GoKawiilNYU mathematics professor Tristan Buckmaster announced three proofs on Tuesday with a preliminary finding on one of the major unsolved problems in theoretical mathematics. The findings, made in collaboration with Anthropic mathematician Levent Alpöge and using both Codex and Claude AI models, are significant in themselves — but they’re also accompanied by an unusual controversy surrounding OpenAI’s attempts to solve the same problem.
“There is another part of this story,” Buckmaster wrote in his statement announcing the proofs, “and one that, honestly, I very much wish I did not have to be concerned with.”
While Buckmaster and Alpöge were finalizing their results, they learned that “information about our progress had been passed to OpenAI.” When they contacted OpenAI about this, they were told that OpenAI had already achieved a full proof of the central problem. But when they asked followup questions about when OpenAI had begun its research into the problem and how much human input was involved, the answers became more evasive.
“It emerged that an entire team had been working on the problem,” Buckmaster says,” and that an insane amount of compute had been used…. Eventually, it was agreed that [the first prompt] had been sent in the past few days, after information about our work had reached OpenAI.”
If true, that would suggest the OpenAI team had become convinced that Buckmaster and Alpöge’s approach was the right one, and decided to use its material advantage in computing resources to reach a formal proof first.
Sebastian Bubeck, who leads OpenAI’s mathematical research, says those claims are “false and inflammatory.”
“To clarify, I came into the discussion following academic norms, and I’m disappointed that it has come to this,” he wrote in a post after Buckmaster’s statement. “Anyone who knows me knows that academic standards are of the highest importance to me.” Bubeck said he would make a more complete statement soon.
The dispute centers on the “Navier-Stokes existence and smoothness” problem, one of the seven “Millennium Prize problems” — a set of major unsolved math problems, each carrying a $1 million bounty from Clay Mathematics Institute for the first person or group to provide a solution. The Navier-Stokes equations are widely used in fluid mechanics but poorly understood in theoretical terms. A solution would represent a significant advance in the collective understanding of mathematical physics.
Although the problem is widely pursued among mathematicians, the specific tactic taken by Buckmaster and his collaborator is far less common. As a result, Buckmaster found it suspicious that OpenAI ended up taking the same approach at the same time.
“The route to the Clay problem through a smooth force, options c and d in Fefferman’s statement of the problem, is the route Luis and Diego opened and the one Levent and I had quietly chosen to attack,” Buckmaster wrote. “Almost nobody else I know of was working on it,” he continued. “It is not the direction one arrives at in a few days by giving a model the problem statement.”
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