OpenAI Says AI System Solved the Navier–Stokes Millennium Prize Problem

OpenAI says an internal multi-agent AI system has produced an analytical proof and Lean formalization showing that a smooth, initially resting fluid can develop a finite-time singularity under a smooth force. The company says the result resolves the Navier–Stokes existence and smoothness problem, one of seven Millennium Prize Problems carrying a $1 million award, although it does not intend to claim the prize. OpenAI’s technical account describes a vortex that spirals inward and stretches while its energy remains finite. The equations, developed in the nineteenth century by Claude-Louis Navier and George Gabriel Stokes, have long challenged mathematicians because it was unknown whether smooth three-dimensional flows could break down in finite time. The Clay Mathematics Institute designated the problem in 2000. OpenAI says its effort began on September 1 after researchers heard rumors that two Millennium Prize Problems had been resolved. It deployed coordinating groups of agents with access to cached internet material and code execution, using approximately 10,000 concurrent agents on the Navier–Stokes task. The system reached its resolution on September 5, about 88 hours after launch; Lean formalization and verification took another 17 hours via GPT-6 Astra. The Navier–Stokes effort generated approximately 2.7 million messages and used about 130 billion output tokens. Across all attempted problems, the system produced 4.9 million messages and used about 300 billion output tokens. OpenAI also reports that nearly 100 agents resolved an unforced Euler-equation regularity problem in approximately 50 hours, helping guide the later Navier–Stokes work. The company says different agent groups explored separate proof and disproof formulations before Codex consolidated useful intermediate results. The announcement has been met with substantial interest because it presents AI as a potential research collaborator on a problem that has resisted generations of mathematicians. It also faces an important limitation: the proof remains a company claim that must be examined by independent experts, and mathematical acceptance is distinct from an AI-generated result or formalization. The announcement was accompanied by controversy involving NYU mathematician Tristan Buckmaster and Anthropic researcher Levent Alpöge, who had been working on related forced-Euler research. OpenAI says it did not access their specific user data and that the proofs differ significantly, while acknowledging that it cannot entirely rule out de-identified data from product use having indirectly helped improve its models. Nature’s report described the claim as a potentially historic computer-assisted breakthrough, while The Washington Post noted that mathematicians will closely scrutinize the released technical paper. The episode therefore combines a major test of automated mathematical reasoning with unresolved questions about verification, attribution, data governance and the scale of compute required for frontier scientific work.