Technology
OpenAI Claims Breakthrough on Navier-Stokes Problem Amid Plagiarism Accusations From NYU Professor
OpenAI has announced that an unreleased artificial intelligence model produced a proof for the Navier-Stokes existence and smoothness problem, a fluid dynamics puzzle that has challenged mathematicians for roughly 90 years and carries a $1 million prize from the Clay Mathematics Institute. The company assigned about 10,000 AI agents to the task, and they generated a proof within 88 hours.
The announcement, made Tuesday, was quickly overshadowed by accusations from Tristan Buckmaster, a mathematics professor at New York University, who published his own related findings just hours earlier. Buckmaster claims OpenAI's system may have drawn on unpublished research that he and his collaborator had been feeding into Codex, OpenAI's coding tool, over several months.
The Navier-Stokes equations describe the motion of liquids and gases and underpin applications such as aircraft design, weather forecasting and blood flow modeling. The central unresolved question was whether these equations could allow a smooth-starting fluid to reach infinite speed in finite time, a condition known as a singularity. OpenAI's proof asserts that this is possible, describing a vortex that spirals inward while accelerating, with its energy remaining finite throughout.
According to OpenAI, the internal model, described as more capable than GPT-6 Astra, began training on August 28. On September 1, after hearing that two Millennium Prize problems might have been solved, the company directed its agents toward the remaining problems. The team working on Navier-Stokes exchanged roughly 2.7 million messages and generated about 130 billion output tokens before completing the proof on September 5. Verifying the proof in the Lean formal language took an additional 17 hours. Estimates for the total computing cost of the project range from $10 million to $22.5 million. OpenAI has said it will not seek the prize money.
Buckmaster contends that no AI model could independently arrive at this specific proof method within four days. Working with Levent Alpöge, a mathematician affiliated with Anthropic in a personal capacity, Buckmaster used both Codex and Claude to prove on August 15 that the Euler equations, a simplified version of Navier-Stokes without friction, do exhibit blow-up behavior. Their method relied on a technique called forcing, developed by Diego Córdoba and Luis Martínez-Zoroa, which Buckmaster says few other researchers were exploring.
Buckmaster published his findings just before midnight on Monday, alongside a statement describing conversations with OpenAI researcher Sébastien Bubeck. He said that when he asked about how the AI model was trained, he received no clear answer. Buckmaster further alleged he was offered sole credit on a paper that would have excluded Alpöge, and that when he threatened to make the matter public, he was warned it could harm his career.
OpenAI has stated that no specific user data was used to solve the problem and that neither its researchers nor its AI agents had access to Buckmaster and Alpöge's work before it was published. However, the company acknowledged it cannot rule out the possibility that anonymized data from users' interactions with its products may have influenced its models. Bubeck has denied requesting that Alpöge's name be removed from any paper, and OpenAI CEO Sam Altman has said the two research approaches appear distinct now that both are public.
The Clay Mathematics Institute has not yet verified either proof, and some mathematicians note that the version OpenAI solved incorporates a term that is often excluded from the traditional formulation of the problem. As a result, neither claim has been definitively confirmed.
The episode has raised broader concerns among mathematicians about the use of AI tools on unpublished research. Terence Tao, who has warned about the implications for the field, compared the situation to outsourcing physical effort: "It's like having machines that can lift weights for you at the gym."