Nature reported that the proof shows the equations, the most commonly used physical model of fluids, can break down: an initially smooth three-dimensional fluid develops a singularity in finite time, with part of it accelerating without bound.

OpenAI said the result establishes statements C and D of the official problem formulation, and released a written proof and a Lean formalization for outside researchers to check.

The effort began Sept. 1, after OpenAI researchers heard rumors that two Millennium problems had fallen elsewhere.

Roughly 100 agents spent about 50 hours disproving regularity for the simpler Euler equations, then some 10,000 agents worked 88 hours on the full Navier-Stokes system, with a further 17 hours for a second model to verify the result in Lean.

The agents exchanged almost five million messages across the project, and executives told reporters the computing bill ran into the millions of dollars.

Quanta Magazine reported that if the result survives scrutiny it would be, by a significant margin, the most important mathematical proof produced by an AI model, possibly marking a turning point in how mathematicians attack difficult problems.

The Clay Mathematics Institute, which set the seven problems in 2000 with a $1 million award attached to each, has accepted nothing and still lists the question as unsolved.

OpenAI logo is seen in this illustration created on June 11, 2026. Photo by Reuters

Tristan Buckmaster of New York University, who published closely related results 12 hours earlier, said word of his unpublished work reached OpenAI three days before the company began its own effort.

Buckmaster said in a statement published late on Sept. 7 that he and Levent Alpöge, a mathematician employed by Anthropic, had proved several related fluid results using models from both companies, including OpenAI's Codex programming tool.

He said he learned on Sept. 3 that word of their progress had reached OpenAI, and that the company turned to the equations only afterwards.

Axios reported that Buckmaster questioned whether private Codex material played a role, and alleged that OpenAI researcher Sébastien Bubeck pushed to drop Alpöge from authorship because he works for OpenAI's chief competitor. When Buckmaster said he would make their exchanges public, Bubeck allegedly replied: "Why would you ruin your career?"

OpenAI said it saw none of the pair's work before its public release and accessed no user data, while conceding it could not rule out that de-identified traces of their tool use had helped train its models.

Bubeck has rejected the allegations as false and inflammatory, and the company maintains the two proofs differ substantially.

Both AI-assisted teams leaned on analytic techniques developed by Diego Córdoba of the Institute for Mathematical Sciences in Madrid and Luis Martínez-Zoroa of CUNEF University, whose approach broke from the computational methods most researchers were using, Quanta reported.

Buckmaster wrote that Martínez-Zoroa deserves a Fields Medal.

Martin Bridson, president of the Clay Mathematics Institute, told Nature it was "certainly an exciting day" for human understanding of mathematics, though the institute's rules require a proposed solution to be published in a qualifying outlet, remain in print for at least two years and win general acceptance among mathematicians before an award is considered.

OpenAI has said it does not intend to claim the money.

Only one Millennium Prize Problem has been resolved. Grigori Perelman's proof of the Poincaré conjecture took years to verify, and Perelman turned down the prize.

The equations were posed by the French engineer Henri Navier in 1822 and given their final form by George Gabriel Stokes in 1845. They describe how gases and liquids move and underpin weather forecasting, aircraft design and the study of blood flow.

The regularity question has stood open for roughly 90 years, since Jean Leray showed in 1934 that solutions exist in a generalized sense without settling whether they always stay smooth.

Terence Tao, the University of California, Los Angeles mathematician who won the Fields Medal in 2006, told The New York Times that "these questions are lighthouses," focal points that draw human effort, and warned that letting AI solve them could hollow out mathematicians' understanding of their own field.