OpenAI claims Navier-Stokes, and the dispute erupts
On September 8, 2026, at 10:00 GMT according to the timestamp in OpenAI’s own news feed, OpenAI published “On the Navier-Stokes Millennium Prize Problem”. Its own description in the official feed: “We’re sharing an AI-generated solution to the Navier-Stokes Millennium Prize Problem, including a writeup and a formal proof in Lean.” An AI-generated solution, then, with a written explanation and a formal proof in Lean, a system that verifies mathematical proofs by machine.
For those not dealing with fluid dynamics daily: the Navier-Stokes equations describe how liquids and gases move, from the water in your tap to the air around an airplane wing. The question of whether their solutions always exist nicely and remain smooth, never going haywire anywhere, is one of seven millennium problems that the Clay Mathematics Institute established on May 24, 2000, with a 1 million dollar prize for each. That question stood open for twenty-six years, and the equations themselves date back to the nineteenth century. Whether this proof holds up, no one knows today: the Clay Institute follows its own procedures with long timelines, and the mathematical world is only just beginning to read. But if it holds, a machine did in four days what generations of mathematicians broke their teeth on.
Four days, 2.7 million messages
What those four days looked like is in OpenAI’s writeup, as Simon Willison summarizes on September 8: the agents, AI systems that work independently without human approval of each step, started on September 1 and needed approximately 88 hours, good for 2.7 million messages and roughly 130 billion output tokens, the text fragments in which a model writes out its work. That scale doesn’t come from nowhere: it is exactly the research acceleration through agents where OpenAI itself published figures last week, made possible by the computing power race in which labs lock in hundreds of billions in capacity. Whoever can afford such a sprint can apparently also direct it at a millennium problem.
The first news wasn’t about mathematics
You might expect that after such an announcement everyone would talk about mathematics. The first big story went about something else. According to Willison’s reconstruction, NYU mathematics professor Tristan Buckmaster and Anthropic mathematician Levent Alpöge had already been working quietly on the same problem for nearly a year, with Claude and Codex (mostly GPT-5.6 Sol), and achieved their breakthrough on August 15. OpenAI would have heard through rumors of their progress and only then started its own attempt. On September 8, TechCrunch recorded Buckmaster’s account.
And then the detail that sticks around, as Willison relays the account: OpenAI excluded Alpöge as co-author, “due to OpenAI’s competitive relationship with his employer”, because of the competitive relationship with his employer Anthropic. Read that sentence again. A mathematician disappears from a scientific publication, not because of what he did, but because of where he works.
Buckmaster goes further, and note: these are his accusations, not established facts. He says that “information about our progress had been passed to OpenAI”, that information about their progress had reached OpenAI. And about the speed of OpenAI’s sprint:
“It is not the direction one arrives at in a few days by giving a model the problem statement.”
In other words: you don’t reach this route in a couple of days by giving a model the problem statement. He also says that OpenAI researcher Sebastien Bubeck pressured him, first with “Why would you ruin your career?” and later with “If you don’t want me to be nice, then I don’t have to be nice.”: if you don’t want me to be friendly, then I don’t have to be. All those quotes come from Buckmaster’s account at TechCrunch. OpenAI firmly denies the core there: “We (the researchers and the agents) did not see any of their work through any means until they released it publicly”, researchers nor agents saw anything of their work before publication, and according to the company no specific user data was accessed to solve the problem. No response from Bubeck himself, and Alpöge doesn’t speak anywhere. Notably, OpenAI covers itself in the writeup as well, as Willison quotes: “While unlikely, we cannot rule out that de-identified data…helped improve our models.” Unlikely, but not to be ruled out that anonymized data helped the models, the company writes in so many words.
The defense also has a file
Yet it is too early to cast OpenAI as the villain. Buckmaster’s account is one-sided: one side of a dispute, told by the party that feels robbed, and for its core, that information was passed on, there is no evidence on the table. Facing the accusation stands an awkward fact: OpenAI put everything online publicly, writeup and formal proof, for everyone to read for free. And a Lean proof can be checked by computer systematically, step by step, without taking the author at his word. That is more transparent than the mathematics tradition ever was, where a proof lies for years with a handful of reviewers behind closed doors. Whoever wants to hide dirty play rarely chooses the most verifiable publication format that exists.
Therefore keep two questions separate. The mathematical question, does this proof hold and will there ever be a prize, will be answered in the coming months and probably years by experts and by the mechanical verification of the Lean proof. The human question is already answered. On the day a machine possibly cracked a millennium problem, the conversation within an hour turned to who gets to be on the title page and who was removed. The machine did the wonder; the people delivered a removed name, and otherwise mostly he-said-she-said without evidence on the table. Remember that image, because you are going to see it more often: the larger the work of the machines, the smaller the behavior around it can look. There apparently is no limit anymore on intelligence; there is one on character.
Update September 10, 2026: a day after this dispute, OpenAI brought alignment pioneer and risk thinker Paul Christiano onto the OpenAI Foundation board.
