OpenAI has claimed a major breakthrough in mathematics after an internal artificial intelligence system produced a solution to the Navier-Stokes existence and smoothness problem, one of the seven Millennium Prize Problems that has challenged mathematicians for nearly 90 years.
The company said its AI agents reached the result in about 88 hours, before another model spent a further 17 hours formally verifying the proof. OpenAI said up to 10,000 AI agents were deployed during the effort, consuming millions of dollars in computing resources.
AI tackles one of mathematics’ toughest questions
The Navier-Stokes equations are fundamental to understanding how fluids such as water and air move. The longstanding problem asks whether apparently smooth solutions to the three-dimensional equations can develop a singularity, where the mathematical description effectively breaks down.
OpenAI says its proof demonstrates that such a finite-time singularity can occur, meaning the equations can produce a situation in which fluid speed becomes unbounded.
The company has released a 165-page mathematical proof along with a formalised version in Lean, allowing researchers to examine and verify the argument independently.
Breakthrough still awaits wider mathematical scrutiny
Despite the significance of the claim, the result has not yet been formally recognised as a solution by the wider mathematical community. The Clay Mathematics Institute, which established the Millennium Prize Problems, requires a proposed solution to undergo extensive expert scrutiny before the $1 million prize can be awarded.
The announcement has also triggered controversy after mathematician Tristan Buckmaster raised questions about whether OpenAI’s work may have benefited from research he and Anthropic researcher Levent Alpöge had been conducting. OpenAI has denied using their unpublished work.
OpenAI has said it does not intend to claim the associated $1 million prize, instead presenting the work as evidence of the rapidly advancing capabilities of AI in scientific research.