What OpenAI's latest controversy reveals about the future of mathematics

OpenAI has announced that its agents have solved the Navier-Stokes existence and smoothness problem, one of the seven Millennium Problems selected by the Clay Mathematics Institute in 2000, each carrying a one-million-dollar prize. Before this announcement, only one other of those seven problems had been solved, which gives a sense of the magnitude of the achievement: the Navier-Stokes equations describe how liquids and gases flow over time, are widely used in fluid dynamics, and until now it was unknown whether they could "break down" under certain conditions and predict physically impossible situations, such as infinite velocity. OpenAI claims to have proved that the full equations can indeed break down, using an unreleased internal model that clearly surpasses Astra, its most recent model, launched just the week before. The company has said it will not claim the prize money.

However, the mathematical achievement has been overshadowed by a controversy over its origin. On the Monday before OpenAI's announcement, New York University mathematician Tristan Buckmaster published on Mastodon a proof that a simplified version of the Navier-Stokes equations can break down, a notable advance in its own right. Buckmaster had worked on the problem for nearly a year alongside Levent Alpöge, an employee of Anthropic, OpenAI's main rival, relying on publicly available models from both companies. According to Buckmaster himself, who contacted an OpenAI employee after hearing rumors about the company's work, OpenAI staff gave him an ultimatum: either he and Alpöge published their advance and OpenAI would present its own solution the following day, or Buckmaster would collaborate with OpenAI on a Navier-Stokes paper that would exclude Alpöge as an author, precisely because of his ties to Anthropic. Buckmaster further states that he asked OpenAI employees whether its agents had had access to the transcripts of the work he and Alpöge had carried out with OpenAI models —which was denied— and whether those models had been trained on those transcripts, a question that went unanswered.

The obvious implication of this account is that OpenAI's models could have benefited, one way or another, from Buckmaster and Alpöge's earlier work. The scenario is plausible: both proofs —Buckmaster and Alpöge's, and OpenAI's— draw on an approach developed by mathematicians Diego Córdoba and Luis Martínez-Zoroa, which according to Javier Gómez-Serrano, a mathematics professor at Brown University, was one of several approaches considered promising for attacking the problem. In other words, it is not impossible that both teams arrived at that approach independently, but neither can it be ruled out that Buckmaster and Alpöge's work influenced OpenAI's. At a press conference, Sébastien Bubeck, a member of OpenAI's technical staff, acknowledged that the team was encouraged to take on the problem after hearing rumors about Buckmaster and Alpöge's efforts, while Mark Chen, OpenAI's chief research officer, again denied that any of the company's agents or employees had accessed the two researchers' transcripts. OpenAI, in short, denies the accusations, but the recent Hugging Face hack has shown that the company itself is not always fully aware of what its agents are doing, which leaves the door open to doubt.

If it turned out that OpenAI's models did benefit from Buckmaster and Alpöge's work, whether through training on their transcripts or improper access to them, the company's inability to clarify the facts and give due credit would leave it in a poor light. But that scenario would carry, according to the article, a small consolation for mathematicians: it would mean that the work of two human researchers —one of them a recognized expert on Navier-Stokes— was essential for OpenAI's agents to solve the problem. Experts have long pointed to so-called "research taste" —the ability to choose promising questions and research directions— as one of the major obstacles for AI in science and mathematics. If OpenAI's agents decided to follow the Córdoba and Martínez-Zoroa approach precisely because Buckmaster and Alpöge had already done so, then that human judgment was decisive for the company's success.

Even so, the broader picture is bleak. The progress achieved by Buckmaster and Alpöge after nearly a year of collaboration with publicly available models demonstrates the potential of human-AI collaboration, but it was not enough to reach a complete solution. OpenAI, by contrast, reached a solution within days using an internal model not available to the public, at an enormous cost: as Bubeck and Chen explained at the press conference, the team only managed to solve the problem by running close to 10,000 agents simultaneously, at a cost of millions of dollars. This is a scale of resources that, as Gómez-Serrano acknowledges, very few mathematicians will ever be able to match: "Whether AI companies will decide to spend their money on one thing or another, I honestly don't know. What is clear is that very few mathematicians will have resources of that magnitude."

The article also includes reflections from UCLA mathematician Terence Tao, who the previous week had posted a thread on Mastodon about the importance of errors, wrong turns and incomplete solutions to the advancement of mathematics. According to Tao, in most cases pure mathematics problems are not posed because the solution itself is desperately wanted, but because experience shows that human efforts to solve them tend to drive the development of the entire field. Solving a problem prematurely through purely AI-based methods —and particularly without transparency about the solution process— could contaminate that process to the point of becoming a net negative for the progress of mathematics as a whole. Humans, even if they take longer than agents to solve a problem, discover along the way new ideas and approaches that can inspire other researchers and even give rise to entire subfields; when AI agents are the ones solving those problems, and when private companies keep their failed paths hidden, those collateral benefits disappear.

The backdrop this episode paints is one of mathematics increasingly dominated by a handful of frontier AI companies, with internal models that far surpass any publicly available tool, virtually unlimited financial resources and little appetite for open academic collaboration. Several researchers have noted that the mathematical community is going through a moment of discouragement, understandable if the world's most advanced mathematics is reduced to a closed race between a couple of private labs. If OpenAI and Anthropic keep competing over ever more spectacular mathematical achievements, there is a chance that no significant open problems will be left for human mathematicians outside those companies, which would radically change the very nature of the field. It remains to be seen exactly what is lost in that process, but the Navier-Stokes episode suggests the cost could be high: not only in terms of credit for those who paved the way, but in the very way mathematical knowledge is generated and shared.