OpenAI proves the Navier-Stokes equations have a finite time singularity

On September 8, OpenAI announced that its internal multi-agent system has successfully proven that the three-dimensional incompressible Navier-Stokes equations can form singularities in finite time, thus solving this century's problem. The proof was driven by a model within OpenAI that is more powerful than GPT-6 Astra and obtained preliminary results on September 5, followed by 17 hours of Lean formal verification.

OpenAI stated that the proof demonstrates that under the influence of smooth external forces, a stationary fluid can evolve into inward spiraling and stretching vortex structures, with energy remaining finite but speed growing unbounded, corresponding to cases "C" and "D" in the Millennium Prize Problems. The entire solving process involved about 10,000 concurrent agents, sent approximately 2.7 million messages, and consumed about 130 billion output tokens. OpenAI acknowledged that this attempt was inspired by rumors of parallel work by Anthropic employee Levent Alpöge and NYU professor Tristan Buckmaster, noting that the two parties arrived at different results regarding the regularity problem of the Euler equations, with OpenAI proving the case without external forces, while the other party proved the case with external forces.