AI Solved a Hard Math Problem. Humans Learned Little

On September 8, OpenAI announced that an internal artificial intelligence system had produced a solution to the Navier-Stokes existence and smoothness problem, one of the seven Millennium Prize Problems and among the most consequential open questions in mathematics. But the achievement, described by the company as a landmark, has so far yielded strikingly little human understanding — and it has ignited a bitter dispute over credit, data and the relationship between artificial intelligence and the people who do science.

According to NPR, mathematicians believe the proof is technically correct but find the dense, 166-page manuscript — drafted by AI and authored simply as “OpenAI” — virtually unreadable. “So far it’s been very difficult to really extract any human understanding from this new AI proof,” said James Maynard, a mathematician at the University of Oxford. “The paper is not written for humans,” said Javier Gómez-Serrano of Brown University.

The problem that everyone expected to fall

The Navier-Stokes equations describe how fluids — air, water, blood — move. They are used daily across physics and engineering, yet at a fundamental level researchers do not fully understand why they work. The Millennium Prize version of the problem, established by the Clay Mathematics Institute in 2000 with a $1 million reward, asks whether smooth, physically reasonable three-dimensional solutions always exist, or whether the equations can “blow up,” with a fluid’s velocity becoming infinite in finite time.

As The Guardian reported, OpenAI says its proof shows the equations do blow up. The community had long sensed that Navier-Stokes was the next Millennium Problem likely to fall: “Among the seven Millennium Problems … Everyone agreed that this would be the next one that got solved,” said Martin Hairer of EPFL, a Fields Medalist who was later named to an OpenAI advisory panel.

The scale of OpenAI’s effort was enormous. The company said it marshalled roughly 10,000 AI agents that worked for 88 hours, sending some 2.7 million messages and consuming about 130 billion output tokens. The Guardian reported that the Navier-Stokes solution alone cost the company somewhere between $6 million and $10 million in compute, with roughly $15 million to $20 million spent overall.

A proof that machines can check but humans struggle to read

Why do mathematicians trust a result they cannot easily read? Because OpenAI also produced a Lean formalization — code in a proof-assistant language that verifies whether a proof is correct. If the Lean code compiles, the proof is generally considered complete. Gómez-Serrano told NPR that the Lean code did indeed compile, and that “the community seems to have the consensus that it is correct.”

That is a genuine milestone: it marks the first time an AI system has been credited with cracking a Millennium Prize Problem. But correctness is not the same as comprehension. “The paper doesn’t explain what parts are important. What parts are routine? How does the idea feed into other places?” Gómez-Serrano said.

A priority dispute erupts

The announcement landed amid a fierce dispute. Approximately 12 hours before OpenAI’s statement, mathematician Tristan Buckmaster of New York University published a public statement describing how, he said, the company had approached him. He and his collaborator Levent Alpöge — a mathematician employed by OpenAI rival Anthropic — had spent about a year working on a closely related problem, using a niche approach, and had stored their drafts in OpenAI’s Codex system.

As ABC Australia detailed, Buckmaster said he was presented with two options: either he and Alpöge post their Euler result while OpenAI posted its Navier-Stokes result, or Buckmaster alone would rewrite OpenAI’s proof — leaving Alpöge off because he works for Anthropic. Buckmaster declined both, saying he would go public if OpenAI proceeded.

“The reply was, ‘Why would you ruin your career?’” Buckmaster recounted. “I replied that I am an academic, and asked why he thought going public would ruin my career… The reply was, ‘If you don’t want me to be nice, then I don’t have to be nice.’”

Buckmaster has been careful with his claims. “I do not know what their model did, or how. I do not know whether our data was used. I am not accusing anyone of anything,” he wrote. Yet he told NPR there was “so much circumstantial evidence that they had far more knowledge of what we were doing than they let on,” adding that “they” might not even mean a person — “it could mean the agents.”

OpenAI has denied the allegations. “We did not rush to publish even though the other team wasn’t communicating with us. The team threatened us with unfounded accusations of plagiarism,” CEO Sam Altman posted on X. OpenAI researcher Sébastien Bubeck said the proofs “differ significantly” and that the company never asked for Alpöge to be removed from his own work. The company also said it “cannot rule out that de-identified data derived from their usage of our products helped improve our models,” while maintaining that no specific user data was accessed.

The profession pushes back

The dispute has widened into a broader reckoning about how AI companies treat mathematics. On September 11, a group of 25 Fields Medalists — mathematics’ most decorated living figures — published a declaration titled “A Severe Misalignment of AI in Mathematics,” warning that treating unsolved problems as benchmarks is “detrimental to the science of mathematics.”

“Solving problems is only a tool and proxy for achieving the primary goal of conceptual understanding and insight,” the declaration states. “Indeed, the mass production at faster and faster pace of ‘true/false’ statements could destroy fertile ground instead of breathing life into new ideas.”

Maynard framed the episode more starkly. “This whole episode could have been a wonderful proof-of-concept of the power of collaboration between humans and AI,” he told NPR. “Unfortunately, because of the decisions of humans involved, it became this scrappy, messy battle.”

A solution — but to the right problem?

Even the result’s scope is now in question. As Scientific American reported, OpenAI’s proof may target an externally forced variant of the problem rather than the physically meaningful full version — exploiting, in effect, a loophole in how the question is framed. That variant is “disconnected from reality and thus less interesting,” the magazine noted, and it remains contested.

OpenAI has moved to address the fallout. On September 10, it announced the creation of an “independent mathematics advisory group” to help review and disseminate new results — while stipulating that the group would not advise on how quickly the company pursues internal progress.

What to watch

The Navier-Stokes episode is a paradox: a machine has apparently proven something humans could not, yet the humans who study the equations understand little more than before. That tension — between producing answers and producing understanding — is the real story. It is one the rest of science is likely to confront as AI systems grow more capable.

The questions that remain are as much about people as about proofs: whether AI-generated results can be turned into genuine mathematical knowledge; whether researchers’ unpublished work is protected when fed into commercial AI systems; and who, ultimately, gets credit when a discovery is made by a machine trained on the work of humans. For now, the Clay Mathematics Institute has yet to recognize any party for a correct solution — and OpenAI says it does not intend to claim the $1 million prize.