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NYU 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：https://cims.nyu.edu/~tristanb/statement.pdf 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 follow-up 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 said, “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.
Sébastian 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：https://www.claymath.org/millennium-problems/ 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.”
While Alpöge is employed by Anthropic, he was not conducting this research on the company’s behalf. As a result, the duo used a mix of models, relying primarily on OpenAI’s Codex in their work. Even so, Alpöge’s affiliation with a rival lab seems to have been a sore point for OpenAI, and Buckmaster alleges that Bubeck asked him to remove Alpöge’s credit as part of a proposed compromise.
When Buckmaster pushed to make the dispute public, he says that Bubeck replied: “Why would you ruin your career?” Buckmaster says that when he pushed back, Bubeck followed up with: “If you don’t want me to be nice, then I don’t have to be nice.”
Buckmaster also raised concerns that, because he used Codex extensively in assembling the project, information from his work could have informed OpenAI’s own efforts to solve the problem. OpenAI reserves the right to train models on Codex interactions,：https://help.openai.com/en/articles/5722486-how-your-data-is-used-to-improve-model-performance although users are able to opt-out. If the OpenAI team used a model trained on Buckmaster’s own Codex interactions, it’s plausible that it could have regurgitated his work when faced with a similar problem.
OpenAI did not respond to a request for comment on this possibility.
Regardless, the issue is likely to reignite the ongoing debate about AI’s role in mathematical research, and OpenAI’s specific incentives. For his part, Buckmaster seems to believe the best answer is to get as much information about the research out into the public eye.
“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. I am stating what I was told, when, and what was proposed to me. I am stating it because the alternative is to let a sequence of announcements say something I know to be false.”
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