CoverTwo-dimensional Navier–Stokes simulation made for this article: vortices after two opposing streams become unstable.
This story is even more delicate.
On 8 September 2026, OpenAI published what it presents as the solution to the Navier–Stokes existence and smoothness problem, one of the Clay Mathematics Institute’s seven Millennium Prize Problems, each carrying a one-million-dollar prize.
What these equations describe
The Navier–Stokes equations describe how fluids move. Water. Air. Turbulence. Currents.
The problem is not that we do not know how to use them: engineers and scientists have worked with them for decades to design wings, forecast the weather or compute the flow through a pipe. The deep mathematical question is a different one, and it has been open for almost a century:
Can these equations develop singularities in three dimensions, or do their solutions always remain sufficiently regular?
What OpenAI claims
That the answer is yes. According to the announcement, an internal model found a configuration in which a vortex tightens and spins ever faster until the solution “blows up” in finite time, while the fluid’s energy stays bounded.
The scale of the effort is part of the story: some 10,000 AI agents worked on the problem for 88 hours and exchanged almost five million messages. Sébastien Bubeck of OpenAI estimates the compute cost at several million dollars.
And one detail sets it apart from any earlier announcement: the proof was formally checked in Lean, a language in which a program verifies every logical step.
Why it is not officially solved yet
Precision matters here: the existence of the proof does not mean the Millennium Problem has been officially closed.
Lean guarantees that what was proved is correct. It does not guarantee that what was proved is exactly what needed proving. That check, that the statement written in Lean is equivalent to the problem mathematicians posed, still has to be done by people.
On top of that, the Clay Mathematics Institute has explicit rules. Before it even considers a solution, it requires three conditions:
- It must be published in a qualifying outlet.
- At least two years must pass after publication.
- It must receive general acceptance in the global mathematics community.
Credit is also under discussion
Charles Fefferman of Princeton, who wrote the Clay Institute’s official description of the problem, said he was “thrilled that the problem was solved”, but named Diego Córdoba and Luis Martínez-Zoroa as the real heroes of the story. Their 2023 technique, an “infinite cascade” of layers combined into a new solution, is the intellectual foundation of the result.
In parallel, Tristan Buckmaster (New York University) and Levent Alpöge (Anthropic) solved several closely related problems, also with AI assistance.
Why it matters even without the seal
Even so, this is huge. If the proof survives review, we would not only be looking at the solution to a problem open for generations: we would be looking at one of the most important examples so far of AI taking a direct part in creating new mathematics.
Sources
- AI Has Solved One of Math’s $1 Million Millennium Prize Problems · Quanta Magazine
- On the Navier–Stokes Millennium Prize Problem · OpenAI
- OpenAI says it has solved one of math’s “Millennium Problems” · CNN Business
- OpenAI claims solution to one of math’s $1 million Millennium Prize problems · The Washington Post
- OpenAI’s historic math solution overshadowed by credit controversy · Axios
- Millennium Prize Problems: Rules · Clay Mathematics Institute