OpenAI says an AI system solved the Navier–Stokes Millennium Prize problem

OpenAI put a math claim on the table that has sat open for about ninety years.

On Sept. 8 the company published “On the Navier–Stokes Millennium Prize Problem.” It said an internal system produced a proof that the equations for fluid motion can start smooth and still blow up into a singularity in finite time.

The company says that internal system is “significantly more capable than GPT-6 Astra.” It posted a writeup PDF and a formalization in Lean, the language used to check proofs by machine.

In plain terms: the Navier–Stokes equations describe how fluids move. OpenAI says an initially smooth fluid at rest, with a smooth force applied, can reach unbounded speed in finite time even though viscosity tends to smooth the flow. Energy, the company says, stays finite the whole way.

That result, OpenAI says, is statement “C,” and also “D,” in the Clay Mathematics Institute’s official problem statement. Clay named existence and smoothness of Navier–Stokes one of seven Millennium Prize Problems in 2000.

OpenAI describes the solution as a vortex — a spinning swirl — that spirals inward and stretches “like spaghetti.” The inner region shrinks and speeds up while its energy stays finite.

“We do not intend to claim the Millennium Prize for this result,” the company wrote. Clay has not announced a prize award in this pass. This desk has not independently verified that outside mathematicians accept the Lean formalization.

OpenAI’s page also describes concurrent work by Levent Alpöge and Tristan Buckmaster. That account is OpenAI’s. It is not independently verified here.