OpenAI Cracks the Navier–Stokes Problem — Along With Accusations That It “Peeked” at Someone Else's Ideas

OpenAI has announced a resolution of the problem of the existence and smoothness of solutions to the Navier–Stokes equations — one of the seven Millennium Prize Problems, each of which has carried a million-dollar reward since May 24, 2000. The work was done by an as-yet-unreleased internal model.

By OpenAI's own account, it all began on September 1, when word reached the company that two of the Millennium Prize Problems had already been solved with the help of language models. Inspired by this — and by a sharp jump in the internal model's capabilities — the team unleashed agents on all of the open problems. They arrived at a solution to Navier–Stokes on September 5, roughly 88 hours after the start; another 17 hours went into formalizing and verifying it in Lean, courtesy of GPT‑6 Astra. By OpenAI's reckoning, across all the problems the agents sent 4.9 million messages and burned through about 300 billion output tokens; on Navier–Stokes alone, 2.7 million messages and on the order of 130 billion tokens. Simon Willison notes that at public API prices for GPT‑6 Astra, that volume of tokens would have cost around 15 million dollars.

Just Who Got Beaten to It

The celebration is overshadowed by accusations. Tristan Buckmaster, a mathematics professor at New York University, together with Levent Alpöge (now at Anthropic), had spent nearly a year wrestling with related problems, making heavy use of Claude and Codex — above all the GPT‑5.6 Sol model. They achieved their breakthrough on August 15. Then the mathematical “rumor mill” kicked in: OpenAI learned that Anthropic had supposedly resolved “a major open problem” and got in touch.

According to Buckmaster, when asked point-blank when their team's first query had been sent, they were kept waiting for an answer for a long time — and in the end it was admitted that this had happened in the last few days, after word of their work had already reached OpenAI. When he asked whether the model had been trained on their Codex sessions, where they had stored all the project's drafts, he was told the model does not access user data; a follow-up question about training went unanswered. OpenAI, for its part, maintains that it did not see anyone else's work and that the proofs differ substantially, though it does not rule out that anonymized data may have influenced the model. Meanwhile Alpöge was not invited to co-author — because of the competition with his employer.

A Rumor Is Enough

Willison sees in the story a troubling parallel with the world of cybersecurity: these days a single rumor of a vulnerability is enough to sic agents on hunting it down. The same, it seems, is true of mathematics — the mere knowledge that an unsolved problem exists can trigger millions of dollars in compute spending, just to get there first. Hence his new favorite question: if you enlist ChatGPT's help in solving a Millennium Prize Problem, what is the probability that your work will feed into the training — and that the next model will help someone else solve the problem before you do?

Source: Simon Willison