That exchange, which Buckmaster described in a detailed public statement this week, sits at the heart of a ballooning controversy around a famous set of mathematical equations called Navier-Stokes.
The Navier-Stokes equations were formulated in the early 1800s by Claude-Louis Navier and George Gabriel Stokes. They're basically Newton's second law of motion for fluids, describing how any fluid moves: Push air with a fan blade, and the equations predict how that air will flow; or heat the atmosphere with sunlight, and they help forecast tomorrow's weather. They describe how planes fly and how blood moves through your arteries.
Imagine the swirl of water circling a bathtub drain. The Navier-Stokes equations describe how it spins faster as it tightens toward the center. A blowup would mean the math predicts that spin is accelerating forever — an infinite speed at a single point. The bathtub can't do that, but if the equations can, that means the model we trusted for years is wrong and we wouldn't even know where it breaks.
OpenAI generated this model of local incompressible motion while pushing the Navier-Stokes equations to their limits. Orange shows faster angular rotation; teal shows slower rotation. The trajectories show inward spiraling and axial stretching. (Image credit: OpenAI)To find the answer, OpenAI used a technique called "forcing" — giving the simulated fluid a smooth, controlled nudge from the outside. Imagine flicking a spinning top with your finger to see if it would fall over — or, in the case of the Navier-Stokes equations, blow up.
After roughly 50 hours, OpenAI's agents found blowups in the Euler equations, OpenAI representatives wrote in a statement. This discovery convinced them that Navier-Stokes was within reach, and they turned their attention to the bigger problem.
OpenAI ran the proof through a program called Lean, software that checks every step of a mathematical argument. The agents "sent 2.7 million messages and used approximately 130 billion output tokens," OpenAI representatives wrote in the statement. At a Sept. 8 news conference, company representatives said a similar effort performed by a customer would have cost $15 million.
The controversy
The two mathematicians had been working on the same strategy (smooth forcing) applied to those same Euler equations. The approach itself was developed by Diego Córdoba, a mathematician at the Institute of Mathematical Sciences in Madrid, and Luis Martínez-Zoroa, another mathematician at CUNEF University in Madrid. Almost no one else in the field was using it.
"When I heard 'forced,'" Buckmaster wrote in his public statement, "it was a bright red flag."
And yet, according to OpenAI, the company launched its Navier-Stokes effort on Sept. 1 — the same day it heard "rumors of other Millennium Prize resolutions." Buckmaster claimed, in his public statement, that those rumors were about his and Alpöge's work specifically.
OpenAI's CEO Sam Altman acknowledged that the company had learned of Buckmaster and Alpöge's work before making its announcement. "We were curious if ours could do it too," he wrote on X. (Image credit: Chip Somodevilla via Getty Images)
OpenAI's statement names both Buckmaster and Alpöge, credits their independent resolution of the forced Euler problem, and says the company offered to co-publish.
"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," he wrote. "I asked again, about training, and I did not get an answer."
In a statement on X, OpenAI representatives denied that company employees or agents saw the pair's work "by any means" until it was publicly released. However, "we cannot rule out that de-identified data derived from their usage of our products helped improve our models," they added.
We were curious if ours could do it too.
It was at that same Sept. 6 meeting that Buckmaster claimed that OpenAI presented him with two options, both of which required cutting Alpöge from authorship. He declined and said he told Bubeck that if OpenAI released its result as proposed, he would go public. Bubeck allegedly replied, "If you don't want me to be nice, then I don't have to be nice."
"I have not seen OpenAI's proof," Buckmaster wrote. "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 has called for an independent investigation into how OpenAI's work began.
The community response
Córdoba, one of the original developers of the forcing technique, said he was stunned by how quickly the result had materialized. "We're a little bit in shock," he told Scientific American.
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OpenAI's proof still awaits formal peer review, and the Clay Mathematics Institute has not yet issued an official verdict confirming or denying OpenAI's proof. But the episode has already surfaced troubling questions for the mathematics community. For example, when a company trains its models on the interactions of millions of users, who really owns the answers that emerge? And what does AI's involvement mean for the future of the field?
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