OpenAI Published Its Navier–Stokes Proof and the Lean Formalization — and Says It Won't Claim the Millennium Prize, Because This Isn't the Classic Version

OpenAI announced a result on the Navier–Stokes existence and smoothness problem on September 8, then on September 10 published the full technical write-up, the paper and a machine-checkable Lean formalization, saying it will not claim the Clay Mathematics Institute's million-dollar Millennium Prize. The scale figures are specific: nearly 10,000 agents working in parallel reached the result on September 5, about 88 hours after the first agents launched, sending 2.7 million messages and producing roughly 130 billion output tokens, after which GPT-6 Astra spent another 17 hours on Lean formalization and verification. The boundary has to be stated plainly: what was proved is that the 3D Navier–Stokes equations develop a singularity under a smooth forcing term applied throughout, which OpenAI frames as establishing statements C and D of the official Millennium Prize formulation rather than the classic problem itself. Separately, NYU mathematician Tristan Buckmaster has raised a priority dispute that OpenAI denies; the two accounts do not currently agree.

"Solved a Millennium Problem" Has to Lose One Term

What gets dropped in retelling is the forcing term. The object here is a fluid starting from rest with a smooth force applied to it throughout, and the conclusion is that it forms a singularity in finite time. The classic Millennium problem asks something else — whether, in the standard unforced setting, solutions of the 3D Navier–Stokes equations remain smooth forever. OpenAI states this more carefully than most retellings do: it says the work establishes statements C and D of the official formulation, and that it will not claim the prize. That self-imposed limit is more notable than the result — a company with every incentive to overstate drew the line itself. Clay's rules also mean this cannot move fast: the prize can only be awarded after peer-reviewed publication and two years of community validation, and it has been awarded exactly once, for the Poincaré conjecture. Even if every dispute were settled, the timetable runs in years.

88 Hours, 10,000 Agents, 130 Billion Tokens

The method-side numbers are the most solid part of this story: nearly 10,000 agents in parallel, a result about 88 hours after the first launched, 2.7 million messages and roughly 130 billion output tokens along the way, then another 17 hours for GPT-6 Astra to write it in Lean and pass verification. Worth noting that this is a different thing from Anthropic's Fermat's Last Theorem formalization last week. That one translated an existing human proof into a machine-checkable form — the novelty was the verification. This one claims a new result, with Lean as the final gate. Together they roughly sketch AI's two current fronts in mathematics: making old conclusions machine-checkable, and parallel search for new proofs on problems with a clear target.

The Priority Dispute Is Unresolved

Buckmaster, a math professor at NYU, alleges OpenAI started only after becoming aware that he and Levent Alpöge (an Anthropic employee) were using a specific method on a related problem, and has raised questions about whether private Codex material played a role. OpenAI's account: it began on September 1 after hearing a rumor, and after completing the proof and Lean verification on September 6 reached out to offer a concurrent release and recognize their priority; neither its researchers nor its agents saw the other work before public release and no specific user data was accessed, though it cannot rule out that de-identified data from product usage helped improve its models; and it says the proofs differ significantly, proving different results in the Euler case (forced versus unforced). Buckmaster's side says they had a Lean-verified proof for the Euler equations by August 22. Observers have also pointed to a clear intellectual debt to earlier work by Córdoba and Martínez-Zoroa. None of these disagreements has a third-party resolution yet, so the honest thing is to place both accounts side by side. One related event landed the same week: after an open letter signed by 771 mathematicians, OpenAI withdrew its sponsorship of the Caltech Mathathon — signatories objected to undergraduates spending $2 million in AI credits on open problems, worried about a flood of unverified "slop mathematics" and about verification costs being pushed onto the mathematical community. Put together, what the field has to work out is not only whether AI can prove things, but who checks, and who pays for the checking.

via: OpenAI, "On the Navier–Stokes Millennium Prize Problem", Quanta Magazine, Scientific American, The Next Web on the published proof and the declined prize