Today may go down as the biggest day in math in at least two decades, if not far longer.

In a development that crystallizes an existential shift in humankind’s oldest intellectual discipline, artificial intelligence has reportedly solved one of the six open “million-dollar” problems in mathematics. The achievement proves that the equations that govern the motion of fluids are fundamentally flawed.

“This is a Deep Blue–Kasparov moment,” wrote mathematician Tristan Buckmaster, referring to the sea change in thinking about machine intelligence that occurred when a supercomputer beat world champion Garry Kasparov at chess in the 1990s, in a statement released late on Monday. “The community needs to have serious and unhurried discussion about where to go from here.”

On supporting science journalism

If you're enjoying this article, consider supporting our award-winning journalism by subscribing. By purchasing a subscription you are helping to ensure the future of impactful stories about the discoveries and ideas shaping our world today.

But according to Buckmaster, the proof’s origins may be murkier than they seem. In the same statement, he wrote that he and Levent Alpöge, a mathematician and an Anthropic employee, made significant progress toward the problem last month. But before the pair could publish, Buckmaster alleged in the statement, rumors of their method reached OpenAI, where researchers used the company’s large language model (LLM) to finish the job with a single prompt. The prompt “had been sent in the past few days, after information about our work had reached OpenAI,” Buckmaster alleged.

OpenAI did not immediately respond to a request for comment; in a press conference later on Tuesday, the firm denied the allegations.

The mathematical problem concerns the Navier-Stokes equations, which govern the flow of fluids—from the whirlpool circling your bathtub’s drain to the turbulent winds El Niño hurls across the North American continent. After two centuries of study, mathematicians haven’t been able to figure out whether these equations perfectly describe reality in every situation or if they sometimes admit aberrant mathematical blips that could never happen in a real fluid. Answering this quandary has been viewed as so beyond the reach of existing mathematics that the Clay Mathematics Institute has offered $1 million to anyone who does it.

Over the past few years, two mathematicians, Diego Córdoba and Luis Martínez-Zoroa, worked out a potential trick to break the equations—or “blow them up,” as such an achievement is described in math. They called the method “forcing.” It focuses on a part of the equations, as written in the Clay problem, that mathematicians previously deemed inconsequential. In fact, most experts typically write the problem without this term, assuming that any blowup should be the same with or without it. But Córdoba and Martínez-Zoroa realized there might be a way to break the equations solely by using this often overlooked piece.

About a year ago, Buckmaster took up Córdoba and Martínez-Zoroa’s approach, working with Alpöge and using LLMs from OpenAI’s rival company Anthropic to parse through the mathematical possibilities. Progress was slow, according to Buckmaster’s statement, until August 15, when he and Alpöge used the method to prove that the Navier-Stokes equations’ simpler, frictionless cousins, the Euler equations, did in fact blow up.

This finding, on its own, is a monumental mathematical achievement that is worthy of any award in math and a key step toward solving the Navier-Stokes problem. Buckmaster and Alpöge were able to verify that the proof was correct using the programming language Lean, but the English-language explanation the LLMs produced was barely readable, according to Buckmaster. The pair started trying to make sense of the mathematical steps one by one and wrote them in a way other researchers would be able to digest.

According to Buckmaster, rumors of their work reached OpenAI at some point in the past week, and the company apparently had a team that was already working on the problem. But after becoming aware of the pair’s work, he alleged, OpenAI focused its attention on “forcing”—the often-ignored piece of the Clay problem. In a major effort over last weekend, the OpenAI team apparently used an internal model to take the result further, blowing up the full Navier-Stokes equations, according to Buckmaster.

Buckmaster wrote in his statement that he spoke on the phone on Sunday with Sébastien Bubeck, who leads OpenAI’s math team, and another unnamed OpenAI employee. Buckmaster alleged that the call became contentious, with OpenAI offering him sole authorship for the Navier-Stokes result in a paper that would acknowledge that the company’s internal model had solved it. Bubeck allegedly wanted to exclude Alpöge, an employee of OpenAI’s competitor, according to Buckmaster. He asserted that he threatened to inform the media and alleged that “the reply was ‘Why would you ruin your career?’” according to the statement.

At 11:58 P.M. EDT on Monday, Buckmaster posted his and Alpöge’s results, along with the statement laying out his account of events. According to the statement, he does not know how the OpenAI team so quickly reconstructed the details of its approach. Buckmaster wrote that he asked Bubeck if the OpenAI LLM was given access to his personal prompts to the company’s model Codex, which he and Alpöge were using throughout the process and thought were private. Bubeck told him no but wouldn’t answer whether the model’s training involved user data, which might have included Buckmaster and Alpöge’s private prompts, Buckmaster alleged.

Beyond the jockeying for credit among the AI vanguard and the implications for the future of mathematics as a human vocation, there’s a mathematical issue. The Clay problem, as written, is solved. But the Clay problem, as many experts imagine it, lacks the piece that the forcing method relies on. Moreover, because this element is so crucial to the method, it’s not clear whether there’s any way to extend the proof to blow up the Navier-Stokes equations without it. Some in the community may say that the problem hasn’t really been solved or that this has only been achieved through a loophole. That leaves the Clay Institute, and the field of mathematics, in a bit of a quandary. In any case, Buckmaster is emphatic that the key ideas be credited to Martínez-Zoroa and Córdoba. “I believe Luis Martínez-Zoroa deserves a Fields Medal,” he wrote in his statement.

Ultimately, math may never be the same. AI companies have fixated on using their technology to solve math problems for more than a year. Because the field’s focus on objectivity aligns with these companies’ goal of developing superhuman intelligence, they’ve been pushing their models to do math that no human can faster than any human can. It appears they’ve succeeded.

“There is a far bigger story here than the one in my statement: the sheer magnitude of what frontier models can now do, and what that means for us all,” Buckmaster wrote on Mastodon a little before 6 A.M. EDT on Tuesday. “I hope the labs can see this and set the petty posturing aside.”

Editor’s Note (9/8/26): This is a breaking news story and will be updated.