OpenAI Claims Breakthrough in Navier-Stokes Equation, but NYU Mathematician Alleges His Work Was Copied
Mathematician Tristan Buckmaster asserts OpenAI cribbed the work he and a collaborator had done, which included significant progress toward solving the equation in August. OpenAI announced its team had solved the equation this week.
In May 2000, to inspire mathematicians and mark the start of the new millennium, the Clay Mathematics Institute in Cambridge, Massachusetts, launched its seven Millennium Prize Problems. Each challenge carried a $1 million award for a solution.
After OpenAI’s reported breakthrough on Navier–Stokes, the possibility of AI attacking multiple Millennium Prize Problems no longer sounds nonsense.
Rumors are now circulating that OpenAI is getting close to verifying a proof of the Hodge Conjecture, while OpenAI or Anthropic… pic.twitter.com/gAQT5GuQyv
— SciTech Era (@SciTechera) September 10, 2026
One of the seven challenges was the Navier-Stokes Equation. The Navier–Stokes equations are among the most useful mathematical tools for describing how fluids move (including liquids such as water and blood, and gases such as air). Their practical value is that they connect pressure, velocity, density, temperature, viscosity, and external forces, allowing scientists and engineers to predict or simulate flow.
Essentially, the Navier–Stokes equations express conservation of momentum for viscous fluids. The Navier–Stokes Millennium Prize Problem asked a very specific question: If you start with a perfectly smooth three-dimensional flow of an ordinary fluid, do the equations always keep producing a smooth, physically sensible flow forever, or can the flow become infinitely extreme at some finite time?
This week, OpenAI announced that it had solved the equation.
OpenAI announced September 8 a solution to the Navier–Stokes existence and smoothness problem, which is so monumental that a $1 million prize has been offered for its resolution. “It’s one of the guiding problems for the field. It is a huge deal to know the answer,” says mathematician Dallas Albritton of the University of Wisconsin–Madison, who was not involved with the new work.
We’re sharing a solution to the Navier-Stokes Millennium Prize Problem, one of the deepest problems at the frontier of mathematics.
The proof was produced by a group of agents, using an OpenAI next-generation model significantly more capable than GPT-6 Astra.
The problem… pic.twitter.com/8zol3BPTL4
— OpenAI (@OpenAI) September 8, 2026
However, New York University mathematician Tristan Buckmaster disputes this claim, as he and another collaborator were making meaningful progress on related work. The mathematician asserts that OpenAI copied his collaborator’s work.
…[P]rofessor Tristan Buckmaster, a British-Australian researcher who trained in Germany, has come forward alleging OpenAI may have copied his work after asking him to collaborate — with one of the company’s scientists allegedly warning he might “ruin his career” if he refused.
Buckmaster had been working with Long Island-raised math prodigy Levent Alpöge, a researcher with OpenAI rival Anthropic, on a different elusive fluid dynamics equation, known as the Euler problem, when he found a solution using AI tools from both Anthropic and ChatGPT.
He said scientists from OpenAI approached him and tried to convince him to jointly announce the solution, claiming they had independently solved Navier-Stokes.
Buckmaster questioned the timing of their finding the solution in a blog post, noting OpenAI had only done so “in the past few days” — after the company became aware of his work.
Buckmaster indicates he feels rolled over in the battle of two AI giants. It appears that when Anthropic solved one or two of math’s “Millennium Prize problems”, OpenAI’s teams were to work on the mathematical challenges themselves.
Dr. Buckmaster said he did not care about the $1 million or even winning the race to come up with the answer. “I love math, and I just wanted to be part of the story,” he said during a two-hour recounting of the events in his N.Y.U. office on Tuesday afternoon.
He is now part of a different story: the acrimonious rivalry between OpenAI and Anthropic, which was founded in 2021 by former OpenAI leaders who said OpenAI had become cavalier about the dangers of artificial intelligence.
“When they are two of the biggest companies in the world, they need to get over it,” Dr. Buckmaster said. “I am just a bystander in these petty games.”
The tension between Dr. Buckmaster and OpenAI also reflects a chasm of distrust between the swashbuckling culture of high-flying, high-tech start-ups and the usually cordial and communal collaboration of mathematicians, who nonetheless increasingly rely on the tools of A.I.
There was an attempt at a compromise, but OpenAI claims it denies that its team ever saw the pair’s work before publication.
Tristan Buckmaster, a mathematics professor at New York University, alleged that OpenAI moved on the problem only after word of his unpublished results reached the company. He had worked for nearly a year with Levent Alpöge, a mathematician employed by rival lab Anthropic.
Both men leaned on OpenAI’s Codex assistant while assembling their own proofs, and the company can train on those sessions unless users opt out.
The pair chose a narrow route through the problem that almost nobody else was pursuing, which made the timing look suspicious to Buckmaster. He says OpenAI mathematician Sébastien Bubeck proposed a compromise that would strip Alpöge of credit, then asked why he would ruin his career.
OpenAI denies that its researchers or agents saw the pair’s work before publication. It says no specific user data went into the effort. The company concedes that de-identified data drawn from product use may have improved its models, though it argues the two proofs differ significantly.
Whether the fluids are smooth or not, the human drama surrounding its alleged solution to the Navier–Stokes equations is anything but.
