Hajra Naz
NYU mathematics professor Tristan Buckmaster announced three proofs on Tuesday. These findings offer preliminary progress on one of the major unsolved problems in theoretical mathematics. The work was done in collaboration with Anthropic mathematician Levent Alpöge. Both Codex and Claude AI models were used in the research. These findings are significant on their own. Still, they come with an unusual controversy attached. That controversy involves OpenAI’s parallel attempt 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.” According to that statement, a parallel effort by OpenAI built on their work before it went public. This led to a tangled mess of academic rivalries and conflicting claims.
OpenAI Publishes a Full Proof Shortly After
Shortly after Buckmaster’s statement, OpenAI published a full proof. It addressed the Navier–Stokes existence and smoothness problem. Buckmaster’s earlier findings had made progress toward the same problem. According to OpenAI, an unreleased next-generation model discovered this proof. That model had tackled several different unsolved problems over the past week. The week-long effort consumed 300 billion output tokens. At current Astra rates, that computing power would cost roughly $22.5 million.
The Navier-Stokes existence and smoothness problem is one of seven Millennium Prize problems. This is a set of major unsolved math problems. Each carries a $1 million bounty from the Clay Mathematics Institute. That reward goes to the first person or group to solve it. The Navier-Stokes equations are widely used in fluid mechanics. Still, they remain poorly understood in theoretical terms. A solution would represent a major advance in mathematical physics.
How the Dispute Unfolded
While finalizing their own results, Buckmaster and Alpöge learned something troubling. Information about their progress had reportedly been passed to OpenAI. When they contacted OpenAI directly, they received a specific response. OpenAI claimed it had already achieved a full proof of the central problem. However, when they asked follow-up questions, the answers grew evasive. Those questions focused on when OpenAI began its research and how much human input was involved.
“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 accurate, this timeline suggests something significant. The OpenAI team may have become convinced that Buckmaster and Alpöge’s approach was correct. The company may have then used its computing advantage to reach a formal proof first.
OpenAI’s own post confirms much of this timeline. Specifically, the company says its latest effort began on September 1. That effort was reportedly inspired by rumors that two Millennium Prize problems had been solved. The post also confirms ongoing conversations with Buckmaster and Alpöge.
Why the Timing Raised Suspicion
Many mathematicians pursue this problem broadly. However, the specific approach taken by Buckmaster and his collaborator is far less common. Because of this, Buckmaster found it suspicious. 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.”
Tension Over Credit and Affiliation
Alpöge is employed by Anthropic. However, he wasn’t conducting this specific research on the company’s behalf. As a result, the two researchers used a mix of AI models. They relied primarily on OpenAI’s Codex in their work. Even so, Alpöge’s connection to a rival lab seems to have bothered OpenAI. Buckmaster alleges that OpenAI mathematician Sébastien Bubeck asked him to remove Alpöge’s credit. This was reportedly part of a proposed compromise.
When Buckmaster pushed to make this dispute public, he says Bubeck responded pointedly. According to Buckmaster, Bubeck asked, “Why would you ruin your career?” Buckmaster says that when he pushed back further, Bubeck followed up with, “If you don’t want me to be nice, then I don’t have to be nice.”
Concerns About Data and Model Training
Buckmaster also raised a separate concern. He used Codex extensively while assembling this project. Because of that, information from his work could have potentially informed OpenAI’s own efforts. OpenAI reserves the right to train models on Codex interactions. Users can opt out of this, though. If OpenAI’s team used a model trained on Buckmaster’s own Codex interactions, it’s plausible that model could have regurgitated his work. This could have happened when the model faced a similar problem.
In its own post, OpenAI pushed back on this possibility. “We (the researchers and the agents) did not see any of their work through any means until they released it publicly—in particular, no specific user data was accessed to solve this problem,” the post reads. “While unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models. However, our proofs differ significantly, and even the precise results proved are different in the Euler case (forced vs unforced).”
A Renewed Debate Over AI in Mathematics
Regardless of how this specific dispute resolves, it’s likely to reignite a bigger conversation. That includes AI’s growing role in mathematical research. It also includes questions about OpenAI’s specific incentives in this space. For his part, Buckmaster seems to believe in one clear solution. He wants to get as much information about this research into the public eye as possible.
