OpenAI, the US artificial-intelligence company behind ChatGPT, says an unreleased AI system has found a proposed answer to one of mathematics’ hardest unsolved problems. The problem, known as Navier–Stokes existence and smoothness, concerns whether equations used to describe the movement of fluids such as air and water can suddenly break down. OpenAI says its system, using roughly 10,000 AI agents working for 88 hours, found a case in which an initially smooth three-dimensional fluid develops a finite-time singularity—a point at which the equations no longer produce a smooth solution—when acted on by a smooth external force. OpenAI says it produced both a conventional written proof and a version checked with Lean, specialist software used to verify mathematical reasoning. It said the Lean verification took a further 17 hours. OpenAI says the result meets two of the official problem’s conditions for showing that fluid-motion equations can break down. No independent verdict yet The claim has not yet been independently accepted by mathematicians. Under the Clay Mathematics Institute’s rules, a proposed solution must be published in an approved academic outlet, then withstand at least two years of scrutiny and gain broad acceptance in the mathematics community before the institute will consider awarding its $1 million Millennium Prize. OpenAI said it does not intend to seek the prize. Dispute over credit The announcement has also prompted a dispute with Tristan Buckmaster, a mathematician at New York University. Buckmaster said OpenAI began its effort after learning of related work by him and Levent Alpöge, a researcher at Anthropic, the AI company behind Claude. OpenAI said it had heard a rumor that it later understood to refer to the researchers’ work. It denied that its researchers or AI agents saw that work before it was made public, or that they accessed specific user data to solve the problem.
