OpenAI says an internal model still in training proved in 88 hours that the Navier-Stokes equations can develop a finite-time singularity, a question the report describes as having remained open for about 90 years and as one of the Millennium Prize problems.
About 12 hours before OpenAI’s paper was released, New York University mathematics professor Tristan Buckmaster and Anthropic researcher Levent Alpöge had already published closely related results. What might have been a straight math breakthrough turned into a public fight over priority and credit.
OpenAI says its internal system found the proof in 88 hours
In its official announcement, OpenAI said an internal model under active training showed that the Navier-Stokes equations, which describe fluid motion, can produce a singularity in finite time. In the plain-language framing used in the report, that means fluid velocity shoots toward infinity at a certain moment and the equation breaks down from that point onward.
The Navier-Stokes equations date back to the 19th century and are described as an extension of Newton’s second law, F=ma, to fluids. Aircraft design, weather forecasting, and blood-flow research all rely on them. The long-standing question is whether a flow that starts out smooth can still suddenly blow up, with velocity becoming unbounded in finite time.
The report notes that in 1934, French mathematician Jean Leray proved the existence of solutions but did not prove that those solutions remain smooth forever. OpenAI said its announcement spells out the technical conditions under which a singularity exists.
The core construction in the announcement is a vortex
The answer, as described in the announcement, is built around a vortex that contracts inward while stretching out, becoming thinner like a strand of noodle while its rotation rate surges. Total energy, however, stays finite until the velocity crosses every bound at a single moment.
The hard part, according to the report, was getting every term in the equation tied to acceleration, pressure, momentum, and viscosity to grow on its own while canceling each other out in just the right way, leaving only a smooth external force. The point was to avoid the shortcut of inserting an infinite force by hand.
OpenAI said the result came from a collective system of about 10,000 agents. It also said the internal model used for the task is clearly stronger than the publicly available GPT-6 Astra, and that training is still ongoing and performance is still improving.
Buckmaster and Alpöge published related work first
The dispute surfaced before publication. OpenAI said in its announcement that on Sept. 1 it heard rumors that two Millennium problems had been solved, which triggered a broad internal review. It later concluded that the source of the rumor was Buckmaster and Alpöge.
According to OpenAI’s account, the two had already proved on Aug. 15 that the forced Euler equations blow up in finite time, and they made that result public on Sept. 7, about 12 hours before OpenAI’s release.
The report says Buckmaster and Alpöge used about 100 agents over roughly 50 hours, relying on Claude and OpenAI Codex together. OpenAI also said publicly that it recognizes their priority on the forced Euler problem and congratulates them.
The dispute centers on access to unpublished work and authorship
Buckmaster’s public version was much less conciliatory. He said he had asked OpenAI whether its model had been trained on, or had access to, their work sessions in Codex, and that he did not receive an answer.
He also alleged that on a Sept. 6 call, OpenAI math lead Sébastien Bubeck offered him a deal: Buckmaster could take sole authorship on the result if he agreed to state that OpenAI’s internal model had solved Navier-Stokes and removed Alpöge, who works at Anthropic, from the paper.
Buckmaster said he rejected that proposal and insisted on going public. He said he was then asked, 「Why would you ruin your own career?」 and told, 「If you’re not polite, I don’t need to be polite either.」
Bubeck rejects the allegations
Bubeck later responded publicly, calling the allegations 「false and inflammatory」 and saying, 「We did not use their prompts or proof to prompt our model or guide our agents.」
The report adds that Bubeck later also publicly acknowledged Alpöge and Buckmaster’s priority on the forced Euler result and congratulated them.