On the Navier–Stokes Millennium Prize Problem

This article announces that an internal AI system developed by OpenAI has resolved the Navier–Stokes existence and smoothness problem, one of the seven Clay Mathematics Institute Millennium Prize Problems. The system produced an analytical proof and a formalization in the Lean theorem prover demonstrating that a smooth, incompressible fluid starting from rest can develop a singularity in finite time under a smooth, finite-energy force. This means the velocity of the fluid can grow without bound, marking a breakdown in the continuum model.

The proof was achieved using a multi-agent system powered by a new internal model more capable than GPT‑6 Astra. The effort involved coordinating up to 10,000 concurrent agents that explored diverse approaches, with insights cross-pollinated between groups. The solution was reached in about 88 hours, with formal verification taking an additional 17 hours. The system also resolved a related "easier" problem concerning the regularity of the forced Euler equations.

The article clarifies that the work began after rumors of progress on Millennium Problems and notes concurrent, independent work on the forced Euler problem by Levent Alpöge and Tristan Buckmaster. OpenAI emphasizes that this milestone demonstrates significant AI progress but states they do not intend to claim the Millennium Prize, framing the result as a snapshot of ongoing development focused on building responsible and steerable AI.

(Source：OpenAI)