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OpenAI says an internal model has cracked one of the hardest problems in math. NYU mathematician Tristan Buckmaster says the story he was told about how it happened came apart while he was on the phone with the company.
Buckmaster says Martínez-Zoroa deserves a Fields Medal
OpenAI said Tuesday one of its internal models had demonstrated finite-time “blow-up” for the forced Navier-Stokes equations, one of seven Millennium Prize Problems, each with a $1 million reward.
Blow-up means that the equations allow the speed of a fluid to become infinite at a point, which is not allowed by physics.
The company says the argument was checked in Lean, a proof-verification language used to formally confirm mathematical proofs.
If it holds up, it would be only the second Millennium problem ever solved, and the first credited to an AI company.
Days earlier, Buckmaster and Levent Alpöge, a mathematician working for Anthropic, had published their results: finite-time blow-up under smooth forcing for 3D incompressible Euler, Boussinesq, and incompressible porous media, made formal in Lean. “Remarkable achievement,” said Fields Medalist Terence Tao.
That part of the story is not contested. Buckmaster himself is quite emphatic in his statement as to the source of the ideas.
The line of attack, he wrote, originates with Diego Córdoba and Luis Martínez-Zoroa, who spent years building forced blow-ups using rough forcing.
He and Alpöge used large language models to push that program to smooth forcing and Euler. Buckmaster went further to say he thought Martínez-Zoroa should be awarded a Fields Medal.
Bubeck once asked, “Why would you ruin your career?”
OpenAI’s announcement came a day after Buckmaster’s account, which describes a scramble that started when a rumor began circulating that Anthropic had solved a major open problem.
He emailed a mathematician at OpenAI on Thursday, September 3rd, to avoid confusion, emphasizing that his work with Alpöge was a personal collaboration and was not supported by Anthropic or NYU. He read through the answer. It was warm and even offered OpenAI compute.
The calls came on Sunday, September 6. Buckmaster says Sébastien Bubeck came on, the two sides had a couple of conversations that afternoon, and Alpöge was not on the line.
He was told that an internal OpenAI model had yielded a ~100-page proof of forced Navier-Stokes blowup, using smooth forcing along “options c and d” in Charles Fefferman’s formal statement of the problem.
That, Buckmaster wrote, was the very narrow way he and Alpöge had secretly selected, one that nearly no one else was following. He said hearing “forced” was a bright red flag.
“Very little human input” went into the OpenAI result, Bubeck told Alpöge, Buckmaster says.
Buckmaster wrote, “it emerged that an entire team had been working on the problem, that this was one of a number of things that was tried, that work had started on the unforced problem, that the team first set the model on easier problems, including Euler, that even the prompt that had been shown to me had been written by prompting Codex, and that an insane amount of compute had been used.”
He says that when he asked when the first prompt was sent, the answer came after a delay, and he eventually put it after the news of his and Alpöge’s work had reached OpenAI.
Buckmaster says he and Alpöge had kept their draft proofs inside OpenAI Codex sessions and that when he asked whether OpenAI’s model might have been trained on that data, he got no answer.
He has plainly stated that he is not claiming it happened; he is only stating that the question went unanswered.
He also alleges Bubeck twice pushed to remove Alpöge as an author due to his Anthropic ties and once asked him, “Why would you ruin your career?”
OpenAI denied the allegations. Bubeck called the claims “false and inflammatory” in a short post on X, said he had entered the debate “following academic norms,” and promised a more complete response.
At a briefing on Tuesday, Bubeck said the model had reached the Euler problem in a different way than the mathematicians, but that the complete Navier-Stokes proof did take a similar route.
The proof was built over the weekend, after the point Buckmaster says news of his result reached OpenAI, he said.
“We did not use their prompt or proofs to prompt our models,” he said.
Both sides draw on Córdoba’s approach, who said the pair are “a little bit in shock” and that a completed proof “will be a big surprise for us.” University of Chicago mathematician Luis Silvestre called it the days of nonstop discussion across the field.
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