We’re sharing a solution to the Navier–Stokes existence and smoothness problem, one of the Millennium Prize Problems.

This proof, produced by an internal OpenAI system, shows that the dynamics of the Navier-Stokes equations for fluid motion can develop a singularity in finite time.

We’re sharing both a writeup of the proof and a formalization in Lean.

The Millennium Prize Problems represent some of the deepest questions at the frontier of mathematics.

The question of whether smooth three-dimensional fluid motion can break down has remained unresolved for roughly 90 years.

A major goal of our work is to empower scientists to advance research and technology that benefits all of humanity.

To solve the Navier–Stokes problem, we used an internal model that is significantly more capable than GPT‑6 Astra.

We believe it is important to inform the world about the pace of AI progress and what to expect from upcoming models.

The problem The Navier–Stokes equations use Newton’s second law of motion (“F=ma”) to describe how fluids move.

Importantly, they treat a fluid as a continuous medium rather than tracking individual molecules.

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