OpenAI has announced a proposed solution to the Navier–Stokes existence and smoothness problem, one of the seven Millennium Prize Problems that has remained unresolved for nearly 90 years.
OpenAI has released both a written proof and a formalised version in Lean, a system used to verify mathematical arguments.
The company said the internal model used for the work is “significantly more capable than GPT-6 Astra”, and that its training remains ongoing. The company said it used approximately 10,000 agents working across 88 hours
“Across all attempted problems, the agents sent 4.9 million messages and used about 300 billion output tokens. In the process of resolving the Navier–Stokes problem, the agents sent 2.7 million messages and used approximately 130 billion output tokens,” said the company in a blog post.
A deeper understanding of the Navier–Stokes equations could, in the longer term, inform areas that rely heavily on fluid-flow modelling, including aerospace engineering, weather forecasting, ocean modelling and biomedical applications.
The company said its internal AI model produced a proof showing that a three-dimensional fluid can develop a singularity in a finite amount of time, meaning its speed can grow without limit even though it starts smoothly and has viscosity.
The underlying question is whether smooth solutions to the equations can remain smooth indefinitely, or whether they can break down through a finite-time singularity.
OpenAI said its solution takes the form of a vortex that spirals inward while becoming increasingly elongated.
The resulting velocity grows without bound while the fluid’s total energy remains finite.
While OpenAI asserts that its result satisfies statements C and D of the Clay Mathematics Institute’s formulation, it has not been peer-reviewed or verified by the Clay Mathematics Institute.
But the Claim Has Triggered a Dispute
However, OpenAI’s claimed breakthrough has become embroiled in a dispute over how the company pursued the result and whether it benefited from unpublished work by mathematician Tristan Buckmaster and Anthropic researcher Levent Alpöge.
Buckmaster detailed his account in a statement released alongside the mathematical proof based on his research, hours before OpenAI announced its breakthrough.
The two had been working on related fluid-dynamics problems for months, using AI models in their research. Their work included a breakthrough on 3D incompressible Euler, a problem closely related to Navier–Stokes, and they were pursuing a route they believed could eventually lead to the forced Navier–Stokes problem.
The dispute began when Buckmaster said he learned that information about his work had reached OpenAI and that he had contacted the company.
Three days later, he spoke twice with OpenAI technical staff member Sébastien Bubeck, who told him that an internal model had produced a Navier–Stokes proof.
That led Buckmaster to ask whether OpenAI's models had been trained on, or had access to, the private Codex (the AI-powered coding platform developed by OpenAI) sessions he and Alpöge used throughout their project.
He also said OpenAI’s first prompt for the problem came only days earlier, after information about their work had reached the company.
Buckmaster also alleged that Bubeck proposed coordinated publication arrangements that would have excluded Alpöge because he works for Anthropic.
However, Bubeck disputed that account.
Bubeck argued in his statement on X that it would be inappropriate to cite Alpöge as an 'independent academic' when he is an Anthropic employee who used Anthropic’s models in the Euler work to author OpenAI’s proof.
Bubeck also said he had considered offering them access to OpenAI’s internal model to help complete their work, but was unsure how to do so while Alpöge worked at Anthropic.
Bubeck said his suggestions were "immediately met with a litany of slander, including direct threats that if we were to announce Navier-Stokes he [Buckmaster] would immediately go to the press with a barrage of unfounded accusations."
"I refuted all these accusations, but he [Buckmaster] replied, 'There is nothing you can do; I simply do not trust you,'" wrote Bubeck.
What OpenAI Says Happened
OpenAI, while announcing its breakthrough, has also officially revealed that their effort did begin after they heard the rumours, which they ‘later realised was related’ to Alpöge and Buckmaster.
OpenAI said that after completing their full project, and believing the rumour that the other two researchers had also found a solution, they reached out to offer a concurrent release of the result.
The company added that they offered visibility into all the prompts they used and the proof.
“We (the researchers and the agents) did not see any of their work through any means until they released it publicly — in particular, no specific user data was accessed in order to solve this problem,” said OpenAI.
However, the company also adds that it cannot rule out that ‘that de-identified data derived from their usage of our products helped improve our models.’
OpenAI also said its proof is substantially different from the pair’s work, including proving a different version of the related problem.
Both OpenAI and Bubeck have said they recognise the priority of Buckmaster and Alpöge's work and congratulate them on their remarkable mathematical achievement.
But What is the Navier Stokes Equation?
The Navier–Stokes equations describe how fluids such as water and air move. They account for factors such as the fluid’s speed, pressure, and viscosity, and are widely used in areas such as aircraft design and weather modelling.
The difficult part is not using the equations to model fluids. Scientists routinely use computers to calculate approximate solutions.
The unanswered mathematical question is whether the equations can ever produce a “blow-up”, a situation where the speed of the fluid becomes infinitely large in a finite amount of time.
Put simply, mathematicians know how to use the equations, but they have not been able to prove that their solutions will always behave properly.
That question became one of the seven Millennium Prize Problems in 2000, with the Clay Mathematics Institute offering $1 million for a valid solution.
OpenAI stated it will not claim the $1 million prize.
The Unintended Consequence
Terence Tao, a Fields Medal-winning mathematician at UCLA, has warned that the race to solve mathematical problems with AI could have an unintended consequence: researchers may stop sharing promising ideas openly.
“We have now seen that even the rumour of someone working on a problem can trigger a massive amount of AI-powered effort to flatten it before the original research project has time to reach its full potential,” he wrote in a post on Mastodon.
That could incentivise mathematicians to keep promising research directions private rather than share them with the wider community.
“In fact, it is now the identification of a promising problem which is the scarce and precious resource,” Tao wrote.
He warned that this could undermine a long-standing principle of open research, where researchers share ideas and problems so others can build on them.
If researchers begin withholding promising problems because they fear an AI model, or another lab with access to one, will solve them first, it could represent a significant shift in how mathematical research is conducted.
Your reaction
Discussion
What's your take on this story?
No comments yet — be the first to share your view.