OpenAI Announces a Navier-Stokes Breakthrough. An NYU Mathematician Accuses the Company of Using His Private Notes

On September 8, OpenAI announced that an unreleased model, backed by nearly 10,000 agents working in parallel for 88 hours at a compute cost north of $40 million, produced a resolution to the Navier-Stokes problem, one of the Clay Mathematics Institute's seven Millennium Prize Problems, each carrying a $1 million award. Before anyone declares the second Millennium Problem ever solved, two things need checking: what OpenAI actually proved, and where the material it worked from actually came from.

Not the problem you think it is

The classic, best-known version of Navier-Stokes concerns the equations with no external force acting on the fluid. OpenAI published a solution to a variant with an added smooth forcing term, in which the agents built a solution that develops a singularity in finite time even as the fluid's energy stays finite throughout. The company itself argues that's enough to formally resolve the Clay Institute's official problem statement, specifically points "C" and "D" of its formulation, which explicitly allow for a forced variant. That's not a stretch of the rules, points C and D genuinely exist in the Clay Institute's official document. The trouble is that for twenty-six years the mathematical community has treated the unforced variant as the "real" challenge, so formal correctness of this move doesn't automatically translate into recognition from the field.

A competing team was working on a related problem at the same time

Tristan Buckmaster of NYU and Levent Alpöge, an Anthropic employee, were independently working on a related but different problem: the forced Euler equation, the zero-viscosity limiting case of Navier-Stokes. OpenAI's own account admits its effort was inspired by a rumor heard on September 1 concerning exactly this pair, and that after reaching its own result on September 6 it reached out to Buckmaster and Alpöge to propose a joint, concurrent announcement recognizing the priority of their work on Euler. OpenAI stated outright: no employee or agent saw the pair's work before it was published publicly, though the company can't rule out that de-identified data from their product usage may have indirectly improved its models.

Buckmaster's version differs on one important point. He says he asked directly whether the model had access to his private working sessions in Codex, OpenAI's AI coding tool, where he and Alpöge had been dropping their notes for the entire duration of the project, and got a clear answer only on the question of live access, not on training. He also describes a conversation with Sébastien Bubeck of OpenAI, in which he says he was given two options: publish part of his results while OpenAI published its full solution the following day, or write a solo paper crediting the use of an OpenAI model, without including Alpöge as a co-author. He declined both.

A denial at the highest level

Both Bubeck and OpenAI CEO Sam Altman responded publicly to the accusations. Bubeck narrowed his denial to one specific point: he flatly denied ever asking for Alpöge's name to be removed from any paper. That denial addresses one specific piece of the allegation, not the whole of Buckmaster's account, which is itself worth noting: both sides largely agree on how the conversation went, and differ over the interpretation of one sentence.

Only one Millennium Problem has ever been solved

Since the list of seven Millennium Problems was announced in 2000, exactly one has been solved: the Poincaré conjecture, proven by Grigori Perelman in 2002-2003. Perelman turned down both the Fields Medal and the $1 million Clay Prize, saying his contribution was no more significant than the earlier work of Richard Hamilton that it built on. Perelman's modesty around credit stands in sharp contrast to what's happening now around Navier-Stokes, where a dispute over whose name belongs on the paper made tech headlines before anyone had published anything for peer review.

Terence Tao: it's a game about production quotas, not mathematics

Terence Tao, widely regarded as one of the greatest living mathematicians, commented on the whole situation without mincing words. He compared it to watching a movie by skipping straight from the first ten minutes to the last ten: technically every plot thread gets resolved, but most of the value of the experience is lost along the way. He also warned of a side effect of this race: the mere rumor that someone is working on a problem can now trigger a wave of AI-powered effort to "flatten" it before the original research project has a chance to reach its full potential, which could push mathematicians toward keeping promising research directions private instead of sharing them with the community. He summed it up in one line: indiscriminate use of AI is turning the subject into a meaningless production-quota game, with little real benefit to the field itself.

The verification is real, it's just the company's own

Contrary to the earliest reports, OpenAI did in fact publish material: a 166-page manuscript with the proof and a Lean formalization, both publicly available to check yourself. That's a real difference from the picture that emerged in the first hours after the announcement, when journalists were writing about the total absence of a preprint. The real difference from Anthropic's formalization of Fermat's Last Theorem, announced earlier that same week, sits somewhere else: the Fermat proof was checked by a separate tool, nanoda, built by an entirely different team with no ties to Anthropic. The Navier-Stokes formalization was checked, over seventeen hours, by GPT-6 Astra, OpenAI's own model. The Lean code itself is checkable by anyone, but the process that produced it and gave it its first pass of verification stayed entirely inside one company.

OpenAI also stated outright that it does not intend to seek the Clay Institute's $1 million prize for this result, which at least partly explains why the company went with the forced variant instead of waiting on the harder, classic version of the problem: this is a demonstration of the model's capability, not a bid for real money. The Clay Institute's rules require a two-year wait after publication before any proof can even be considered for the prize anyway, so any official recognition is a long way off regardless of who's right in the authorship dispute. This is already the second time this month a flagship number from OpenAI has turned out to be more complicated than the first headline suggested: with GPT-6 Astra it was test conditions inflating a benchmark score, here it's the scope of the claim itself and who actually verified it.

What's mathematically true regardless of the dispute

It's worth separating the authorship dispute from the question of the mathematics itself. Even OpenAI's critics, Buckmaster included, are questioning the circumstances around how the result came together and got announced, not its mathematical correctness. Using a swarm of ten thousand agents to solve a research problem of this class in under four days is, on its own, a significant demonstration of capability, regardless of who ultimately deserves credit. The real problem is how the company managed the whole process around that help, not whether AI can support high-level mathematics at all.