the two formulations should not be quoted as one position. Not verified and therefore not repeated here: the attribution of a “several million dollars” quotation to Mark Chen in the exact words used by the source, which we give as reported in the form first-tier coverage carries. Labelled as contested rather than settled: everything said in private between Bubeck, Buckmaster and Alpöge, on which the parties disagree and no independent record exists. Disclosure: Levent Alpöge works at Anthropic, whose models this site is built with and whose Claude our knowledge base recommends, and the pair's work used both OpenAI's Codex and Anthropic's Claude.
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