OpenAI Says 10,000 Agents Solved Navier–Stokes

OpenAI published a claimed solution to the Navier–Stokes existence and smoothness problem on September 8, one of the seven Clay Mathematics Institute Millennium Prize Problems and an open question for roughly 90 years. The company says an internal system produced both a written proof and a machine-checkable formalization in the Lean proof assistant, showing that a three-dimensional incompressible fluid starting smoothly at rest can develop a singularity — a point where fluid speeds grow without bound — in finite time, while its total energy stays finite.

The model behind it has not shipped. OpenAI describes it only as an internal system "significantly more capable than GPT-6 Astra", the model it released on September 3, and says training began on August 28 and is still running. The company frames the release as a status report on capability rather than a product announcement, and states plainly that it does not intend to claim the Millennium Prize money.

Ten thousand agents, 88 hours

The process OpenAI describes is as notable as the result. On September 1, company researchers heard rumors that two Millennium Prize problems had been resolved. They responded by pointing a system of coordinating agents — powered by the internal model, with access to a cached copy of the internet and the ability to run code — at every open Millennium problem at once, splitting agents into communicating groups and prompting different groups with different variants of each problem statement. The group that cracked Navier–Stokes ran on the order of 10,000 concurrent agents.

Along the way the agents were also handed a set of deliberately easier warm-up problems. One was the regularity question for the Euler equations, Navier–Stokes with the viscosity term removed. Roughly 100 agents working for about 50 hours disproved it in the unforced case, which OpenAI says surprised the team. That Euler result was then fed back to the Navier–Stokes groups as a stepping stone. The agents reached their resolution on September 5, about 88 hours after launch; formalizing and verifying it in Lean took another 17 hours using GPT-6 Astra. Across every problem attempted, the agents exchanged 4.9 million messages and burned roughly 300 billion output tokens, with 2.7 million messages and about 130 billion tokens spent on Navier–Stokes alone.

What it settles, and what it does not

Several outlets covering the announcement argued that OpenAI solved the wrong problem, because its fluid has an external force applied to it rather than evolving freely. That framing does not survive a reading of the official problem statement. Charles Fefferman's description of the problem for Clay asks for a proof of any one of four statements. Statements (A) and (B), which assert that smooth solutions always exist, require the forcing term to be identically zero. Statements (C) and (D) — the breakdown statements — explicitly ask for a smooth divergence-free initial velocity field together with a smooth force satisfying prescribed decay bounds. OpenAI says its proof establishes (C) and (D), and forcing is permitted there by design.

The genuine caveats are different ones. Nobody outside OpenAI has yet checked the argument, and a Lean formalization certifies that a proof follows from its stated definitions, not that the definitions faithfully encode the Clay problem — auditing that correspondence is exactly the work the mathematical community has not had time to do. A result of this size will take weeks of scrutiny, and the company's decision not to pursue the prize means no Clay adjudication will force the issue.

The result OpenAI was racing

The rumor that set OpenAI running was real. NYU mathematician Tristan Buckmaster and Levent Alpöge, a researcher at Anthropic, posted three preprints with accompanying Lean formalizations establishing finite-time blowup with smooth forcing for the incompressible porous medium equation, the two-dimensional Boussinesq system, and the three-dimensional incompressible Euler equations. Their code sits in a public repository, and the work builds on an approach developed by Diego Córdoba and Luis Martínez-Zoroa.

Terence Tao, who has worked on these equations for years, called it a remarkable achievement and wrote that he sees no fundamental obstacle to extending the method to full Navier–Stokes. Their proofs were heavily AI-assisted too — Buckmaster has said they leaned on Claude and on OpenAI's Codex, and that the first machine-generated writeup was the worst he had ever read, taking weeks of rewriting to reach publishable quality.

A dispute over who knew what

Buckmaster has published an account alleging that OpenAI behaved badly in the days before both releases. By his telling, he emailed an OpenAI mathematician on September 3 about the rumors, then met on September 6 with that researcher and with Sébastien Bubeck, twice and without Alpöge present. He says he was told an internal model had produced a roughly 100-page blowup proof for forced Navier–Stokes, and was offered two options: post the Euler result and let OpenAI post Navier–Stokes a day later, or author a paper presenting OpenAI's Navier–Stokes result himself, crediting the model. Buckmaster writes that Bubeck twice pushed for Alpöge to be dropped from authorship because of his Anthropic employment, and that when he refused and threatened to go public he was asked why he would ruin his career. He has also raised the possibility that private Codex sessions exposed the pair's unpublished direction, while stressing that he has not seen OpenAI's proof and is not accusing anyone, only recounting what he was told.

OpenAI rejects the characterization. Bubeck called the allegations false and inflammatory. Chief research officer Mark Chen told reporters that no people or AI systems searched user data and that he was disappointed by the claims. The company's own post says it reached out on September 6 to offer a concurrent release and a joint announcement, that it did not see the pair's work through any means before publication, and that its Euler proof differs from theirs — OpenAI's agents did the unforced case, Buckmaster and Alpöge the forced one. It also concedes that while unlikely, it cannot rule out that de-identified data derived from product usage helped improve its models, and it explicitly recognizes the pair's priority on forced Euler.

What is left is a governance question no lab has answered. Both results depended on frontier tools, and one of the two teams was using a competitor's product to work on an unpublished proof that the competitor then raced them to. Nothing in the current arrangement lets an outsider verify a lab's assurances about what its systems did or did not see; the assurance is the evidence.