This week, OpenAI declared that it had solved the Navier-Stokes problem, one of the seven famous challenges posed by the Clay Mathematics Institute in 2000, and the response from mathematicians has not been one of celebration. Each of these seven problems comes with a $1 million reward, yet the announcement has instead provoked widespread unease throughout the discipline.
The dispute revolves around how OpenAI reached its milestone. Two mathematicians who sit at the center of recent disputes told The Verge that the company’s actions have shaken up the field and prompted questions about how it handles individual researchers. According to OpenAI, it spent roughly 10,000 agents, tens of millions of dollars of compute, and 88 hours to arrive at the solution. It says it first learned of the problem through a rumor that researchers were making headway.
The Navier-Stokes Problem and the Millennium Prizes
Fluid motion is the subject of the Navier-Stokes problem, a mathematical challenge of great difficulty that has engaged researchers across the globe for many years. In 2000, the Clay Mathematics Institute announced its Millennium Prize problems, a set of seven questions judged to be among the hardest in the field.
A reward of $1 million attaches to each prize. Just one has been collected since the quarter-century began: the Poincaré conjecture, which deals with three-dimensional spheres from the field of topology.
The company heard about the issue through word that researchers were making headway on it. OpenAI then chose to test whether one of its advanced, unreleased models might be able to do the same. That model was able to.
Buckmaster’s Account of What Happened Next
An NYU professor named Tristan Buckmaster and Levent Alpöge, a researcher at Anthropic, were working on the Navier-Stokes problem together. Neither of them had finished a proof at that point. The two sides largely agree on the order of what followed, even though some of the specifics remain hotly disputed.
OpenAI learned about Buckmaster’s work and began racing toward its own solution, which prompted Buckmaster to reach out to the company. He described conversations with OpenAI researcher Sébastien Bubeck as contentious and, in his view, threatening. The company presented him with an option for moving forward — one that left Alpöge out.
“All I had to do was throw Levent under the bus,” Buckmaster told The Verge in a phone interview. He said he flatly rejected Bubeck’s offer, which he viewed as a “bribe,” and also began questioning whether OpenAI may have benefited from his use of Codex, one of the company’s AI tools he had been using to tackle the problem.
His office, he said, had been transformed into something of a “war room,” with colleagues helping scrutinize his mathematics, coordinate outreach, and even get in touch with lawyers.
OpenAI’s Response to the Allegations
OpenAI spokesperson Laurance Fauconnet strenuously denied that material from Buckmaster’s prompts had played a role in the solution. “We can say categorically that it is impossible for Dr. Buckmaster’s Codex prompts over the last two months to have influenced the system in any way, including training,” he said.
Buckmaster remains unconvinced. “Given their behavior up until this point, one should take such statements with great skepticism,” he said.
OpenAI has denied that anyone or any agent accessed Buckmaster’s specific user data. The company acknowledged until its more recent comments that it could not rule out the possibility that data derived from his use of the products was used to improve the model.
Bubeck’s Version of the Story
Sébastien Bubeck has pushed back against Buckmaster’s account of the conversations. He acknowledged offering OpenAI’s resources to help Buckmaster complete his own proof or to have him take over the writing of the company’s. He said OpenAI had made similar arrangements with other mathematicians, though he did not identify them.
Bubeck also noted that Alpöge’s affiliation with Anthropic was a sticking point. From OpenAI’s perspective, he said, the question was how the company could have an internal project with an Anthropic employee.
Why Mathematicians Are Unease
Careful and deliberate is how mathematics usually comes across. Not like this. OpenAI has spent vast sums of compute power and agents on a single problem, and it reads to many as an impossibly well-resourced interloper charging into problems researchers have dedicated their lives to studying.
To them, the company looks more like an interloper than a newcomer with enthusiasm, showing little respect for established norms or the people left behind by its actions. The feeling is that OpenAI has different reasons for doing mathematics.
Mathematicians want to advance the field. OpenAI wants to win.
“Given their behavior up until this point, one should take such statements with great skepticism.”
What Comes Next for Mathematics
OpenAI’s actions have unsettled the field and made everyone afraid of what it might do next in its determination to trounce its rivals. The Navier-Stokes solution stands as a historic achievement in any normal circumstance. It now comes amid dispute and accusation.
How the company handles individual researchers and whether it respects long-standing norms in the field are now open questions. Those queries are part of the record.