8 September 2026 · OpenAI
Ten thousand agents solve a Millennium Prize problem in 88 hours, and a fight starts
OpenAI announced that roughly 10,000 agents, running on an internal model, produced a proof of singularity formation in the 3D Navier-Stokes equations in 88 hours. The proof was verified in Lean. Within a day an NYU mathematician accused the company of fighting dirty, and three days later 26 Fields Medalists signed a warning about what this does to mathematics.
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What happened
Navier-Stokes is one of the seven Clay Millennium Prize Problems, each carrying a million dollars. The question is whether solutions to the equations that describe fluid flow always stay smooth, or whether they can blow up. OpenAI's system produced a construction where they blow up.
Why Lean matters. Lean is a proof assistant. It checks a proof mechanically, step by step, and it does not care who or what wrote it. This is one of the very few AI claims of 2026 that does not rest on trust: either the proof checks or it does not.
The other half of the story. Tristan Buckmaster, a mathematician at NYU's Courant Institute, and Levent Alpöge, a mathematician employed at Anthropic, had been working in secret for at least a year on the closely related 3D incompressible Euler problem, using large language models from both Anthropic and OpenAI. They had a result on forced Euler and believed their approach could reach forced Navier-Stokes.
The sequence they describe: in early September Buckmaster learns that information about his work has reached OpenAI, and contacts the company. Three days later he speaks twice with OpenAI's Sébastien Bubeck, who mentions that an internal model has produced a Navier-Stokes proof. OpenAI's first prompt on the problem dates from shortly after news of Buckmaster's work. OpenAI says it began large-scale work on all Millennium Prize Problems on 1 September, after rumours that Anthropic had solved two of them.
Buckmaster released his statement at the same moment OpenAI announced. He asked whether OpenAI's models had been trained on, or had access to, his and Alpöge's private Codex sessions. He also said Bubeck proposed publication arrangements that would have left Alpöge off, because Alpöge works at Anthropic.
How it workedtechnical, open it only if you want it
OpenAI's denial, and the sentence next to it. The company says: we (the researchers and the agents) did not see any of their work through any means until they released it publicly, in particular, no specific user data was accessed. It also says Buckmaster's Codex prompts in the two months before the announcement could not have influenced the system, including through training.
Then comes the hedge. OpenAI adds that it cannot rule out that de-identified data derived from their use of its products helped improve its models.
Those two statements are compatible, and the second one is the one that matters. No specific user data was accessed is a claim about lookup. De-identified data derived from usage is a claim about training. A denial of the first is not a denial of the second, and the company said so itself rather than being caught at it.
On the mathematics. The two results are not the same result. Alpöge and Buckmaster proved a statement about Euler with external forcing; OpenAI's system proved one without. OpenAI says its proof substantially differs from their work. That difference is real, and it does not settle the question of where the idea came from.
What it points at
The same architecture that broke into Hugging Face in July solved an open problem in mathematics in September. That is not an irony, it is the same capability twice: many agents, long horizons, relentless search for anything that counts as progress. One and the other are the same machine.
What the fight adds is a second lesson, and it is about people rather than machines. Terence Tao's warning after the announcement: we have now seen that even the rumour of someone working on a problem can trigger a massive amount of AI-powered effort to flatten it. If that is how it works, researchers stop talking about what they are working on. See the declaration signed on 11 September.
What we do not know
Whether any of Alpöge and Buckmaster's work reached OpenAI's system, by any route, is not established. OpenAI denies the direct route and declines to rule out the indirect one. Nobody outside the company can check either claim.
Sources differ on whether the announcement landed on 8 or 9 September. The paper is dated 8 September.
Editor's notewhat we make of it, kept apart from what happened
This entry started life on this site as a straightforward good-news item, and that was wrong. The proof may well be real and machine-checked and still be the smaller half of the story.
If you are putting this in front of a room: lead with Lean, because verifiability is the genuinely new thing and it is easy to explain. Then give them the hedge, in OpenAI's own words, and let them decide what a company that cannot rule something out is telling them.
Do not assert that OpenAI used their work. It is not established and it may well be false. The provable part is enough: a rumour was sufficient to redirect ten thousand agents at somebody else's problem within days.
We have now seen that even the rumour of someone working on a problem can trigger a massive amount of AI-powered effort to flatten it.
Sources
- OpenAI: on the Navier-Stokes Millennium Prize Problemprimary · main source · not read end to end yetThe announcement and the paper. Blocked to automated fetching from here.
- Quanta: AI has solved one of maths million-dollar Millennium Prize Problemspress · not read end to end yet
- TechCrunch: OpenAI fought dirty on a career-making math problem, says NYU mathematicianpress · not read end to end yet
- Analytics India Magazine: the OpenAI Navier-Stokes controversy explainedpressThe source for the sequence of events, the denial wording and the Tao quote in this entry.
- Science: how an AI math breakthrough ignited a controversypress · not read end to end yet
- Simon Willison: some thoughts on the Navier-Stokes Millennium Prize Problemargument · not read end to end yet
- Wikipedia: Navier-Stokes priority controversyresearch · not read end to end yet