AI appears to have achieved another major milestone, potentially one of its most significant breakthroughs yet. On Tuesday, OpenAI announced that an unreleased AI model had solved the Navier-Stokes existence and smoothness problem, one of the seven Millennium Prize Problems that have remained among mathematics’ most challenging unsolved questions since 2000. However, New York University professor Tristan Buckmaster has alleged that OpenAI may have relied on data from his research with Levent Alpoge, a mathematician at Anthropic, while reaching its result.
OpenAI said it used a new, unreleased AI model to tackle the problem. According to the company, the model is significantly more capable than even GPT-6 Astra – which some have described as AGI. For the Navier-Stokes problem, OpenAI said it deployed up to 10,000 AI agents working collectively for 88 hours and processing billions of tokens.
The Navier-Stokes equations, formulated in the 19th century by Claude-Louis Navier and George Gabriel Stokes, explain the movement of liquids and gases. They have applications in fields such as aircraft design, weather prediction and research into blood flow.
The Millennium problem asks whether smooth solutions to the equations describing three-dimensional fluid motion always remain smooth, or if they can eventually break down. OpenAI said its proof demonstrates that “an initially smooth fluid at rest can develop a singularity in a finite time.” In simple terms, a spaghetti-like vortex continually shrinks as it spins faster, with its velocity increasing without bound in a phenomenon mathematicians refer to as finite-time blowup.
Mathematician argues OpenAI may have used his data
The announcement has quickly turned into both a major mathematical development and a controversy. If the proof is ultimately accepted, it would mark only the second Millennium Prize Problem to be solved and the first to be solved by AI.
However, even before mathematicians had an opportunity to examine the paper, NYU professor Tristan Buckmaster alleged that OpenAI’s description of its approach bore similarities to separate research he was conducting with Levent Alpoge using Codex and Claude AI tools. Buckmaster said OpenAI only started attempting to solve the equation after receiving information about their research.
The development led Buckmaster to question whether OpenAI had used their work as training data. “I asked whether the model had been trained on, or had access to, our sessions in Codex, into which we had been putting all our drafts for the whole of this project,” he said in a statement. “I was told the model did not look up user data. I asked again, about training, and I did not get an answer.”
Buckmaster stressed that he was only recounting what he had been told. “I am stating it because the alternative is to let a sequence of announcements say something I know to be false,” he added.
OpenAI later responded to the allegations in a post on X. “We (the researchers and the agents) did not see any of their (Buckmaster and Alpoge) work through any means until they released it publicly — in particular, no specific user data was accessed in order to solve this problem,” the company said.
However, OpenAI acknowledged that it could not completely rule out the possibility that Codex-related data had contributed to improving its models. “While unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models,” it said.
How did OpenAI’s new AI model solve Navier-Stokes problem?
In a blog post, OpenAI said its next-generation AI model had significantly stronger mathematical abilities than Astra. The company claimed Astra managed to solve around 10 per cent of questions on its internal mathematics benchmark, whereas the unreleased model achieved nearly 50 per cent. “Basically, you throw at it almost any open problem,” OpenAI researcher Sebastien Bubeck said, “and it’s a coin flip whether the model can solve it.”
OpenAI said training of the model began in late August. On September 1, amid growing speculation in the mathematics community that Anthropic had solved two Millennium Prize Problems, OpenAI tasked the model with attempting all six remaining unsolved problems, including the Riemann hypothesis, P versus NP and Navier-Stokes.
“We didn’t expect it to solve any,” researcher Noam Brown said. After roughly 50 hours, however, the AI had progressed far enough on a Navier-Stokes-related problem for OpenAI’s human researchers to step in. They redirected resources from the other five problems and focused the available computing power on Navier-Stokes.
OpenAI then expanded the number of AI agents from 100 to as many as 10,000. The company said the effort generated 2.7 million messages exchanged between agents and 130 billion output tokens, which it compared to approximately one million books. OpenAI said the work was completed on Saturday and verified computationally on Sunday, using “millions of dollars” worth of computing resources.
OpenAI said the race to solve a Millennium Prize Problem had become so competitive that it chose not to share the proof with external mathematicians before publication. Humans nevertheless remained involved throughout the process. OpenAI researcher Dan Roberts characterised the researchers’ role as “a bumble bee cross-pollinating across different groups and delivering different bits of information.”
The significance of the development is closely tied to the importance of the problem itself. In 2000, the Clay Mathematics Institute identified seven Millennium Prize Problems and offered a $1 million reward for each valid solution. So far, only one had been solved: the Poincare conjecture, which was proved by Grigori Perelman, who subsequently declined the prize money.
