For more than 25 years, the Clay Mathematics Institute has offered $1 million each for solutions to seven of the hardest problems in mathematics. Only one had ever been solved, by Grigori Perelman, who famously turned the money down. In September 2026, a second one fell, and the solver was not a person.
On September 8, OpenAI announced that an unreleased internal model had resolved the Navier Stokes existence and smoothness problem, one of the Millennium Prize Problems that has defeated physicists and mathematicians since the question was formally posed in 2000. Two weeks later, the company said the same model had knocked out more than 100 other long standing open problems.
As per geekblog.net research, it is the biggest claim an AI lab has ever made about pure research. It is also the start of the most heated fight between the tech industry and academic mathematics in memory. Here is what actually happened, what it means, and what to watch next.
What the Navier Stokes Problem Actually Asks
The Navier Stokes equations describe how fluids move. Water in a pipe, air over a wing, blood through an artery, smoke curling off a candle: engineers use these equations every day, and they work extremely well in practice.
The trouble is that nobody could prove they always behave. The Millennium Prize question asks, roughly, whether a smooth, well behaved flow of an incompressible fluid in three dimensions stays smooth forever, or whether it can suddenly “blow up,” producing a point where the fluid’s speed becomes infinite in a finite amount of time.
A proof in either direction wins the prize. For decades, most experts assumed the answer was “stays smooth.” As the Spanish mathematician Diego Córdoba told Quanta Magazine, “Ten years ago, nobody believed there was a singularity for Navier Stokes.”
What OpenAI Says Its AI Proved
According to OpenAI, its model proved the opposite of the old consensus: the equations can break down. The construction is a spinning vortex of fluid that spirals inward while stretching longer and thinner, which OpenAI’s own write up compares to spaghetti. Even though viscosity normally smooths flows out, this vortex concentrates until the velocity becomes infinite in finite time, while the total energy of the fluid stays finite the whole way through.
In the language of the official Clay problem statement, OpenAI says the result settles statements C and D, the “breakdown” versions of the question.
How the Proof Was Found
The method matters as much as the result. OpenAI did not ask a chatbot a question and wait for an answer. It says it ran a swarm of roughly 10,000 AI agents working at the same time, with internet access and the ability to write and run code. Over about 88 hours, those agents exchanged some 2.7 million messages and generated around 130 billion output tokens, with groups of agents sharing and merging promising code as they went.
Quanta Magazine reports the total computing bill ran to several million dollars.
How It Was Checked
Here is the part that separates this from earlier AI hype. The proof was then translated into Lean, a formal proof language in which every logical step is checked by a computer. OpenAI says that formalization took another 17 hours. A formally verified proof does not depend on a human referee catching every error in hundreds of pages; if the formal statement matches the real problem, the machine has checked the logic.
That “if” is where human experts still come in, and it is one reason the result is described as subject to further scrutiny rather than settled history.
The Humans Behind the Machine
The story is not simply “AI beats mathematicians.” The blowup approach the model exploited builds directly on years of human work, especially by Córdoba (Institute of Mathematical Sciences, Madrid) and Luis Martínez Zoroa (CUNEF University), whose research made a Navier Stokes singularity look plausible in the first place.
Princeton’s Charles Fefferman, the mathematician who wrote the official Clay description of the problem, was blunt about where the credit belongs. “The heroes of the story,” he told Quanta, “are Córdoba and Martínez Zoroa.”
A Race, a Rival, and a Credit Dispute
OpenAI was not alone. Just before midnight on September 7, hours ahead of OpenAI’s announcement, NYU mathematician Tristan Buckmaster and Levent Alpöge, a researcher at Anthropic, released their own singularity result, produced with help from an internal Anthropic model. How the two results compare, and who reached what first, is still being argued.
Things got messy quickly. TechCrunch reported that Buckmaster accused OpenAI of pressuring him not to credit his Anthropic collaborator and questioned whether his team’s work had fed into OpenAI’s proof. Buckmaster has also been candid about the raw quality of machine output, calling the first AI generated proof his team saw “the most horrendous I have ever read.”
OpenAI, for its part, has said it does not intend to claim the $1 million prize. Under Clay rules, a solution must be published in a qualifying journal and survive two years of scrutiny before any award is considered anyway.
25 Fields Medalists Push Back
On September 11, a group of 25 Fields Medal winners, the highest honor in mathematics, published a declaration arguing that the AI race to solve famous problems is misaligned with what mathematics is for. Terence Tao shared it on his blog under the title “A Severe Misalignment of AI in Mathematics.”
Their core argument: “Solving problems is only a tool and proxy for achieving the primary goal of conceptual understanding and insight.” They warn that results “announced in a rush” leave no time for proper write ups, for isolating the new ideas, or for citing the earlier work they depend on, and that commercial competition rewards secrecy over the open culture that math runs on.
OpenAI’s Response: 100 More Problems and an Advisory Group
On September 21, OpenAI announced two things at once. First, the internal model, which began training on August 28, has now resolved “more than 100 long standing open problems across most areas of mathematics.” The company has not yet published a list, the fields involved, or how each result was verified.
Second, it created an independent, unpaid Advisory Group on Mathematics and Artificial Intelligence, hosted at the Institute for Advanced Study in Princeton. Its nine initial members include Fields medalists Timothy Gowers and Martin Hairer, physicist Edward Witten, Camillo De Lellis, Ravi Vakil, Ulrike Tillmann, Nikhil Srivastava, Melanie Matchett Wood and François Charles.
The group will advise on how new results are reviewed, assessed and released. Its limits are explicit, though. OpenAI says the group will not advise on how fast the company moves on math, and the Institute stated plainly that “we do not have decision making power at any AI company.”
Why This Matters Beyond Math
It is easy to file this under “interesting for academics.” That would be a mistake, for three reasons.
- It shows what agent swarms can do. Ten thousand coordinated agents running for four days is a preview of how AI labs plan to attack hard problems in chemistry, materials, chip design and software security. Math is simply the field where success can be checked with certainty.
- Formal verification changes trust. The pairing of AI discovery with machine checked proof is a template. If it spreads to code and hardware, “the AI says it works” could become “the AI proved it works.”
- The credit fight is a preview for every profession. Who gets recognized when a model builds on decades of human research, and who decides when a result is ready to announce, are questions that will soon reach medicine, law and engineering too.
What to Watch Next
- Independent review. Expect mathematicians to pick apart the formal statement to confirm it matches the Clay problem exactly.
- The list of 100+ problems. Until OpenAI publishes which problems were solved and how, that number is a claim, not a result.
- The Buckmaster and Alpöge paper. Its reception will shape how credit is divided for the first Millennium Prize result reached with AI.
- Rules of the road. The Fields medalists’ letter and OpenAI’s advisory group are the opening moves in setting norms for how AI labs publish research.
The Bottom Line
A problem that stood for a quarter century appears to have fallen to a machine, backed by a computer checked proof. That is historic. But the mathematicians raising alarms are not wrong either: an answer is not the same as understanding, and the way this result arrived, in a race, with a public credit dispute, shows the field has no rulebook yet for AI discovery. The proof may be done. The hard conversation is just starting.
Source: geekblog.net