On the Navier–Stokes Millennium Prize Problem
The proof establishes that smooth 3D fluid motion can develop finite-time singularities, accompanied by a verified Lean formalization.
- A multi-agent network of approximately 10,000 concurrent agents powered by an unreleased internal frontier model produced the proof in 88 hours.
- The Navier-Stokes resolution consumed 2.7 million agent messages and roughly 130 billion output tokens.
- Lean formalization and proof verification were completed in 17 hours using GPT-6 Astra.
- The multi-agent system also proved a disproof of unforced Euler equation regularity in approximately 50 hours.
- OpenAI published the analytical writeups and open-sourced the Lean proofs on GitHub without claiming the Clay Institute prize money.
Mathematicians and fluid dynamicists have formal proof that continuum approximations of fluid motion break down into singularities, marking a major milestone in autonomous AI theorem proving.
