Has the Navier–Stokes proof been checked by anyone?

The Navier–Stokes proof carries a Lean formalization, which Scientific American reports all but guarantees its correctness; OpenAI says formalization and verification took an extra 17 hours using GPT-6 Astra. Mathematicians quoted in that piece, including Diego Córdoba and Luis Silvestre, say the community is still working through what the result implies.

Is OpenAI claiming the $1 million Millennium Prize?

OpenAI states it does not intend to claim the Millennium Prize for this result. The Clay Mathematics Institute attaches a $1 million award to each of its seven Millennium Prize Problems. OpenAI frames the Navier–Stokes release as a report on the pace of AI progress rather than a prize submission.

What is the dispute over credit about?

NYU professor Tristan Buckmaster and Anthropic researcher Levent Alpöge had been working on related fluid-dynamics questions. OpenAI says its effort began on September 1 after hearing a rumour, that it offered a joint announcement, and that it then learned the pair had resolved the forced Euler problem. OpenAI states its researchers and agents saw none of their work and that the proofs differ.

Can I read the Navier–Stokes proof myself?

OpenAI published the Navier–Stokes writeup as a PDF, a second PDF for the Euler regularity result, and the Lean certificates in the openai/NavierStokesAndEuler repository under Apache-2.0. That repository was created on September 8, 2026 and is written in Lean, so anyone with a Lean toolchain can re-check the formal argument.

What else did the agents solve?

Alongside Navier–Stokes, the agent groups disproved regularity for the unforced Euler equations, the same blowup question with the viscosity term removed. Nearly 100 agents worked about 50 hours on that one. Across every problem attempted, the agents exchanged 4.9 million messages and used roughly 300 billion output tokens.

OpenAI · 2026-09-08 · seismic

Navier–Stokes — OpenAI agents produce a Lean-checked blowup proof

OpenAI says an internal model, running as about 10,000 coordinated agents, produced a proof that 3D Navier–Stokes fluid flow can blow up in finite time. The proof ships as a paper plus a Lean formalization on GitHub.

OpenAI announcement graphic for the Navier–Stokes Millennium Prize proof

OpenAI's agents produced, and Lean verified, a finite-time blowup proof for the 3D Navier–Stokes equations.

Key specs

Concurrent agents	~10,000
Time to proof	88 hours

Quick facts

Maker	OpenAI
Model used	Unreleased internal model, more capable than GPT-6 Astra
Result	Finite-time singularity for 3D Navier–Stokes (statements C and D)
Verification	Lean formalization, 17 hours via GPT-6 Astra
Artifacts	Proof paper (PDF) + Lean certificates (Apache-2.0)
Bonus result	Euler regularity disproof, ~100 agents in ~50 hours
Prize	OpenAI says it will not claim the $1M Millennium Prize

What is it?

OpenAI published a proof that the 3D Navier–Stokes equations can develop a singularity in finite time — fluid speeds growing without bound even though viscosity tends to smooth motion out. The solution is a vortex that spirals inward and stretches like spaghetti, shrinking and speeding up while its energy stays finite. The Clay Mathematics Institute named this question one of seven Millennium Prize Problems in 2000, and it had stood open for roughly 90 years.

How does it work?

A system of coordinating agents ran on an internal OpenAI model that has been training since August 28 and is described as significantly more capable than GPT-6 Astra. Agents were split into groups with tools to run code and read a cached copy of the internet, and each group was prompted with a different variant of the problem. Codex then merged the most useful findings across groups. About 10,000 concurrent agents worked the Navier–Stokes case; a separate group of nearly 100 agents first disproved regularity for the unforced Euler equations, and that result was fed back as a hint.

Why does it matter?

The Lean formalization is what separates this from a plausible-looking argument: the proof is machine-checked, so its correctness does not rest on trusting a model's prose. For working mathematicians it means an AI system now produces results at the level of an open century-old problem, with the writeup and the Lean certificates published for anyone to review. OpenAI says it will not claim the $1 million prize and frames the release as a progress report on how fast its models are improving.

Who is it for?

mathematicians and AI researchers

Frequently asked questions

Has the Navier–Stokes proof been checked by anyone?: The Navier–Stokes proof carries a Lean formalization, which Scientific American reports all but guarantees its correctness; OpenAI says formalization and verification took an extra 17 hours using GPT-6 Astra. Mathematicians quoted in that piece, including Diego Córdoba and Luis Silvestre, say the community is still working through what the result implies.
Is OpenAI claiming the $1 million Millennium Prize?: OpenAI states it does not intend to claim the Millennium Prize for this result. The Clay Mathematics Institute attaches a $1 million award to each of its seven Millennium Prize Problems. OpenAI frames the Navier–Stokes release as a report on the pace of AI progress rather than a prize submission.
What is the dispute over credit about?: NYU professor Tristan Buckmaster and Anthropic researcher Levent Alpöge had been working on related fluid-dynamics questions. OpenAI says its effort began on September 1 after hearing a rumour, that it offered a joint announcement, and that it then learned the pair had resolved the forced Euler problem. OpenAI states its researchers and agents saw none of their work and that the proofs differ.
Can I read the Navier–Stokes proof myself?: OpenAI published the Navier–Stokes writeup as a PDF, a second PDF for the Euler regularity result, and the Lean certificates in the openai/NavierStokesAndEuler repository under Apache-2.0. That repository was created on September 8, 2026 and is written in Lean, so anyone with a Lean toolchain can re-check the formal argument.
What else did the agents solve?: Alongside Navier–Stokes, the agent groups disproved regularity for the unforced Euler equations, the same blowup question with the viscosity term removed. Nearly 100 agents worked about 50 hours on that one. Across every problem attempted, the agents exchanged 4.9 million messages and used roughly 300 billion output tokens.

Try it

git clone https://github.com/openai/NavierStokesAndEuler