For the second time ever, someone has solved one of the seven Millennium Problems—math’s biggest targets, each worth a $1-million prize. But unlike the first time, that someone is an artificial intelligence start-up.
Today OpenAI announced that its internal model has proved that the Navier-Stokes equations, which mathematicians use to study how fluids move, are fatally flawed. The reveal comes after mathematician Tristan Buckmaster alleged that OpenAI had tackled the proof after the company became aware that Buckmaster and his colleague Levent Alpöge had been using a specific method to break a related problem. OpenAI has denied the allegations.
OpenAI’s proof shows that, on rare occasion, the Navier-Stokes equations “blow up,” meaning they dictate that a fluid’s speed becomes infinite at some points, something that is impossible in nature. The company says the proof has been certified using the programming language Lean, which all but guarantees its correctness.
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“We’re a little bit in shock,” says Diego Córdoba. He and Luis Martínez-Zoroa originally developed the approach the AI used to solve the problem, which is called “forcing.” “If it’s done, that will be a big surprise for us,” Córdoba says.
The night before OpenAI announced the proof, Buckmaster posted on social media that he and Alpöge, who is also a mathematician and an employee of Anthropic, had blown up the Euler equations, widely seen by the field as a step toward solving the Navier-Stokes problem. In a statement released on Monday, Buckmaster alleged that OpenAI had found out about the pair’s progress in the last week and had adopted the same method as Buckmaster and Alpöge had been using to solve the full problem.
In a press briefing on Tuesday, OpenAI mathematician Sébastien Bubeck denied the rumors about the proof’s origins. Bubeck said OpenAI’s internal model had independently solved the Euler problem by totally different means than those employed by Buckmaster and Alpöge. OpenAI’s solution to the full Navier-Stokes problem, however, did follow a similar method as that used by the two mathematicians. And the proof was developed over the weekend, according to Bubeck—that is, after the time that Buckmaster claims the news of his and Alpöge’s result had reached OpenAI. Bubeck emphatically denied that the two mathematicians’ work had influenced OpenAI. “We did not use their prompt or models or proof,” he said.
It will likely take time to hash out the differences between the two proofs, and the progeny of AI’s mathematical ideas are famously difficult to trace. “Yesterday and today are crazy days,” says University of Chicago mathematician Luis Silvestre. “We’re all, in the community, discussing the implications of this.”
Editor’s Note (9/8/26): This is a breaking news story and may be updated.