OpenAI presents proof for the Navier–Stokes problem

Summary

OpenAI says it has resolved the three-dimensional Navier–Stokes existence and smoothness problem, one of the most important open mathematical questions about fluid behaviour. The company has released a 166-page analytical proof and a formalisation in Lean, a language used to verify mathematical proofs by computer.

The result constructs a fluid initially at rest that, under a smooth external force, develops unbounded velocity in finite time while keeping its kinetic energy bounded. This corresponds to alternatives C and D in the Clay Mathematics Institute’s official formulation: under the conditions constructed in the proof, a smooth solution to the equations can cease to remain smooth.

In practice

Navier–Stokes equations describe fluid motion and underpin models used in aerospace, weather forecasting, ocean science and blood-flow research. The central question was whether viscosity — the effect that tends to smooth fluid motion — always prevents singularities from forming in three dimensions.

OpenAI’s construction uses a vortex that spins, contracts and stretches. As the central region shrinks, velocity grows without bound while total energy remains finite. The challenge was not simply producing diverging velocity; it was ensuring that the external force stayed smooth. The proof uses oscillations within the fluid to cancel terms that would otherwise make that force singular.

Context

According to OpenAI, the proof was found by an internal system more capable than GPT-6 Astra, coordinating roughly 10,000 agents in parallel. Work began on September 1, reached a Navier–Stokes resolution around 88 hours later, and used approximately 130 billion output tokens. Lean formalisation and verification took an additional 17 hours.

The company says it does not intend to claim the one-million-dollar prize associated with the problem. It has released the proof and formalisation code for external mathematical scrutiny. Independent review by specialists remains decisive: a proof of this scale is only fully established after examination by the mathematical community.

Why it matters

• This is a demonstration of AI applied to frontier mathematical research, rather than only solving exercises or assisting with software development.
• The use of thousands of coordinated agents shows a form of automated research at a very different scale from a single model answering a prompt.
• If the proof withstands scrutiny, it resolves a question open for roughly 90 years and changes the mathematical understanding of when these equations stop describing a continuous fluid.