OpenAI announced it has solved the Navier–Stokes existence and smoothness problem, one of seven Millennium Prize Problems. The claim comes with a million-dollar prize attached, though the company says it will not claim the money. The announcement is overshadowed by accusations that OpenAI used the work of NYU mathematician Tristan Buckmaster and Anthropic employee Levent Alpöge as a starting point without giving them credit. OpenAI denies the claims.
The Navier–Stokes problem
The equations describe how fluids like water and air move over time. They are widely used in fluid dynamics but have never been fully understood. Physicists and mathematicians did not know if the equations could break down and predict impossible states, such as a fluid having infinite velocity. On Monday, Buckmaster posted a proof on Mastodon showing a simplified version of the equations can indeed break down. He and Alpöge worked on the problem for nearly a year using publicly available models from OpenAI and Anthropic. Today, OpenAI presented a proof that the full equations can break down as well. The proof was obtained using an internal model that dramatically outperforms the Astra model, which was released last week. The company says it does not plan to claim the million-dollar prize.
The controversy
Along with the proof, Buckmaster posted a document detailing his interactions with OpenAI employees after he heard rumors about their work and reached out to one of them. According to him, OpenAI employees presented two possibilities. Either he and Alpöge could post their work and OpenAI would post their Navier-Stokes solution the following day, or he could work with OpenAI on a Navier-Stokes paper that excluded Alpöge from authorship, due to his affiliation with Anthropic, OpenAI’s biggest rival. Buckmaster also wrote that he asked the employees whether the agents had obtained access to transcripts of the work that he and Alpöge had done with OpenAI models, which they denied. He asked whether OpenAI models had been trained on those transcripts, to which they offered no response. MIT Technology Review reached out to Buckmaster for comment but did not hear back before publication.
The clear implication of the document is that OpenAI’s models somehow made use of Buckmaster and Alpöge’s work. That scenario is plausible. Both the Buckmaster/Alpöge and OpenAI proofs make use of an approach to the Navier-Stokes problem pioneered by the mathematicians Diego Córdoba and Luis Martínez-Zoroa. According to Javier Gómez-Serrano, a mathematics professor at Brown University, this approach was one of several that was thought to hold promise for solving the Navier-Stokes problem. While it is by no means impossible that both teams could have arrived at this approach independently, it is also conceivable that Buckmaster and Alpöge’s work could have influenced OpenAI’s.
In the press briefing, Mark Chen, OpenAI’s chief research officer, again denied that any agents or OpenAI employees accessed Buckmaster and Alpöge’s transcripts. Given what has been revealed about the Hugging Face hack, it is clear that OpenAI is not always entirely aware of what its agents are doing.
What it means for researchers
If OpenAI’s models did train on Buckmaster and Alpöge’s work, or if its agents somehow gained access to it, then the company’s failure to track down the truth and assign those researchers appropriate credit reflects poorly on it. But there might be a thin silver lining to that version of the story for mathematicians, because it would suggest that the hard work of two humans, one of whom is a prominent expert on Navier-Stokes, was essential to the agents’ ability to solve the Millennium Problem. Experts have long identified “research taste,” or the ability to choose promising research questions and directions, as a major obstacle for AI in science and mathematics. If the OpenAI agents did indeed choose to follow the Córdoba–Martínez-Zoroa approach because Buckmaster and Alpöge had done the same, then human research taste played an essential role in OpenAI’s success.
Even so, the bigger picture here is sobering. The progress that Buckmaster and Alpöge made over almost a year of collaboration with publicly available models speaks to the promise of human–AI collaboration. But they were not able to achieve a full solution. Meanwhile, OpenAI brute-forced a solution in a few days using an internal model, and their successful solution came at an astronomical cost. In the press briefing, Bubeck and Chen said the team was only able to solve the problem by running about 10,000 agents concurrently, at a cost of millions of dollars.
Over the past few months, several researchers have said mathematicians are becoming depressed. It is not difficult to see why. Mathematics is quickly becoming the province of frontier AI companies with impressive internal-only models, money to burn, and a lack of collaborative spirit. “Whether AI companies will decide to spend their money on doing one thing or another, I truly don’t know,” says Gómez-Serrano. “What is clear is that very few mathematicians will have resources of that scale.”
If OpenAI and Anthropic keep striving for more and more impressive mathematical accolades, there might not be any open problems left for human mathematicians outside of those companies to wrestle with. That would dramatically change the field of mathematics. Last week, UCLA mathematician Terence Tao wrote a Mastodon thread describing how important mistakes, wrong directions, and incomplete solutions are for the field. “In most cases in pure mathematics, the problems are posed not because we desperately want the solution to these problems in and of themselves, but because we have seen from past experience that human-directed efforts to solve these problems tend to spur further development of the field,” Tao wrote. “Prematurely solving the problem by purely AI-powered methods—particularly without full transparency into the solution process—can contaminate this process to the point where it actually becomes a net negative for the progress of mathematics as a whole.”
Humans might take longer than agents to solve mathematical problems, but in the process, they uncover new mathematical approaches and ideas that might inspire their peers and even birth their own subfields. But when AI agents solve those problems instead—and when private companies keep the agents’ wrong turns from public view—those benefits disappear. It remains to be seen what else will vanish in the process.
