OpenAI Publishes Solution to Navier-Stokes Millennium Problem and Lean Proof

OpenAI has announced that its internal system has provided a solution to the existence and smoothness problem of the Navier-Stokes equations: under smooth external forces, an initially at rest and smooth three-dimensional incompressible fluid can form singularities in a finite amount of time, while the total energy remains finite throughout the process. This result corresponds to propositions C and D as stated by the Clay Mathematics Institute, and the paper along with the Lean formal proof has been made publicly available, pending independent review by the academic community. The project utilized an internal model significantly stronger than GPT-6 Astra, with about 10,000 parallel agents forming the proof in approximately 88 hours, sending 2.7 million messages and generating around 130 billion output tokens, with an additional 17 hours spent on Lean formalization and verification. OpenAI has no intention of claiming the Millennium Prize. Initially, OpenAI mistakenly believed that Anthropic employee Levent Alpöge and New York University professor Tristan Buckmaster had also solved the problem, but later confirmed that they had addressed the forced Euler equations problem and acknowledged their priority on that issue.

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