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 “Million Dollar” problems in mathematics, proving that the equations that govern the motion of fluids are fundamentally flawed.
“This is a Deep Blue–Kasparov moment,” mathematician Tristan Buckmaster wrote in a statement published late on Monday, referring to the sea change in thinking about machine intelligence that occurred when a supercomputer beat the human world champion Gary Kasparov at chess in the 1990s. “The community needs to have serious and unhurried discussion about where to go from here.”
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But according to Buckmaster, the proof’s origins may be murkier than they seem. In the same statement, Buckmaster wrote that he and Levent Alpoge, a mathematician and Anthropic employee, made significant progress toward the problem last month. But before the pair could publish, rumors of their method reached OpenAI, Buckmaster alleges, 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 alleges in the statement.
OpenAI did not immediately respond to a request for comment.
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 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 offered a million dollars to anyone who does it.
Over the last few years, two mathematicians, Diego Córdoba and Luis Martínez-Zoroa, worked out a potential trick to break the equations—or “blow them up,” as it’s called in math. They called the method “forcing.” It focuses on a part of the equations as written in the Clay problem that mathematicians have 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 Alpoge, using large language models from OpenAI’s rival company Anthropic to parse through the mathematical possibilities. Progress was slow, according to Buckmaster’s statement, until August 15, when they 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, worthy of any award in math, and a key step toward solving the Navier-Stokes problem. They 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 write 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 last week. OpenAI apparently had a team already working on the problem, according to Buckmaster. But after becoming aware of the pair’s work, he alleges, OpenAI focused their 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 writes 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 alleges 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 Alpoge, an employee of OpenAI’s competitor, according to Buckmaster. Buckmaster states that he threatened to inform the media, and alleges 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 Alpoge’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 their approach. Buckmaster asked Bubeck if the OpenAI LLM was given access to his personal prompts to the company’s model Codex, which he and Alpoge 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 Alpoge’s private prompts, Buckmaster alleges.
Beyond the jockeying for credit between 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, since 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 it only has 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 the statement.
Ultimately, math may never be the same. AI companies have fixated on using their technology to solve math problems for over a year. Because the field’s focus on objectivity aligns with their goal of developing superhuman intelligence, they’ve been pushing their models to do math 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 6:00 A.M. EDT on Tuesday. “I hope the labs can see this and set the petty posturing aside.”
Editor's note: This is a breaking news story and will be updated.