AI-assisted Navier-Stokes breakthrough sparks credit fight between OpenAI and Anthropic-linked researchers

A dispute over one of the most celebrated unsolved problems in mathematics erupted on Tuesday after NYU mathematician Tristan Buckmaster published a statement accusing OpenAI of rushing to claim credit for work that built on his own research — and of pressuring him to cut his collaborator from authorship because of ties to a rival AI company.

The Math

Buckmaster and Levent Alpöge, a mathematician employed by Anthropic, spent roughly a year using AI tools to extend a research program pioneered by Diego Córdoba and Luis Martínez-Zoroa. Their work, published late Monday along with formal verification in the Lean proof assistant, demonstrated that several fluid equations closely related to Navier-Stokes — including the 3D incompressible Euler equations — can develop singularities in finite time under smooth forcing. Fields Medalist Terence Tao called the results "a remarkable achievement" and said he sees "no fundamental obstacle" to extending the methods to the full Navier-Stokes equations.newscientist+2

The Navier-Stokes equations, which describe how fluids move, are one of the seven Millennium Prize Problems identified by the Clay Mathematics Institute, each carrying a $1 million reward. Mathematicians have struggled for two centuries to determine whether the equations' solutions always remain smooth or can blow up into singularities.scientificamerican

The Dispute

According to Buckmaster's public statement, rumors of his and Alpöge's progress reached OpenAI around September 3. Days later, OpenAI scientist Sébastien Bubeck told him an internal model had produced a roughly 100-page proof extending the result to the full forced Navier-Stokes equations. Buckmaster alleges the company's effort began only after learning of his work, and that it followed the same narrow technical approach — targeting the "forcing" term in the Clay problem's formal statement — that he and Alpöge had been quietly pursuing.officechai+1

Buckmaster says Bubeck twice pushed to exclude Alpöge from authorship, citing his employment at Anthropic as a complication. When Buckmaster said he would go public, he alleges Bubeck replied, "Why would you ruin your career?" He also says he asked whether OpenAI's model had been trained on his private Codex session data, which contained draft proofs. He says he received no clear answer.scientificamerican+1

Bubeck responded on X, calling the allegations "false and inflammatory" and saying he had acted "following academic norms." He promised a fuller statement.officechai

What Remains Unresolved

OpenAI's claimed Navier-Stokes proof has not been made public or verified by outside mathematicians. Even the verified Buckmaster-Alpöge results address forced versions of related equations, not the unforced Navier-Stokes problem as most experts interpret the Millennium Prize. David Silvester at the University of Manchester noted the current work may nonetheless be enough to claim the prize, while Camilla Nobili at the University of Surrey cautioned that extending to full Navier-Stokes will require more than simply rerunning the same technique.glitchwire+3

"Nothing will change in the applications where Navier-Stokes is used because of this result," Silvester said. "It's a mathematical nicety, honestly." The larger question — whether competing AI labs will distort the norms of mathematical discovery — may prove harder to resolve than the equations themselves.newscientist