Proved in Just 88 Hours! AI Closes In on the Navier-Stokes Millennium Problem

Hi, it's me, Shiichan! Today's story is on a scale that genuinely surprised me.

OpenAI News openai.com

What was announced?

OpenAI News announced a research result related to the Navier-Stokes equations, one of the Clay Millennium Prize Problems. A multi-agent system built on an OpenAI internal model produced a significant result on this problem.

Why it matters

The Navier-Stokes equations, derived by Navier and Stokes in the 19th century, are the fundamental equations describing fluid motion. The question of whether a smooth, three-dimensional incompressible fluid flow can develop a singularity — a point where the equations break down — in finite time has remained open since Leray's partial proof in 1934. In 2000, the Clay Mathematics Institute named it one of seven Millennium Prize Problems.

What changes

What OpenAI announced is a result showing that a fluid starting at rest, subjected to a smooth external force, can develop a singularity in finite time. This corresponds to the forced variants of the official problem statement (Statement C and D). The resulting solution takes the shape of a vortex-like structure that spirals inward and accelerates "like spaghetti," while its energy stays finite, consistent with physical law.

However, OpenAI explicitly states that it does not intend to claim the Millennium Prize for this result. What was solved is the forced version, distinct from the original unforced version, which is considered the harder open question. Even so, OpenAI chose to share it as evidence of how far its AI models' capabilities have progressed.

Dive Deep

The discovery process was pretty extraordinary too. Starting August 28, OpenAI trained a new internal model, and nearly 100 agents working together first resolved the regularity question for the unforced Euler equations (the simpler equations you get by dropping the viscosity term from Navier-Stokes) in about 50 hours. That result became a strategic guide for the later Navier-Stokes effort.

For the main event, roughly 10,000 concurrent agents explored different approaches and cross-pollinated ideas, arriving at a resolution on September 5 — about 88 hours after the first agents were launched. From there, formalizing and verifying the proof in Lean (a proof assistant) took another 17 hours, with the full project wrapping up on September 6. That verification step used GPT-6 Astra.

The resources consumed were massive too: across both the Euler and Navier-Stokes efforts combined, the system sent 4.9 million messages and produced roughly 300 billion output tokens. For Navier-Stokes alone, that's 2.7 million messages and about 130 billion output tokens.

Wrap-up

• A multi-agent system built on OpenAI's internal model proved a result on the Navier-Stokes Millennium Prize Problem: finite-time singularity formation in the forced version
• Solving the unforced Euler equations' regularity question first served as a stepping stone
• The resolution took about 88 hours, plus another 17 hours for formal verification in Lean
• OpenAI explicitly says it won't claim the Millennium Prize, since the result covers the forced version of the problem
• A must-read if you're interested in open problems in mathematics, or how far AI reasoning and symbolic math capabilities have come