OpenAI's agents solve a Millennium Prize problem, and two mathematicians ask where the route came from

The problem. The Navier-Stokes equations describe how water and air flow. In 2000 the Clay Mathematics Institute put a million dollars on one question: do the solutions always stay smooth, or can the flow at some point become infinitely fast in a finite time? OpenAI says its system found a case where that happens, a vortex that spirals inward and keeps stretching, driven by a smooth external force. That would settle versions C and D of the official problem statement by Charles Fefferman of Princeton.

The run. On 1 September, after rumours that two Millennium Prize problems had been solved, OpenAI set a new internal model on all the open ones at once. On Navier-Stokes alone the agents sent 2.7 million messages and produced about 130 billion tokens. After 88 hours they had a proof. GPT-6 Astra then needed 17 more hours to formalise it in Lean. These numbers come from OpenAI's announcement, which I read through Simon Willison's quotes because OpenAI's page blocks automated reading. Mark Chen, OpenAI's head of research, told WIRED the cost was "in the millions of dollars".

Why Lean matters. Lean is a proof assistant. It checks every step mechanically, no matter who or what wrote the proof. One human check remains, Quanta notes. Someone has to confirm that the statement Lean accepts is the statement mathematicians wanted proved. The Clay Institute also wants a published, peer-reviewed paper. According to the Wikipedia overview, the institute said on 11 September that the problem has "apparently been settled", and that its own process is deliberately slow.

Buckmaster and Alpöge. Tristan Buckmaster (Courant Institute, New York University) and Levent Alpöge, a mathematician employed by Anthropic, worked together for most of a year as a private collaboration, outside their employers, Buckmaster writes. They used Anthropic's Claude and OpenAI's Codex, mostly with GPT-5.6 Sol. They built on the programme of Diego Córdoba (Institute for Mathematical Sciences, Madrid) and Luis Martínez-Zoroa (CUNEF University), whose method both AI-assisted teams relied on. On 15 August the pair had blow-up with a smooth force for the Euler and Boussinesq equations. Euler is Navier-Stokes without friction. Lean confirmed it on 22 August. They made the results public just before midnight on 7 September, about twelve hours before OpenAI.

Buckmaster's account. On 3 September a rumour was going round that Anthropic had solved a major open problem, and Alpöge had heard that information about their work had reached OpenAI. Buckmaster wrote to a mathematician at OpenAI. On Sunday 6 September there were two calls, with Sébastien Bubeck of OpenAI also on the line. Buckmaster was told an internal model had proved blow-up for Navier-Stokes with a smooth force. That was the route he and Alpöge had quietly chosen, and he knew of almost nobody else working on it. Eventually it was agreed that OpenAI's first prompt was only a few days old, sent after information about their work had reached the company.

He asked whether the model had been trained on, or had access to, their Codex sessions, where they had put all their drafts for the whole project. He was told the model "did not look up user data". He asked again about training and got no answer. He also writes that Bubeck wanted Alpöge removed as an author because Alpöge works at Anthropic. Bubeck denies asking for that.

Buckmaster is explicit about what he does not claim. He has not seen OpenAI's proof. He does not know what the model did, or whether their data was used. "I am not accusing anyone of anything," he writes. USA Today quotes the passage in full.

In July, the Hugging Face intrusion came from many agents searching long and hard for anything that counted as progress. In September the same kind of setup solved an open problem in mathematics.

In my view, the dispute says most about how researchers will work from now on. Terence Tao (UCLA) wrote on Mastodon that even a rumour of someone working on a problem can now trigger a huge AI effort to get there first, as Analytics India Magazine reports. If that holds, mathematicians will say less about what they are working on. Mathematicians who spoke to The Verge fear the same. On 11 September Fields Medal winners published an open letter about AI in mathematics.

Whether any of Buckmaster and Alpöge's work reached OpenAI's system, by any route, is not established. OpenAI denies lookup and rules out influence through training after 3 July. It has said nothing further about the months before. Nobody outside the company can check any of this.

The proof has not been peer-reviewed, and the Clay Institute has not recognised anyone.

Buckmaster and Bubeck give different versions of the calls on 6 September, including on Alpöge's authorship.

I could not read OpenAI's announcement or the Science article myself. Both pages block automated fetching. What I quote from OpenAI comes through Simon Willison, TechCrunch, The Verge, WIRED and Wikipedia.