On the Navier–Stokes Millennium Prize Problem

OpenAI announced a proof addressing the Navier–Stokes existence and smoothness problem, one of the seven Millennium Prize Problems, using an internal system more advanced than GPT-6 Astra. The proof demonstrates that initially smooth fluid motion can develop a singularity in finite time, resolving a question unresolved for nearly 90 years. The solution includes both a written proof and a formalization in the Lean theorem prover, marking a significant advancement in mathematical and computational research.

The Navier–Stokes equations, dating to the 19th century, describe fluid motion using Newton’s second law and treat fluids as continuous media. A longstanding question has been whether these equations can break down, leading to infinite fluid speeds within finite time despite viscosity. The proof shows such a singularity can occur, requiring the equations to model fluid behavior at the particle level beyond the singularity point.

The solution was produced by a multiagent system of approximately 10,000 concurrent agents, which explored diverse approaches and consolidated insights to arrive at the proof. The agents operated under strict safeguards and used tools including cached internet access and code execution. The effort began on September 1 and culminated in the proof on September 5, with Lean formalization completed the following day.

The proof resolves statements C and D of the Millennium Prize formulation by demonstrating a vortex that spirals inward, elongates, and accelerates while maintaining finite energy. The agents also resolved a related Euler equations problem, further validating their approach. OpenAI coordinated a concurrent release with researchers who independently developed a similar solution.