OpenAI said on Tuesday that an internal, unreleased artificial intelligence model had solved the Navier-Stokes existence and smoothness problem, one of mathematics' seven Millennium Prize Problems, a claim that has since been overshadowed by a public dispute with two mathematicians over how the result came together.
The Navier-Stokes equations describe how fluids move and are used in fields ranging from weather prediction to blood flow modeling. The specific question OpenAI says it resolved is whether these equations can "blow up," meaning break down mathematically, under certain extreme conditions.
What OpenAI Claims
OpenAI said the proof was produced by a multi agent system built on a model it described as significantly more capable than GPT-6 Astra, with as many as 10,000 agents working on different parts of the problem at various points. The company said the effort began on September 1 and reached its result on September 5, about 88 hours later, at a compute cost the company put in the millions of dollars.
The Navier-Stokes problem was one of seven Millennium Prize Problems set by the Clay Mathematics Institute in 2000, each carrying a $1 million award for a correct solution. Only one of the seven had previously been solved. OpenAI also published a formal proof written in Lean, a system used to verify mathematical arguments.
The Credit Dispute
NYU mathematician Tristan Buckmaster and Anthropic researcher Levent Alpöge, who was working independently and not on Anthropic's behalf, had been pursuing closely related research using AI tools including Codex and Claude. Shortly before OpenAI's announcement, the two posted three preprints, verified in Lean, establishing finite-time blowup for a related set of equations.
In a statement accompanying those preprints, Buckmaster said that while he and Alpöge were finalizing their results, they learned that information about their progress had reached OpenAI. He said OpenAI later told him its own model had already produced a roughly 100 page proof of the forced Navier-Stokes problem, but gave inconsistent answers about when its effort had actually started and how much human input was involved.
"It emerged that an entire team had been working on the problem," Buckmaster said, adding that "an insane amount of compute had been used." He also alleged that OpenAI researcher Sebastien Bubeck asked him to remove Alpoge's name from credit as part of a proposed compromise, and warned him against making the dispute public.
Bubeck denied the allegations in a post on social media platform X, calling them "false and inflammatory" and saying he had approached the matter "following academic norms." In a briefing with reporters, he said OpenAI recognized "the priority of Levent Alpoge and Tristan Buckmaster's work" and offered them "nothing but congratulations."
OpenAI has said its effort began after researchers heard rumors that two Millennium Prize problems had been solved, and that neither its researchers nor its agents accessed Buckmaster and Alpoge's unpublished work by any means before it was made public. The company added that it could not entirely rule out that de-identified usage data from its products may have influenced its models.
A Wider Debate in Mathematics
The dispute has drawn comment from beyond the two sides involved. Fields Medalist Terence Tao said the episode reflects "frenetic competition" now surrounding AI and mathematics, warning that "the indiscriminate use of AI is turning the subject into a meaningless production quota game that ultimately is of very little benefit, either to mathematics or to the world."
The Clay Mathematics Institute's official problem page continued to list the Navier-Stokes existence and smoothness problem as unsolved as of September 8, meaning OpenAI's claim has not yet received the institute's formal review, the process that would ultimately determine whether the $1 million prize is awarded.
