New Delhi: OpenAI Tuesday announced that one of its artificial intelligence systems had found a solution for the Navier-Stokes problem, a fluid-dynamics puzzle that has defeated mathematicians for about 90 years. The problem is one of the seven Millennium Prize Problems, a list of the most important unsolved questions in mathematics drawn up by the Clay Mathematics Institute in 2000, each carrying a $1-million reward. Before this week, only one of the seven had been solved, and never by a machine.
And that is what has stunned the field—that AI has now made serious progress on a problem the world’s best mathematicians have been unable to crack.
In a statement announcing his own related work, NYU mathematician Tristan Buckmaster called it “a Deep Blue-Kasparov moment,” a reference to the day a computer first beat a reigning world chess champion.
He said the real significance of this breakthrough is that a single mathematician working with AI can now do in weeks what once took years, a shift, he said, that will force the field of mathematics to rethink how it trains students, assigns credit and checks results.
But the announcement has been overshadowed by a bitter dispute: Buckmaster says he and a colleague reached a closely related result first, and has accused OpenAI of chasing the problem only after learning of their work and then pressuring him over who gets the credit.
What is the puzzle, and why it matters
The problem is simple to state but has proved fiendishly hard to solve. The Navier-Stokes equations, written in the 19th century, are the rulebook for how fluids move, whether water, air or oil. Scientists and engineers rely on them every day to design aircraft and ships, forecast the weather, model ocean currents and simulate blood moving through arteries. When a computer predicts how air will flow over a wing, it is solving these equations.
The unsolved question is about how far that rulebook can be trusted. If a fluid starts out in a smooth, calm state, will the equations always keep producing sensible, finite answers no matter how long the fluid keeps moving? Or can they, at some moment, produce what mathematicians call a “blowup”, a point where the calculation says the fluid’s speed becomes infinite?
Real water in a real pipe never reaches infinite speed. If the equations throw up infinity, it means the equations themselves have broken down and stopped describing reality at that moment. The question is not whether a tap will suddenly explode, but whether the mathematics that sits underneath vast areas of science and engineering is sound, or whether it hides points where it fails.
These equations underpin weather and climate models, aircraft and engine design, and the study of turbulence, one of the oldest unsolved puzzles in physics. Knowing whether they can break down, and under what conditions, would tell scientists where their most important tool can be relied on and where it cannot. Settling the question either way has long been seen as one of the deepest challenges in mathematics, which is why the Clay Institute announced a $1-million prize for it.
What OpenAI announced
OpenAI said the proof was the work of a swarm of AI agents rather than a single model answering a question. “We’re sharing a solution to the Navier-Stokes Millennium Prize Problem, one of the deepest problems at the frontier of mathematics,” the company wrote on X. In a briefing it said about 10,000 AI agents had run for 88 hours, at a cost of millions of dollars, using an unreleased model more advanced than its current systems.
OpenAI said the effort began around 1 September, after it heard rumours that two millennium problems had been solved; it also said it does not intend to claim the $1-million prize.
The mathematics establishment welcomed the news but urged caution. Martin Bridson, president of the Clay Mathematics Institute, called it “an exciting day” for human understanding of mathematics. The institute, however, has not accepted OpenAI’s proof, and Navier-Stokes still sits on its list of unsolved problems.
Two things explain the caution. First, OpenAI’s result covers a version of the problem in which the fluid is pushed by an outside force, whereas the version most people think of as the prize concerns a fluid left to itself, held together only by its own internal friction, or viscosity. Second, the full proof has not been independently checked by mathematicians outside the company, and several said they would treat it as a claim until it was.
The rival claim
Hours before OpenAI’s announcement, Buckmaster published a four-page statement saying he and a collaborator had reached a closely related result first, using the same line of attack.
Buckmaster works at NYU’s Courant Institute and won the Clay Research Award in 2019, shared with Vlad Vicol and Philip Isett, for earlier work on fluid equations, where he helped show that some of their solutions can behave far more wildly than had been assumed.
Buckmaster’s current collaborator, Levent Alpoge, is a mathematician employed at Anthropic, an OpenAI rival, though both men say they worked together privately, with no company or university behind them.
According to Buckmaster’s statement, the pair spent about a year on the problem using several AI tools, including Anthropic’s Claude and OpenAI’s Codex.
This week they released three related papers, on the incompressible porous media equation, the Boussinesq system and the 3D Euler equations, a simpler, friction-free cousin of Navier-Stokes.
On 15 August, the statement says, they proved that the Euler equations can “blow up”, and they had the proof checked by Lean, a programme that verifies every logical step of a proof, on 22 August.
The work drew praise from Terence Tao, a Fields Medal winner and one of the most respected living mathematicians, who called it “a remarkable achievement” and said the same methods had a strong chance of reaching the full Navier-Stokes problem.
Buckmaster was blunt about the three papers, which he said were rushed out under pressure. He described the first AI-written proof as “the most horrendous I have ever read” and called one write-up “AI slop”.
The dispute
According to Buckmaster’s statement, a rumour began spreading on 3 September that Anthropic had solved a major unsolved mathematics problem. Worried it referred to his own work, he emailed a mathematician at OpenAI to explain that the project was a personal collaboration with “no formal agreement behind it”. He said the reply was friendly and even offered free computing power.
On 6 September, Buckmaster said, he spoke twice with OpenAI scientist Sebastien Bubeck and another representative, and was told that an internal OpenAI model had produced a roughly 100-page proof, taking the argument all the way to the full Navier-Stokes problem, using the same route he and Alpoge had been quietly following. Buckmaster said he has not seen that proof.
Buckmaster said he was first told the model had solved the problem with “very little human input”, but that during the calls it emerged a full team had worked on it, that several approaches had been tried, that the model had first been set easier problems including Euler, and that “an insane amount of compute” had been used.
He said he and Alpoge had stored their draft work inside OpenAI’s Codex tool throughout the project, and that when he asked whether the company’s model had used that material he was told it “did not look up user data”, but got no answer when he asked about training.
Buckmaster said he was offered two options: That the pair post their Euler result and OpenAI post its Navier-Stokes result the next day, or that Buckmaster alone write up the Navier-Stokes result and credit an OpenAI model.
He said Bubeck twice argued that Alpoge should be left off the paper because he works at Anthropic, and that when he said he would go public he was asked, “Why would you ruin your career?” He set limits on his account. “I am not accusing anyone of anything,” he wrote. “I am stating what I was told, when, and what was proposed to me.”
OpenAI and Bubeck respond
OpenAI rejected the account. In a statement it congratulated Buckmaster and Alpoge on their work and said its researchers and AI agents “did not see any of their work” before it was published, and that no specific user data was used. It added that it could not fully rule out that anonymised data from their use of its products had indirectly helped train its model, and stressed that the two proofs are different.
Bubeck disputed Buckmaster’s version in two posts. He first called the allegations “false and inflammatory” then wrote, “I never ever asked for Levent to be removed from authorship of his own work.” He added that the discussion had been about whether Buckmaster could lead a rewrite of OpenAI’s own proof.
He admitted the remark about Buckmaster’s career was an “extremely poor choice of words”, and said he apologised and withdrew it at once. OpenAI chief executive Sam Altman defended him, saying Bubeck “acted with integrity and generosity throughout”.
(Edited by Viny Mishra)
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