On Tuesday, NYU mathematics professor Tristan Buckmaster announced three proofs with a preliminary finding on one of the most significant unsolved problems in theoretical mathematics. The Navier-Stokes proof controversy began when Buckmaster revealed that OpenAI may have used information from his research to publish its own competing solution.
The Navier-Stokes existence and smoothness problem is one of the seven Millennium Prize problems, each carrying a $1 million bounty from the Clay Mathematics Institute for the first person or group to provide a valid solution. These equations are widely used in fluid mechanics but remain poorly understood in theoretical terms. A complete solution would represent a monumental advance in mathematical physics, making the stakes incredibly high for everyone involved.
Working alongside Anthropic mathematician Levent Alpöge, Buckmaster used both OpenAI’s Codex and Claude AI models in their research. Their findings are significant in their own right, but they also come with an unusual controversy surrounding OpenAI’s parallel efforts to solve the same problem. The Navier-Stokes proof controversy centers on allegations that OpenAI built on their work before it became public, leading to conflicting claims and academic rivalry.
OpenAI Publishes Its Own Proof
Shortly after Buckmaster’s statement, OpenAI published what it claimed was a full proof of the Navier–Stokes existence and smoothness problem. According to OpenAI, the proof was discovered by an unreleased next-generation model that had tackled a range of unsolved problems over the previous week. The week-long effort consumed 300 billion output tokens, amounting to $22.5 million worth of compute at current Astra rates.
While Buckmaster and Alpöge were finalizing their own results, they learned that “information about our progress had been passed to OpenAI.” When they reached out to OpenAI for clarification, they were told the company had already achieved a complete proof of the central problem. However, follow-up questions about when OpenAI began its research and how much human input was involved were met with evasive answers.
Suspicious Timing and Shared Methodology
The specific mathematical approach taken by Buckmaster and Alpöge is far from common. In his statement, Buckmaster wrote: “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.” He continued, “Almost nobody else I know of was working on it. It is not the direction one arrives at in a few days by giving a model the problem statement.”
This raises serious questions at the heart of the Navier-Stokes proof controversy: why did OpenAI suddenly pursue the same unconventional approach at the same time? Buckmaster found this timeline highly suspicious. It emerged that an entire team had been working on the problem at OpenAI, and an extraordinary amount of compute had been deployed. Eventually, it was agreed that the first prompt had been sent in the past few days, after information about Buckmaster and Alpöge’s work had reached OpenAI.
If true, this would suggest that OpenAI became convinced Buckmaster’s approach was correct and decided to leverage its material advantage in computing resources to reach a formal proof first. The Navier-Stokes proof controversy highlights how AI companies with massive computational resources could potentially race ahead of academic researchers who cannot match their scale.
Allegations of Intimidation and Credit Disputes
The tension escalated further when Buckmaster alleged that Sébastien Bubeck from OpenAI asked him to remove Alpöge’s credit as part of a proposed compromise. While Alpöge is employed by Anthropic, he was not conducting this research on the company’s behalf. The duo relied primarily on OpenAI’s Codex in their work, making Alpöge’s affiliation with a rival lab a sore point for OpenAI.
When Buckmaster pushed to make the dispute public, he claims 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.” These allegations paint a troubling picture of how major AI companies might behave when competing for prestigious scientific recognition. Buckmaster’s willingness to speak openly about the dispute, despite potential career risks, highlights the high stakes involved in the Navier-Stokes proof controversy.
Did Codex Training Data Play a Role?
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. OpenAI reserves the right to train models on Codex interactions, although users can opt out. If the OpenAI team used a model trained on Buckmaster’s own Codex interactions, it’s plausible the model could have regurgitated his work when faced with a similar problem.
In its own post, OpenAI downplayed 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 in order to solve this problem.” However, the statement acknowledged, “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).”
This aspect of the Navier-Stokes proof controversy raises broader questions about data rights and intellectual property in AI research. If academic researchers contribute data to AI systems that are then used to compete against them, the entire ecosystem of scientific collaboration could be undermined.
Implications for AI in Mathematical Research
Regardless of the outcome, the Navier-Stokes proof controversy is likely to reignite the ongoing debate about AI’s role in mathematical research and OpenAI’s specific incentives. The Millennium Prize problems have long been considered the ultimate test for mathematical genius. If AI systems can solve them, it raises fundamental questions about the nature of discovery and credit in mathematics.
For mathematicians and AI researchers alike, this controversy serves as a cautionary tale about competition, intellectual property, and ethics in the age of advanced AI. The question of who truly solved the Navier-Stokes problem—and whether AI companies should be allowed to use user data to gain competitive advantages—will likely be debated for years to come.
Buckmaster’s response has been to push for as much transparency as possible, getting information about the research out into the public eye. His statement concluded: “There is another part of this story, and one that, honestly, I very much wish I did not have to be concerned with.” The Navier-Stokes proof controversy demonstrates that as AI becomes more powerful, the lines between collaboration and competition in scientific research may become increasingly blurred.
