OpenAI Announces Navier-Stokes Proof as Clay Prize Review Continues

OpenAI announced a claimed solution to the three-dimensional Navier-Stokes problem, a $1 million Millennium Prize question about whether smooth fluid motion can develop unbounded velocity despite viscosity. The Clay Mathematics Institute described the problem as apparently settled, but its formal review has not concluded.

The equations model the movement of fluids. The open question asks whether smooth three-dimensional flow can develop unbounded velocity in finite time despite viscosity. It is one of seven Millennium Prize Problems, each carrying a $1 million prize.

The paper presents a solution for every positive viscosity, beginning from rest and reaching unbounded velocity in finite time while keeping kinetic energy bounded. It addresses alternatives C and D in the official problem statement. A formalization in Lean, a programming language used to check mathematical proofs, followed the result.

The effort began Sept. 1, and the system reached its result Sept. 5 after about 88 hours. Formalizing and checking the proof in Lean took an additional 17 hours. OpenAI reported about 10,000 concurrent agents, which exchanged 2.7 million messages and generated about 130 billion output tokens on the effort.

The announcement follows other AI-assisted mathematics claims. On Aug. 1, OpenAI said an internal version of Astra produced arguments for 10 longstanding problems in mathematics and theoretical computer science. Humans prepared the manuscripts, and the arguments were formalized in Lean. On Sept. 21, the company announced that a new internal model had resolved more than 100 open problems across most areas of mathematics, without naming the problems or publishing proofs in that announcement.

OpenAI also established an independent mathematics advisory group to advise on reviewing and communicating emerging results, their significance, and academic and professional standards. The group will not advise on the pace of the company’s internal math research.

Questions about research credit have also surfaced. Mathematicians Tristan Buckmaster and Levent Alpöge had released results on forced blowup for the incompressible porous media, Boussinesq and three-dimensional Euler equations. Buckmaster said he had not seen OpenAI’s proof, did not know what the model did or how, and did not know whether his and Alpöge’s data had been used. He said he was not accusing anyone of anything.

The Clay Mathematics Institute said Sept. 11, “The process is deliberately unhurried, but we will provide updates.” Its prize review has not concluded.