OpenAI's $1m math claim sparks controversy

OpenAI announced Tuesday (local time) that its AI had solved the Navier-Stokes equation problem, one of the Millennium Prize Problems carrying a $1 million reward. But the announcement has snowballed into controversy after a rival research team — which includes a mathematician from Anthropic — revealed it had already proven closely related results. Allegations have also surfaced that OpenAI tried, days before the announcement, to strike a deal to have the Anthropic mathematician's name removed.

The Navier-Stokes equations describe the motion of fluids — liquids, gases and plasmas. They are widely used in fluid dynamics, engineering and other applied fields. However, except in cases drastically simplified by additional assumptions, it is extremely difficult to derive exact answers using the equations alone. Much about their mathematical properties in general cases also remains unknown.

Specific instances of the equations have virtually limitless practical applications. But the Navier-Stokes equation problem itself matters not for practical reasons but for mathematical ones.

The Navier-Stokes equation problem asks researchers to prove — or find a counterexample to — basic properties of the equations' solutions, such as existence, smoothness and uniqueness, in fairly general cases.

The Clay Mathematics Institute put up $1 million prizes in 2000 for solving each of seven such problems, including this one. Only one of the seven has been solved to date, leaving the remaining six — including the Navier-Stokes problem, whose resolution has yet to be verified — officially unsolved.

The Clay Mathematics Institute's website describes the problem this way: "This is the equation that governs the flow of fluids such as water and air. But there is no proof for even the most basic questions one can ask about it: Do solutions exist, and are they unique? Why ask for a proof? Because proofs give not only certainty but understanding."

Because of the problem's nature, there is no simple yes-or-no standard for determining whether it has been solved. The Clay Mathematics Institute has instead laid out more detailed criteria in a paper written by Charles Fefferman, a Princeton University mathematics chair professor and leading authority in the field.

OpenAI Chief Research Officer Mark Chen announced Tuesday that the company had solved the Navier-Stokes equation problem. He called it "a critical milestone in AI research that shows great promise for the world, in that it will let us find answers to more of the most difficult problems we face."

OpenAI said an AI model it began training Aug. 28 solved the problem within just days. The model, used internally, performed considerably better than GPT-6 Astra, the company's commercial flagship model launched Thursday, according to OpenAI.

OpenAI said its AI proved that a singularity can form within finite time in fluid motion governed by the Navier-Stokes equations. The presence of a singularity means the AI found a counterexample to the equations' "smoothness" property.

OpenAI said its internal model reached the solution on Saturday, 88 hours after the first agents began operating. Formalizing and verifying the proof using the Lean proof assistant, run through GPT-6 Astra, took another 17 hours, with the proof completed Sunday.

OpenAI said its agents exchanged about 2.7 million messages and used roughly 130 billion tokens to solve the Navier-Stokes problem. The company added that it attempted all six of the Clay institute's unsolved problems, including this one, with its agents exchanging a combined 4.9 million messages and using about 300 billion tokens in the process.

OpenAI said the result satisfies the criteria the Clay institute set for solving the Navier-Stokes equation problem.

The claim, however, has not yet been reviewed by outside experts or recognized by the Clay institute. The institute's criteria require that a solution be widely accepted as correct by the mathematics community, a process that takes at least two years from the time the work is published in a peer-reviewed journal.

But a priority dispute erupted Tuesday, the day of OpenAI's announcement, after a rival researcher raised questions, and controversy has grown following allegations that OpenAI attempted to strike a deal before the announcement.

Tristan Buckmaster, a professor at New York University's Courant Institute of Mathematical Sciences, and Levent Alpoge, a mathematician at Anthropic, posted a preprint — a draft paper released before peer review — on Monday claiming to have solved three problems closely related to the Navier-Stokes equation problem.

Hours after posting the preprint, at 11:58 p.m. Eastern time, Buckmaster published a statement online. He said that beyond the three results verified with Lean and posted in the preprint that day, he believes he has also obtained a result proving finite-time blowup for the hypo-dissipative Navier-Stokes equations, but has not yet posted that as a preprint because the Lean verification is not complete.

He said he had made progress on the related problems since Aug. 15 and had been working intensively on them. Buckmaster said that after rumors spread that Anthropic had solved a major unsolved problem, Alpoge told him there were rumors that OpenAI had obtained information about their progress. He then emailed a prominent mathematician working at OpenAI on Thursday.

OpenAI proposed setting up a meeting the same day he sent the email, and the actual conversation took place on Sunday afternoon, Buckmaster said. He said OpenAI put forward two proposals for the two sides to coordinate on releasing their research findings, both of which he rejected.

Buckmaster said that during the negotiations, Sebastien Bubeck, a mathematician at OpenAI, twice asked him to remove Alpoge, the Anthropic mathematician, from the list of paper authors.

Buckmaster's claim that OpenAI made a remark that could be interpreted as a threat has caused a particularly large stir. Buckmaster said, "I said that if OpenAI released its results in the manner it proposed, I would disclose what had happened. The response was, 'Why would you want to ruin your own career?' I pointed out that I was an academic and asked, 'Why would disclosing the truth ruin my career?' The answer to that was, 'If you don't want me to look good, then I don't need to make you look good either.'"

He added, "I want to be clear about what I am not claiming. I have not seen OpenAI's proof. I do not know what their model did or how it did it. I do not know whether our data was used." He said he was simply disclosing when things were said and what was proposed to him.

He said, "The reason I am saying this is that otherwise I would be letting a series of announcements from OpenAI put out content I know to be false." He added that if OpenAI had truly succeeded in solving the problem, it should simply say so loudly and directly.

In response to the allegations, OpenAI told US media outlets that there was no misappropriation of unpublished data or research results, and that it never asked for anyone to be removed from the author list. The company did apologize for the remark about ruining one's career, calling it inappropriate.

OpenAI added, "We cannot rule out the possibility that de-identified data generated as they (Buckmaster and Alpoge) used our products helped improve our models."