OpenAI Claims AI Solved Navier-Stokes Problem; Mathematicians Raise Data Concerns

Artificial intelligence may have just pulled off its most ambitious feat yet - though not everyone is convinced it did so cleanly. On Tuesday, OpenAI announced that one of its still-unreleased models had cracked the Navier-Stokes existence and smoothness problem, one of the seven notoriously difficult Millennium Prize Problems that have stumped mathematicians since the year 2000. Almost immediately, though, New York University professor Tristan Buckmaster pushed back, suggesting the company's model may have drawn on research he had been developing alongside Anthropic mathematician Levent Alpoge.

According to OpenAI, the model used to crack the problem is a successor to GPT-6 Astra, a system some have already floated as a candidate for AGI. Solving Navier-Stokes, the company said, took roughly 10,000 AI agents running in parallel for 88 hours and burning through billions of tokens in the process.

The Navier-Stokes equations date back to the 19th century, credited to Claude-Louis Navier and George Gabriel Stokes. They're the mathematical backbone behind how liquids and gases behave, and they show up everywhere from aircraft engineering to weather models to simulating blood flow through the body. The specific Millennium question asks whether smooth, well-behaved solutions to these equations - when applied to three-dimensional fluid motion - always stay smooth over time, or whether they can eventually spiral out of control. OpenAI says its proof demonstrates that "an initially smooth fluid at rest can develop a singularity in a finite time." The company described the phenomenon as a spaghetti-like vortex that shrinks while spinning ever faster, its speed racing toward infinity in what's known in the field as finite-time blowup.

A researcher raises questions about where the data came from

The announcement instantly became two stories in one - a landmark in mathematics, and a controversy over how it was reached. If OpenAI's proof checks out, it would mark only the second Millennium Prize Problem ever solved, and the first cracked by an AI system rather than a human.

Before the broader mathematics community had even gotten through the paper, Buckmaster flagged that OpenAI's description of its process bore an uncomfortable resemblance to work he had been doing privately with Alpoge, using a mix of Codex and Claude. Buckmaster says OpenAI's push on the equation only began after the company had somehow gained insight into what he and Alpoge were working on.

That timing left Buckmaster uneasy about whether his data had made its way into the model's training. "I asked whether the model had been trained on, or had access to, our sessions in Codex, into which we had been putting all our drafts for the whole of this project," he said in a statement. "I was told the model did not look up user data. I asked again, about training, and I did not get an answer."

Buckmaster was careful to frame his statement as a recounting of what he'd been told rather than an accusation. "I am stating it because the alternative is to let a sequence of announcements say something I know to be false," he added.

OpenAI responded to the allegations in a post on X. "We (the researchers and the agents) did not see any of their (Buckmaster and Alpoge) work through any means until they released it publicly - in particular, no specific user data was accessed in order to solve this problem," the company said. It did, however, leave the door open on one point, acknowledging that "while unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models."

Inside the effort to solve it

OpenAI says the new model represents a significant jump over Astra when it comes to mathematics. Astra, the company noted, managed to solve around 10 percent of problems on an internal math benchmark, while the newer, unreleased system solved close to half. OpenAI researcher Sebastien Bubeck summed it up bluntly: "Basically, you throw at it almost any open problem, and it's a coin flip whether the model can solve it."

Training on the model reportedly began in late August. Then, on September 1, with rumors swirling that rival lab Anthropic had already solved two Millennium Prize Problems, OpenAI decided to point the system at all six of the remaining open questions at once - including the Riemann hypothesis, P versus NP, and Navier-Stokes. Researcher Noam Brown admitted the team wasn't optimistic going in: "We didn't expect it to solve any." Around the 50-hour mark, though, the system had made enough headway on a Navier-Stokes-adjacent line of reasoning that the human team stepped in, pulled resources away from the other five problems, and threw everything at this one.

From there, OpenAI scaled its agent count from 100 up to 10,000. The final push, the company said, involved 2.7 million messages exchanged between agents and 130 billion tokens of output - comparable, by OpenAI's own estimate, to the length of roughly a million books. The proof was finished on Saturday and verified by computer on Sunday, at a cost the company described only as "millions of dollars" in compute.

Competitive pressure was apparently intense enough that OpenAI skipped its usual step of circulating the proof among outside mathematicians before going public. Human researchers weren't entirely sidelined, though - OpenAI's Dan Roberts likened their role to that of "a bumble bee cross-pollinating across different groups and delivering different bits of information."

Why it matters

The Clay Mathematics Institute named the seven Millennium Prize Problems back in 2000, attaching a $1 million reward to each. Only one - the Poincaré conjecture - had been solved before now, by Grigori Perelman, who famously turned down the prize money. OpenAI says it has no plans to claim the reward for Navier-Stokes either. "Our goal in releasing this result is to report on the substantial progress of our AI models," the company said. "We do not intend to claim the Millennium Prize."

The announcement lands amid a broader wave of AI-driven progress in mathematics. OpenAI had earlier said Astra made headway on ten major open math problems, and an Anthropic staffer subsequently claimed its Fable model solved five of those same problems in a single day.

ABOUT THE AUTHOR

Kahekashan

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Kahekashan is a passionate technophile with a keen eye for cutting-edge gadgets, emerging technologies, and everything in the digital realm. Raised in a Defence family with strong values and a background in literature, she has consistently pursued excellence in every endeavour. Her last full-time assignment involved content writing with the Indian School of Business.

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