OpenAI says it has solved the existence and smoothness problem for the Navier-Stokes equations, one of the seven Millennium Prize problems worth one million dollars. The result was announced while two mathematicians were working on the same route, and it is already overshadowed by a dispute over authorship.
According to the company, the proof was produced by an internal model that OpenAI describes as "significantly more capable than GPT-6 Astra". The work was coordinated by about 10,000 AI agents over 88 hours. The proof was formalized in Lean, the language that lets a computer check mathematical proofs. Research lead Mark Chen said the compute alone cost "in the millions of dollars".
Image: OpenAI — https://openai.com/about/
The problem is more concrete than it sounds. The Navier-Stokes equations describe how fluids such as water and air move. They have been used for decades, but it was not known whether, under some conditions, they could predict impossible states, such as infinite velocity. OpenAI's proof addresses exactly that point. The Clay Institute, however, requires a complete answer, and independent review will be needed to confirm it.
The dispute stems from the history of the work. NYU mathematician Tristan Buckmaster says he had worked for over a year on the solution with Levent Alpöge, an Anthropic researcher. Buckmaster says OpenAI twice asked him to remove Alpöge from authorship because he works at a rival company. When he refused, an OpenAI researcher allegedly asked him: "Why would you ruin your career?" The company rejected the accusations, and in a press briefing said its team had itself been inspired by rumours about the two researchers' work. Whether OpenAI's models actually used that material remains unclear.
On the scientific side, the case calls for caution. An aerospace engineering professor noted that the proof may concern a very specific scenario, and that many claims of having "solved Navier-Stokes" are overblown. Independent verification by the mathematical community will be the decisive test.
The episode also shows how AI is changing research. When work requires enormous amounts of compute, the competitors are mostly a few companies with very large resources, and the informal norms of academic collaboration matter less. For mathematicians, the next question is how to work alongside systems like these.
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