Whether OpenAI’s proof was truly derived independently will determine if this revelation becomes a genuine scientific milestone or a cautionary tale about an increasingly ruthless race for AI supremacy.
In the meantime, a million-dollar prize has exposed a far more troubling equation: the unintended consequences of pairing extraordinary computational power with intense corporate rivalry, devoid of the spirit of collaboration that has long been central to scientific and mathematical progress.
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Comments
Always be generous with credit. It pays you back when others share their work with you.
Did Buckmaster solve the equation or did he not? In mathematics, difficult equations can go unsolved literally for centuries. But once work starts, rarely do mathematicians start over – they build on what has gone before (“standing on the shoulders of giants” at times). It doesn’t matter how close someone came to solving a problem. If you didn’t solve it, you didn’t solve it. The person (or entity) that solves the problem gets the credit. It’s understood that he relied on the work of everyone before him, even if it only informed him of what rabbit holes to avoid.
Buckmaster himself was almost certainly building on the work of others. If he had solved the equation first, he could give those who came before credit for their work, but he would have rightly claimed the credit for reaching the solution.
As I understand it, Buckmaster did not solve the equation (nobody has solved the equation), but he had made significant progress toward an eventual solution (e.g., solving intermediate steps). He is claiming that the AI found his work somewhere on the Internet and simply appropriated it – which if you think about it, is almost all that most AIs do all day. It is, in fact, the mode in which they operate more than 95% of the time.
It looks like I misunderstood a great deal of what went on as described in this article. Clearly OpenAI is claiming they have solved these equations. As I understand it now, Buckminster is claiming the overall solution required and used the solution to a subproblem he solved but had not published at the time. OpenAI denies their solution required his.
Does anyone have a copy of Fluid Dynamics for Dummies I could borrow?
I used Landau and Lifschitz
The Navier-Stokes (NS) equations has non-linearities that make them very difficult to solve.. NS admits solutions with vortices that get stretched with higher speeds at smaller and smaller scales, but there is a point at which very small turbulent eddies get dissipated due to friction and converted instantly to thermal energy.. Dissipation due to friction is what keeps the equations smooth in nature without blowing up.
There is also a natural constraint due to the speed of light, which prevents infinite speeds that can result from the pure application of Navier-Stokes without constraint.
What this team did was to contrive a system in which an (unnatural) geometry was assumed that caused extreme nonlinear amplification to exceed viscous dissipation. To keep the amplification going until it blew up, they included a forcing term (i.e. boundary conditions) that injected energy beyond what would normally be lost by dissipation.
So this was a brute force approach that started with an extreme but smooth initial condition with energy injection from outside the system and which was specified to exceed the dissipation rate. So they used a ton of processors to integrate the Navier-Stokes equations from an unrealistic initial condition with unreasonable boundary conditions until something went infinite.
This is interesting to the math community because it proved the existence of a singularity. It is not very interesting to the physics community, because it requires extreme assumptions. It would be far more interesting if they found a singularity using initial and boundary conditions that are consistent with what we see on planet Earth.
Sorry for the long meandering post. This was me looking at various sources, trying to piece together what they did and why it did not seem very appealing at first glance. After understanding their solution, it was not very interesting at second glance either.
As to the issue with Buckmaster, it is hard to know whether they “stole” his approach. I have seen multiple instances where different groups converged on the same solution independently.
The real breakthrough was made in 2014 by Terrence Tao, the famous UCLA mathematician. He designed a “toy” model in which linearized Navier Stokes equations were modeled like an electronics circuit with “logic gates” of fluid vortices—where one vortex swirl transfers energy to another. He proved that these fluid logic gates could be programmed into a self-replicating loop that pumps kinetic energy from large scales to infinitely smaller scales faster than viscosity can dissipate it, thereby causing the toy model to blow up. Tao’s toy model proved that a mathematical singularity was possible in the full Navier-Stokes equations. This methodology was adopted by both Buckmaster and OpenAI.
The next step was taken by Buckmaster and Anthropic to apply Tao’s idea to non-linearized Navier-Stokes equations, except without friction ( called the Euler equations) using AI assistants. This required specialized mathematical techniques to accomplish the representation with AI. Buckmaster proved that the model blew up when there was no viscous dissipation of the eddies (i.e. no viscosity).
The last step which won the prize was application of the full Navier-Stokes equations (i.e. with frictional dissipation). OpenAi’s simulation was extremely similar to what Buckmaster did, except that they added in the viscous term that Buckmaster left out, They also made the external forcing term large enough that it would overcome the viscous dissipation term.
So it does seem like the heavy lifting was done by Tao and Buckmaster.
There’s the Komogorov spectrum based on the speculation that the rate of energy dissipation in turbulent medium does not depend on viscosity. Whatever the viscosity, flow goes to smaller and smaller scales until it reaches the size that viscosity dissipates. That results in a -5/3 spectrum or -11/3 depending on whether all 3 dimensions are lumped together or not.
In particular nothing goes singular. But there’s no forcing term either.
Viscosity by the way is transfer of the x momentum in the y direction etc, which is how large scale turbulence can look like viscosity to even larger flows, when actual viscosity is orders of magnitude smaller scale.
Good job, kelly and rh.