# 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.

## Why it matters

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.

## Sources

- [OpenAI: On the Navier–Stokes Millennium Prize Problem](https://openai.com/index/navier-stokes-solution)

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Summarized by dstilled on 2026-09-08. https://dstilled.ai/story/db7b030c-dce3-4c9e-ba6f-ae5168e730f3