Last week OpenAI reported that an internal artificial intelligence (AI) system has produced a solution to the Navier–Stokes existence and smoothness problem, one of the Millennium Prize Problems. The Millennium Prize Problems are considered the hardest and most important problems to solve in mathematics.
The problem is this. Take a fluid whose motion is governed by the Navier–Stokes equations. The flow starts smooth: the initial velocity field is smooth and well-behaved, and any external force is a smooth function of space and time. Must the flow remain smooth for all time, or can it “blow up” in a singularity, meaning that the velocity becomes infinite somewhere in finite time? This would be a sort of “spontaneous explosion” (so to speak) of the fluid. If you want the mathematics (but be warned, it’s complicated), take a look at “The Navier-Stokes Problem in the 21st Century” (2024), by Pierre Gilles Lemarié-Rieusset.
OpenAI’s result is that blow-up can happen, at least when a carefully chosen smooth force is allowed to drive the fluid. The construction is an engineered mathematical example. Even as a model of a real liquid it would fail before the singularity: water is made of molecules, the incompressibility of the fluid is an approximation, the continuum equations ignore the cutoffs that appear at small scales, and relativistic and quantum corrections are ignored.
So don’t worry, the water in your glass isn’t about to explode. The Navier-Stokes result does not change the weather, or your plumbing. It does change what we know about the equations themselves: they are not guaranteed to stay a faithful description of a fluid, even starting from perfectly nice data.
AI systems have been solving lesser open problems in mathematics for months, but this is arguably the first major problem that an AI system is the first to solve. The news has been discussed and commented on by major media outlets. For example, Fox News has aired a 20 minute segment titled “AI cracks one of math's biggest unsolved problems,” with well-known mathematician Steven Strogatz.
An unreleased internal model, run as a swarm of roughly 10,000 coordinating agents, had produced an analytical proof and a Lean formalization showing that the 3D incompressible Navier–Stokes equations can develop a finite-time singularity. OpenAI has released a 170 pages technical writeup titled “Finite Time Blowup for Navier-Stokes.”
The construction starts from rest, applies a smooth force, keeps energy finite, and still blows up. OpenAI framed this as resolving statements C and D of the Clay Mathematics Institute’s official formulation of the Millennium Prize Problem. The company said the agents reached the result in about 88 hours, with another 17 hours of Lean verification via GPT-6 Astra, and that it would not claim the $1 million prize.
“One of the most amazing moments for me in OpenAI history was watching this happen over the past week,” OpenAI CEO Sam Altman posted to X.
The drama
Researcher Sébastien Bubeck said the company launched the effort after seeing X rumors that Anthropic-linked work had resolved Millennium problems (also reported by Science), and that the team at first assumed there must be a mistake.
The announcement came hours after Tristan Buckmaster, an NYU mathematician, posted a statement with Levent Alpöge (an Anthropic researcher acting independently). The pair had spent much of a year, using Claude and OpenAI’s Codex, to continue work started by mathematicians Diego Córdoba and Luis Martínez-Zoroa, who for several years have been exploring the construction of forced blow ups. “We took their work as a starting point, using Large Language Models to push their program to completion,” said Buckmaster. He added that OpenAI learned of their progress, then described to him a ~100-page forced Navier–Stokes proof that followed the same unusual route almost nobody else was pursuing.
He alleged that in calls with Bubeck he was offered either a coordinated release or sole authorship on a write-up that credited an OpenAI model and dropped Alpöge because of his employer. When he said he would go public, he reported being asked why he would “ruin your career” and told “If you don’t want me to be nice, then I don’t have to be nice.” He emphasized he had not seen OpenAI’s proof and was not formally accusing anyone of data theft. OpenAI and Bubeck called the allegations false and inflammatory. Buckmaster was then interviewed by The New York Times (open copy).
OpenAI insisted neither people nor agents searched user data, congratulated Buckmaster and Alpöge on their related Euler work, and said the proofs differed. It added that it could not rule out that de-identified product usage had helped improve its models. In other words, perhaps the interactions of Buckmaster and Alpöge with OpenAI Codex could have been part of the data used to train the unreleased internal model. Then OpenAI amended the statement: these interactions “could not have influenced the system in any way, including through training.” The reasonable doubt, of course, remains.
Reactions and commentaries
The Clay Mathematics Institute, which awards the Millennium Prizes, stated that it “shares in the excitement of the global mathematical community as we contemplate the announcement that the Navier-Stokes problem has apparently been settled,” with a warning that the award process “is deliberately unhurried.”
