OpenAI has published a 165-page proof claiming a solution to the Navier-Stokes existence and smoothness problem, one of the seven Millennium Prize Problems. The paper constructs an example for 3D incompressible fluid where the initial state is fully smooth, the fluid starts at rest, external forces remain smooth, and kinetic energy stays finite, yet velocity becomes unbounded in finite time, producing a singularity. According to OpenAI, this establishes variants C and D from the official Millennium Prize formulation. The proof was also formally verified in Lean.

The solution was developed by a multi-agent system of approximately 10,000 agents, consuming around 130 billion output tokens across 2.7 million messages over roughly 88 hours for the proof search, plus 17 hours for Lean formalization and verification. OpenAI notes the primary proof was obtained by an internal model substantially more capable than GPT-6 Astra and still in training.

Navier-Stokes Solution

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