OpenAI has published a formal proof related to the Navier-Stokes problem, days after NYU mathematician Tristan Buckmaster raised concerns that the lab had drawn on information about his group's near-complete work to speed up its own effort.
The Navier-Stokes equations describe how fluids such as water and air move, and understanding their behaviour underpins fields ranging from weather forecasting to aircraft design.
A specific unsolved question about these equations, concerning whether solutions can break down in finite time, is one of the Clay Mathematics Institute's seven Millennium Prize Problems, a set of challenges considered among the most significant unsolved problems in mathematics.
Only one of the seven has ever been solved, which is part of why a credible proof, by a human mathematician or an AI system, would represent a genuinely significant result rather than an incremental academic milestone.
Two groups, two claims
Buckmaster and Anthropic researcher Levent Alpöge separately announced three AI-assisted proofs of their own, saying they used both Codex and Claude during their work.
OpenAI says an unreleased next-generation model delivered its own full proof after a concentrated run consuming 300 billion output tokens, at an estimated computing cost of $22.5 million.
Where the disagreement lies
Buckmaster has framed the timing of the two efforts as more than coincidence, saying, "There is another part of this story, and one that, honestly, I very much wish I did not have to be concerned with."
He has said OpenAI personnel asked him to remove Alpöge's credit from related material, and that an OpenAI researcher suggested he could "ruin his career" if he went public with his concerns.
OpenAI disputes that account, saying its internal team began its own effort on 1 September, that its researchers "did not see any of their work," and that "no specific user data was accessed" during the process.
The company has acknowledged, however, that it cannot entirely rule out its model benefiting indirectly from de-identified usage data, while noting that its published proof differs from Buckmaster's own.
What the dispute is really about
Beyond the specific timeline, the disagreement touches on broader questions the AI research community has yet to settle: how credit should be assigned when multiple teams use AI tools to accelerate the same problem.
There is also a discussion as to whether access to enormous computing resources gives some labs an unfair advantage in a race for prestige, and who ultimately benefits when AI systems begin contributing to genuine mathematical discovery rather than simply assisting with it.
For now, both sides agree on one thing: that a landmark problem in mathematics has, in some form, moved closer to resolution, even as who deserves the credit for that progress remains unresolved.