Fields Medalist Terence Tao (the Field Medal is considered analogous to the Nobel Prize in mathematics). Tao’s critique: rumor of work in progress now triggers industrial-scale compute that can flatten a research program before it matures. “There is no speed limit,” he said, and the incentives now point toward secrecy rather than the centuries-old habit of sharing promising directions, New Scientist reports (the article is paywalled, but the full text is in the page source). Talking to CNN, Tao compared the result to skipping from the first ten minutes of a film to the last ten: the plot is resolved, but most of the value is lost.
Nature (open copy) treats OpenAI’s claim as the first time a computer has solved a truly major open problem in mathematics. The commentary quotes Clay president Martin Bridson calling it “an exciting day” for human understanding of mathematics, and Luis Martínez-Zoroa’s verdict that OpenAI’s result is “truly remarkable.”
Ben Goertzel notes that OpenAI “spent something like $22 million worth of compute” on finding the solution. He argues that proving named theorems is now close to “solved” the way chess is, but that is not AGI: creatively conjecturing new ideas remains the harder, more creative step.
Several high-profile reactions treat this mathematical achievement as potentially historic while treating the process - speed, compute scale, rumor-driven racing, credit negotiations, and the risk that unpublished drafts in commercial tools shaped the outcome - as a warning about how AI is changing mathematics in ways that are not always good.
Many renowned mathematicians and Field medalists have signed a declaration, warning that this “is detrimental to the science of mathematics, and to the mathematical community” and that society “confront similar problems in many other forms of intellectual work.” It’s worth noting that not all top mathematicians agree with the declaration.
A few notes of my own
The part of this story I keep thinking about is not the proof itself, but the possibility that interactions with users helped an AI make a breakthrough. I leave the box checked: my chats can be used to train future models, and I’m happy to contribute to AI progress. That attitude is easy for me because I deem it very unlikely that I could stumble onto a breakthrough anytime soon. If I believed I were close to one, I might feel differently about feeding unpublished drafts into a system owned by a major AI lab. The Buckmaster–Alpöge episode is interesting precisely because it sits on that line. Even if no one “looked up” their transcripts, the question remains: once frontier labs ingest the working notes of the people closest to an open problem, who deserves credit the last mile?
I’m not entirely persuaded by Ben Goertzel’s assertion that creative conjecturing is the thing AI cannot do. Ask a large model for a thousand weird conjectures about the Riemann hypothesis or quantum gravity and you will get a thousand conjectures. Nine hundred and ninety-nine will be useless. But that is also what happens if you ask me, or almost any human mathematician or physicist. So I don’t blame AI for having difficulties when it comes to generating creative conjectures - for I’m guilty of the same!
The hit rate on raw speculation has always been very low. If one of those thousand ideas later turns out to matter, I am not sure why its origin - human or AI model - should change the scientific status of the idea.
The reported compute cost makes another shift visible. Twenty-two million dollars is not pencil-and-paper money. Mathematics is starting to look like the rest of Big Science: a field in which some of the decisive experiments require expensive apparatus. Cleverness still matters, but it is no longer the only input.
I also doubt that machines finishing famous problems will empty the seminar rooms. Chess was “solved” in the narrow sense that machines outplay every human - and yet more people play, study, and watch chess than before. Engines did not kill the game; they changed what human excellence looks like. I expect something similar in mathematics. The prestige theorems may fall one by one. That need not mean people stop caring about mathematics. It may mean they care about different parts of it - explanation, taste, new questions, the parts of the subject that are not a race to a named prize.
I‘m not entirely unsympathetic to the worry that this will demotivate brilliant scientists and bruise egos. A life spent circling a problem, only to watch a swarm finish it over a weekend, is hard to swallow. But suppose the issue is not a mathematical proof but an AI-generated cure for cancer. In this case, I don’t think the ego of scientists should weigh more than the lives that would be saved. Mathematics is closer to chess than to oncology, but even so, if a result is true and useful, the fact that a human did not get there first is a real loss for someone, and still a second-order loss.
There are rumors that AI-generated solutions to other Millennium Prize Problems could be announced soon. These rumors are unconfirmed at the time of writing. Regardless, my guess is that the remaining Millennium Prize Problems, and then large stretches of physics and the other formal sciences, will go the same way: not all at once, and not without fights over credit and data, but steadily, as the power of AI keeps growing and the cost of throwing AI at a well-posed target keeps falling.
When the dust settles, mathematics (and then all science) will be seen as a joint endeavor of us humans and our AI mind children.