On September 8, 2026, OpenAI announced that an internal artificial intelligence model solved the forced version of the Navier-Stokes existence and smoothness problem. The announcement, accompanied by a 165-page proof with formal verification in Lean, sparked an academic controversy within hours over data use and authorship. The company called the result a milestone for AI research.
What is the Navier-Stokes problem?
The Navier-Stokes equations describe the motion of fluids such as water, air, or blood. They are used in aeronautics, meteorology, and blood circulation studies, among many other fields.
The central Millennium question is whether a fluid starting with finite energy can develop a singularity, meaning a velocity that grows without bound in finite time. OpenAI claims its model demonstrated that such a situation is possible.
The announcement and its scope
The company deployed about ten thousand artificial intelligence agents over eighty-eight hours. The system consumed one hundred thirty billion tokens on this problem alone and completed formal verification in Lean in seventeen additional hours. The model used has not been released to the public and, according to OpenAI, is significantly more capable than GPT-6 Astra.
Despite the technical scope, the Clay Mathematics Institute keeps the problem listed as open. Its rules require publication in a qualified venue, a minimum period of two years, and explicit acceptance by the international mathematical community.
Buckmaster’s allegation
Hours before the announcement, Tristan Buckmaster, a mathematics professor at New York University, stated that he and Levent Alpöge, a researcher at Anthropic, had spent nearly a year working on the same problem with Codex, OpenAI’s tool.
Buckmaster alleged that on September 3 he discovered that information about their progress had been transmitted to OpenAI. He added that the company proposed he publish the result alone, excluding Alpöge from authorship.
OpenAI’s response
OpenAI denied accessing specific data from the researchers before publication. In the same statement, it acknowledged that it cannot rule out that anonymized data derived from the use of its products may have helped improve its models.
The company called Buckmaster and Alpöge’s work extraordinary and stated that its approach was significantly different. A spokesperson added that the allegations of pressure are categorically false.
A precedent for research
The controversy goes beyond the specific case. It raises the question of who owns the ideas a researcher develops while using artificial intelligence tools as support.
The case also sets a precedent on the use of anonymized data in model training and on the transparency that artificial intelligence companies must offer to the academic community.
What remains to be verified?
The Clay Institute has not validated the proof. Community review may take years, and only then will it be recognized whether the solution is correct.
OpenAI stated it will not claim the one-million-dollar prize. The controversy over data use and authorship continues without independent resolution.
