OpenAI says its AI has solved the Navier-Stokes Millennium Prize problem
San Francisco, September 9: OpenAI said on Tuesday that an unreleased internal AI system has produced a proof of the Navier-Stokes existence and smoothness problem, one of mathematics' seven Millennium Prize Problems. If the proof holds, it would be the second of the seven problems to be solved, and the first by an AI system.
The proof claims that an initially smooth fluid at rest, driven by a smooth force with its energy staying finite, can develop a singularity in finite time. In OpenAI's account, a vortex spirals inward and stretches, its core shrinking as it spins faster. The Clay Mathematics Institute poses the problem in four versions. OpenAI said its system established the two versions that amount to a disproof. The company published the proof with a formal verification in the Lean proof assistant.
OpenAI said the model behind the proof has been in training since late August and is significantly more capable in mathematics than its current GPT-6 Astra system. On September 1, after rumours spread that two Millennium Prize problems had been resolved, OpenAI set the model on all six remaining problems. The list included the Riemann hypothesis and P versus NP. "We didn't expect it to solve any," researcher Noam Brown said.
The effort scaled from roughly 100 agents to as many as 10,000 concurrent AI agents, working for about 88 hours and exchanging 2.7 million messages. OpenAI said the Navier-Stokes work used about 130 billion output tokens and millions of dollars in computing resources. A Lean verification took another 17 hours, the company said.
The announcement quickly drew a dispute over training data. Tristan Buckmaster, a mathematics professor at New York University, said he and Levent Alpöge, a mathematician at Anthropic, had worked for nearly a year on closely related problems. They used OpenAI's Codex and Anthropic's Claude for the work. He said OpenAI's effort began only after the company received information about their work.
Buckmaster said he asked OpenAI whether the model had been trained on, or had access to, their Codex sessions, which held their project drafts. "I was told the model did not look up user data. I asked again, about training, and I did not get an answer," he said in a statement. He said he was making no accusation, and that he was speaking because "the alternative is to let a sequence of announcements say something I know to be false."
OpenAI said its researchers and agents did not see any of the pair's work until it was released publicly. It said no specific user data was accessed to solve the problem. The company also said it cannot rule out that de-identified data from the pair's use of its products helped improve its models. OpenAI said it recognises the priority of their work on the Euler equations, a related problem.
The company said it will not claim the million-dollar Clay Mathematics Institute prize. "Our goal in releasing this result is to report on the substantial progress of our AI models. We do not intend to claim the Millennium Prize," OpenAI said. The Clay Institute named seven problems in 2000, offering one million dollars for each. Only the Poincare conjecture has been solved, by Grigori Perelman, who declined the money.
The Navier-Stokes equations, written in the 19th century by Claude-Louis Navier and George Gabriel Stokes, describe how fluids move. They are used in aircraft design, weather forecasting and the study of blood flow. The Clay Institute continues to list the problem as unsolved on its website.
