OpenAI said its internal system has produced a solution to the Navier–Stokes existence and smoothness problem, one of the Clay Mathematics Institute’s Millennium Prize Problems, and has released both a paper and a formalized proof in Lean. According to the company, the result shows that under smooth forcing, a three-dimensional incompressible fluid with smooth initial data at rest can develop a singularity in finite time while keeping total energy finite throughout the process. OpenAI said this corresponds to statements C and D in the Clay Institute’s official formulation, but stressed that the work still needs independent review by the academic community. The company also disclosed details of the computation behind the result, saying it used an internal model that is significantly stronger than GPT-6 Astra. Around 10,000 parallel agents worked for roughly 88 hours to produce the proof, exchanging 2.7 million messages and generating about 130 billion output tokens, with Lean formalization and verification taking another 17 hours. OpenAI added that it does not plan to claim the Millennium Prize and clarified a separate misunderstanding involving Levent Alpöge and Tristan Buckmaster, whose work was on the forced Euler equations rather than the Navier–Stokes problem.
OpenAI said its internal system has produced a solution to the Navier–Stokes existence and smoothness problem. The company described the result as follows: under smooth forcing, a three-dimensional incompressible fluid with smooth initial data at rest can develop a singularity in finite time, while total energy remains finite throughout the process.
OpenAI said the result matches statements C and D in the Clay Mathematics Institute’s official formulation of the problem. It has released both a paper and a Lean formalized proof, though it said the work still awaits independent scrutiny from the academic community.
The company said the project used an internal model that is significantly stronger than GPT-6 Astra. About 10,000 parallel agents worked for roughly 88 hours to produce the proof, sending 2.7 million messages and generating about 130 billion output tokens. Lean formalization and verification took another 17 hours.
OpenAI also said it does not intend to claim the Millennium Prize.
On related work announced around the same time, OpenAI said it had initially believed that Anthropic employee Levent Alpöge and New York University professor Tristan Buckmaster had also solved the same problem. It later said the two had in fact solved a forced Euler equations problem, and acknowledged their priority on that result.
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