OpenAI claims an AI proof that fluid equations can blow up, but only with a designed outside force
OpenAI says its AI proved fluids can break down, part of a $1 million prize problem. The force-free case is still open; credit is disputed.
The Navier–Stokes equations describe how fluids such as water and air move. Whether their solutions can suddenly break down (“blow up”) is one of the Clay Mathematics Institute’s $1 million Millennium Prize problems. On 8 September 2026 OpenAI said an unreleased model had proved they can, in three dimensions, for a fluid that starts at rest and is pushed by a smooth, specially designed outside force.
The 166-page paper comes with a Lean version, a form of the proof a computer can check. OpenAI says it meets two of the breakdown options, (C) and (D), in the official problem statement, and that it will not claim the prize. By OpenAI’s own figures, the run used at least 10,000 AI agents working together for about 88 hours. OpenAI’s Noam Brown said it cost millions of dollars.
On 11 September the Clay institute said the problem appeared to have been settled, but that its review would be deliberately slow and follow its prize rules, including a two-year wait. No prize has been awarded.
Credit is disputed. The night before, Tristan Buckmaster (NYU) and Levent Alpöge, an Anthropic employee, had posted computer-checked blow-up results, also with an outside force, for related fluid equations. Buckmaster accused OpenAI of moving onto their method after contacts in early September; OpenAI denied using their prompts or data. In mid-October Scientific American reported that several expert groups think OpenAI’s final construction differs from the approach it was first said to follow.
What it means
If it holds up, an AI helped settle one officially accepted form of a famous prize problem. It says nothing about whether a fluid left alone can break down, the version experts care most about. Until people have checked the full argument, it remains a claim under review.
Caveats
- The force is artificial and designed. Whether a fluid left alone (Clay’s options A and B) can blow up is still open.
- No independent human check of the full argument had been published. On 13 October Scientific American reported that, more than a month on, no one fully understood the paper.
- Bastounis, Circelli and Hansen argue the Lean version doesn’t faithfully match the written proof; in one place the paper claims more than the Lean version proves. Kevin Buzzard disagreed and put the risk of a mismatch in the headline statement at essentially zero.
- Constantin, Ignatova and Vicol showed that constructions with the key features of OpenAI’s cannot work with a real-analytic (very smooth) force, so the method can’t simply be extended to the force-free problem.
- The compute figures are self-reported by OpenAI. Sources differ on the page count (165 or 166) and on the timeline of contacts between OpenAI and Buckmaster and Alpöge.