On Tuesday local time, OpenAI dropped a bombshell.
This AI giant announced that an internal, unreleased AI model solved the existence and smoothness problem of the Navier-Stokes equations in just 88 hours— one of the seven Millennium Prize Problems designated by the Clay Mathematics Institute in 2000, with a $1 million prize, and which has puzzled the mathematical community for nearly a century.
The Navier-Stokes equations, developed in the 19th century by French engineer Claude-Louis Navier and Irish mathematician George Gabriel Stokes, describe the motion of fluids such as air, water, and blood . From aircraft design to weather forecasting, from blood flow to ocean current simulation, these equations permeate nearly all scientific and engineering fields involving fluids .
But a fundamental question remains unresolved: Do these equations always yield physically reasonable answers, or do they “blow up” under certain conditions—producing infinite velocities?
OpenAI's answer is clear: it will explode.
OpenAI explicitly stated in the official announcement that its system generated a proof and a Lean formal verification demonstrating that an initially smooth, stationary fluid can develop a singularity in finite time. This resolves statements “C” and “D” of the Millennium Prize Problem.
What does this look like physically? OpenAI describes it as a “vortex” — fluid spirals inward like spaghetti, stretching continuously, with the central region contracting and accelerating, while energy remains finite. The technical challenge lies in the Navier-Stokes equations: the terms describing motion — acceleration, pressure gradient, momentum transfer, and viscosity — must all become enormous yet cancel each other out with perfect precision.
The derivation process is extremely costly:
OpenAI deployed up to 10,000 parallel AI agents. First, 1,000 agents solved the Euler equation (a "close relative" of the Navier-Stokes equations, with the viscous term removed) in 50 hours; then, all 10,000 agents spent 11 hours extending the solution to the full Navier-Stokes problem. The agents exchanged nearly 3 million messages and consumed approximately 130 billion output tokens, with computational costs reaching millions of dollars.
According to New Scientist, if customers want to reproduce this result, OpenAI has quoted approximately $15 million.
Martin Bridson, President of the Clay Mathematics Institute, commented: “This is indeed an exciting day, as we are witnessing a major advancement in humanity’s understanding of mathematics.”
But this discovery has been shrouded in controversy from the start.
They stole our research!
On the day before OpenAI's announcement—September 7—New York University mathematics professor Tristan Buckmaster and Anthropic researcher Levent Alpöge had just published a paper announcing a major breakthrough in related fluid equations.
Both researchers also leveraged AI: they used Anthropic’s Claude model and OpenAI’s Codex and Astra models. Renowned mathematician and Fields Medalist Terence Tao called this work an “extraordinary achievement” on social media.
However, Buckmaster subsequently issued a statement accusing OpenAI of hastily reallocating resources to publish their results after learning of their research progress.
Buckmaster detailed the events in the statement:
On September 6, Buckmaster received a call from OpenAI informing him that their internal model had completed a breakthrough proof of the "externally driven" version of the Navier-Stokes equations, with a manuscript approximately 100 pages long.
When Buckmaster pressed further on when OpenAI began researching the issue and how many people were involved, the responses became increasingly vague. Eventually, OpenAI admitted that the first prompt was sent “in the past few days,” and it was during this same period that news of Buckmaster and Alpöge’s research progress reached OpenAI.
What angered Buckmaster even more was that OpenAI attempted to remove Alpöge from the list of authors.
Buckmaster claimed that OpenAI offered him a “collaboration proposal”: he could be listed as the author of a paper announcing that OpenAI’s internal models had solved the Navier-Stokes problem, but Alpöge’s name could not appear—on the grounds that he was affiliated with the competitor Anthropic. Buckmaster rejected this proposal.
When Buckmaster said he intended to make these concerns public, the other party allegedly responded: “Why would you ruin your own career?” and “If you don’t want me to be nice, then I don’t have to be nice either.”
Buckmaster also noted that the research path he and Alpöge chose—attacking the Clay problem via a “force-driven” approach—is a very niche direction. “As far as I know, almost no one else is working on this,” he wrote, “it’s not a direction you can come up with in a few days just by giving a model a problem statement.”
