OpenAI announced Tuesday that it had solved the Navier-Stokes existence and smoothness problem, one of the seven Millennium Prize Problems in mathematics that has remained open since 1900. The claim, made in a blog post and first reported by The New York Times and Wired, says the company used an internal AI model more powerful than its recently released GPT-6 Astra, paired with 10,000 concurrent agents, to produce a proof. But the announcement landed in the middle of a dispute with two mathematicians who say they were working on the same approach and question whether OpenAI had access to their research.

What the Navier-Stokes Problem Actually Asks

The Navier-Stokes equations describe how fluids and gases move. They are fundamental to physics, engineering, and climate modeling. The Millennium Prize version of the problem asks whether smooth, globally defined solutions always exist for these equations in three dimensions, or whether singularities can form. A $1 million prize from the Clay Mathematics Institute awaits anyone who produces a rigorous proof or counterexample. OpenAI says its model produced a proof and that the company does not plan to claim the prize.

A Competing Claim Surfaces

One day before OpenAI's announcement, New York University mathematician Tristan Buckmaster published findings on a related problem in collaboration with Levent Alpöge, a researcher at Anthropic. Buckmaster says he contacted OpenAI after learning the company was aware of their progress. He found that OpenAI had produced a proof for the Navier-Stokes equation using a mathematical route he and Alpöge had been developing, partly through OpenAI's Codex and Anthropic's Claude.

Buckmaster's central concern is whether OpenAI's model was trained on or had access to their Codex sessions, where they had been storing drafts of their work throughout the project. In his public statement, he says he asked OpenAI directly whether the model had been trained on those sessions and did not receive a clear answer on the training question.

OpenAI's Response and the Training Data Question

OpenAI stated that no specific user data was accessed to solve the problem. The company added that while unlikely, it cannot rule out that de-identified data derived from Buckmaster and Alpöge's usage of its products helped improve its models. Sebastien Bubeck, a member of technical staff at OpenAI, said the team did not see any of Buckmaster and Alpöge's work until it was released publicly, and that the proofs differ significantly in both method and the precise results proved.

Buckmaster responded on Mastodon, arguing that OpenAI was "openly admitting they used training data from a period after we found our result." The exchange highlights a tension that will likely recur as AI companies use their own platforms for research: the line between de-identified training data and direct access to a researcher's working sessions is not always obvious, and the stakes increase dramatically when the work involves a million-dollar prize problem.

What Comes Next

The mathematics community will need to verify the proof regardless of its origin. Navier-Stokes is among the most scrutinized open problems in the field, and any claimed solution will face years of peer review. For developers and AI researchers, the more immediate question is about process. As AI labs increasingly use their own tools for frontier research, the boundary between a model's training data and a user's active workspace will require clearer standards. OpenAI's admission that it cannot fully rule out training data contamination, even in a de-identified form, suggests the company knows this conversation is just beginning.