OpenAI Claims Navier-Stokes Proof as Credit Dispute Erupts

Every headline today says an AI solved a million-dollar maths problem. Read the fine print and you find something narrower, and a second story that may matter more than the first. On September 8, OpenAI published a paper claiming finite-time blowup for the forced three-dimensional Navier-Stokes equations, produced by an internal model. The same morning, NYU mathematician Tristan Buckmaster released a four-page statement describing two Sunday phone calls with OpenAI, an offer to drop his co-author from a paper, and a question about his own data that he says went unanswered.

Two results, one day

Buckmaster and Levent Alpöge, a mathematician employed by Anthropic, worked as a purely personal collaboration with no institutional involvement from either employer. Their results: finite-time blowup with smooth forcing for the incompressible porous media equation, for Boussinesq, and for 3D incompressible Euler. In plain terms: under a smooth external force, an initially calm fluid can develop a singularity in finite time. The breakthrough came on August 15, and Lean machine verification followed on August 22. Buckmaster credits the underlying programme to Diego Córdoba and Luis Martínez-Zoroa, and says the pair pushed it through with heavy help from LLMs: Claude, Codex, and Astra for write-ups and auditing.

OpenAI's result goes one step further, to forced Navier-Stokes itself. By the company's own account, a model "significantly more capable than GPT-6 Astra" was run as roughly 10,000 concurrent agents starting September 1 and reached the result 88 hours later, on September 5, consuming about 130 billion output tokens and 2.7 million messages on this problem alone. GPT-6 Astra formalised the proof in Lean in 17 hours. No independent mathematician has checked it yet. OpenAI has published the Lean code and says it does not intend to claim the Millennium Prize.

Is the million-dollar problem solved?

Not in the sense people mean. The Clay Mathematics Institute's official problem statement lists four options, and options C and D permit a smooth external force; OpenAI says its theorem matches the text of those two. But the problem mathematicians actually mean by "Navier-Stokes", the unforced version, remains open. Fields Medallist Terence Tao called the Buckmaster-Alpöge work "a remarkable achievement" and said he sees no obvious obstacle to pushing the methods to Navier-Stokes, with enormous technical difficulties remaining. He then had to clarify that his comments were being read as confirming a breakthrough that had not happened.

What was said on Sunday

Buckmaster's timeline starts on September 3. With a rumour circulating that Anthropic had cracked a major problem, and with Alpöge receiving tips that news of their progress had reached OpenAI, Buckmaster emailed a prominent mathematician at the company. The reply came the same day: details would help "to avoid competing here", and OpenAI would be happy to provide compute. He asked to talk the following week, twice. On Sunday September 6 at 12:45 he was asked to meet "at any point today". Two calls followed that afternoon, with research executive Sébastien Bubeck on the line.

He was first told the model had simply been given the problem statement with "very little human input". As the call went on, a different picture emerged: a whole team had worked on it, easier problems including Euler had been tried first, even the prompt he was shown had been written by Codex, and "an insane amount of compute" had been used. When he asked when the first prompt was sent, it was eventually agreed this happened after information about their work reached OpenAI.

He asked whether the model had access to their Codex sessions, where they had put every draft of the project. He was told the model does not look up user data. He asked whether those sessions had been used in training, and says he got no answer. Then came two proposals: post the Euler result and let OpenAI post Navier-Stokes the next day, or have Buckmaster alone write a Navier-Stokes paper acknowledging an internal OpenAI model. He says Bubeck twice asked for Alpöge to be removed from authorship and remarked that it was "so annoying" that Alpöge works at Anthropic. When Buckmaster said he would go public, the reply he reports is "Why would you ruin your career?", followed by "If you don't want me to be nice, then I don't have to be nice."

Buckmaster is careful about what he is not claiming: he has not seen OpenAI's proof, does not know whether his data was used, and is "not accusing anyone of anything". OpenAI's statement says its researchers and agents did not see the pair's work "through any means" until it was public, and that while it cannot rule out de-identified usage data having helped model improvement, it considers that unlikely. Bubeck called the allegations "false and inflammatory" and said the internal model solved the problem by "totally different means".

Three things to take from this if you never touch a fluid equation

First, Buckmaster's most important question was not about mathematics: were the sessions holding our drafts used in training? If your team puts code, contracts or customer data into an AI tool, that answer belongs in your contract, and "the model does not look up user data" does not settle it. Second, 10,000 agents for 88 hours is now a research method; the scale is out of reach for most companies, but the price of pointing a hundred agents at a problem overnight drops every month. Third, when the model does most of the work, credit stops being an academic question. Write down who owns AI-assisted output built with a partner, who may publish it, and whose name goes on it, before there is anything worth arguing over.

Sources: OpenAI, Tristan Buckmaster's statement (PDF), TechCrunch, The Decoder