OpenAI responds: "No theft occurred."
In response to these allegations, OpenAI quickly responded.
OpenAI mathematician Sébastien Bubeck firmly denied all allegations at the press conference, stating that OpenAI’s internal model independently solved the Euler problem (a “cousin” of the Navier-Stokes problem) using a method entirely different from that of Buckmaster and Alpöge.
For the solution to the full Navier-Stokes problem, Bubeck acknowledges that a method similar to that of two mathematicians was indeed used, but he emphasizes that it was entirely an independent discovery.
“We did not use their prompts or proofs to prompt our model or guide our agents,” said Bubeck.
Regarding the timeline, OpenAI's explanation is: On August 28, they began training a new AI model with advanced mathematical capabilities . On September 1, the company heard rumors that Anthropic had made progress on the Navier-Stokes problem and decided to allocate additional computational resources to this issue.
“No personnel or AI system searched user data to resolve this issue,” said Mark Chen, Chief Research Officer at OpenAI, at the press event. Bubeck also emphasized: “Before our release, we had not seen any of their work (Buckmaster and Alpöge). In hindsight, we can see that our proofs differed significantly, and even the specific results of the proofs were different.”
OpenAI also acknowledged, “Although unlikely, we cannot completely rule out that de-identified data from their use of our products could help improve our models.”
Regarding the accusation of "removing Alpöge's name," Bubeck also refuted it on social media.
Hashpower rules—who gets sacrificed?
The essence of this dispute goes far beyond a personal feud between two individuals or two competing companies.
Scientific American states: "Tracing the origins of AI mathematical ideas is extremely difficult." When AI models are trained on datasets containing vast amounts of human research, how do we define the boundary between "original" and "derivative"? When one company has access to another company’s employees’ Codex, how can we ensure research is not “borrowed”?
Another point of contention is that OpenAI’s proof may not meet the original requirements of the Millennium Prize Problem. The Clay Mathematics Institute’s definition of the Navier-Stokes problem is based on equations without external forces, whereas OpenAI’s proof addresses the version with external forces. According to BBC, OpenAI’s solution addresses two of the four statements required by the Millennium Prize, but OpenAI does not intend to claim the prize.
But there is another, deeper concern from Fields Medalist Terence Tao.
Just before the controversy erupted, Terence Tao had discussed the potential impact of AI on mathematical research, using the Navier-Stokes equations as an example.
Terence Tao is concerned that if AI completes an entire exploration in a closed environment, humans may ultimately receive only a finalized result, with many valuable intermediate pathways never entering the mathematical community. Open problems like the Navier-Stokes equations have, for decades, driven the development of new methods, new questions, and new researchers.
He devised a thought experiment: “What if AI had arrived 20 years earlier?” In that world, Zhang Yitang might have been outcompeted by AI. In 2005, mathematicians developed the GPY method; in 2013, Zhang Yitang proved that the upper bound for prime gaps is 70 million. Terence Tao then launched the Polymath8 project, enabling mathematicians worldwide to collaborate and ultimately leading to major breakthroughs such as the Maynard sieve.
But what if AI had brute-forced this problem back in 2005? Those methods might never have been discovered, and the entire field of number theory could have stalled. Terence Tao warned that if AI brute-forces solutions merely to climb leaderboards, it might “kill the goose that lays the golden eggs,” causing mathematics to stop growing.
But computing power is the real currency of this era.
OpenAI can accomplish in days what mathematicians might take months or even years to achieve, not through smarter algorithms, but through 10,000 agents and millions of dollars in computing resources.
Buckmaster compared this event to a "Deep Blue versus Kasparov moment"—like how IBM's Deep Blue supercomputer defeated chess world champion Kasparov in 1997. But unlike chess, which has clear rules and referees, the rules for academic competition in the AI era are still being written.
Terence Tao's thought experiment makes it clear: when AI companies can brute-force any mathematical problem with sheer computational power, what may be truly sacrificed is mathematics itself—the new methods, tools, and problems born through exploration, along with the entire developmental path of a generation of mathematicians.
Perhaps in this era, the one who finds the answer first wins, and no one cares what was sacrificed along the way.
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