On 8 September OpenAI published a proof, formalised in Lean, that the Navier-Stokes equations for fluid flow can develop a singularity in finite time. In the company's words this resolves one of the seven Millennium Prize Problems of the Clay Institute. The page cites no check by independent mathematicians or by the institute itself, so for now this is OpenAI's claim.
- The solution was found by an internal model OpenAI has been training since 28 August and describes as significantly more capable than GPT-6 Astra.
- The group of agents that reached it was on the order of 10,000 concurrent agents; the work took about 88 hours and the Lean formalisation another 17.
- OpenAI says it will not claim the prize and recognises the priority of Alpöge and Buckmaster for a separate result on the forced Euler equation.
According to OpenAI, its agents needed about 88 hours to answer a question that has been open for roughly 90 years.
The question is old and beautiful. The Navier-Stokes equations describe how fluid moves: air around a wing, water, blood. They treat it as a continuous medium, with no individual molecules. And nobody had proved whether a fluid that starts smooth can, in finite time, reach a point where its speed grows without limit. If it can, the description breaks.
What we actually have
We have a proof written for people, and the same proof written in Lean, a language in which a machine checks every step. That is far more than a press release with a chart. Formalisation means anyone with time and a machine can run the check themselves.
What we don't have is the thing that settles more than anything in mathematics: people outside the company who have sat down with the text and say it holds. The page carries not one such voice. Not from mathematicians, not from the Clay Institute, which announced the problem in 2000.
And mind the direction. The announced answer is a negative one: not that the fluid always stays smooth, but that it can break. The construction is a vortex that spirals inward and stretches out like spaghetti, while the speed at its centre grows without bound and the energy stays finite.
The scale is the other story
On the order of ten thousand agents at once in a single group. Two million seven hundred thousand messages between them. If the numbers are right, a whole swarm was at work here, split into groups that trade intermediate results until something holds.
OpenAI writes that the point of publishing is to show how fast its models are advancing. I believe that sentence. Which is why, for now at least, I read the post as a report on speed.
If the proof holds, we will learn it from the mathematicians who first try to break it. Until then we write exactly what is on the page: according to OpenAI.