On September 8, 2026, the mathematics world was rocked by both a major breakthrough and a brewing controversy. NYU professor Tristan Buckmaster, alongside Anthropic mathematician Levent Alpöge, announced three new proofs that advanced understanding of the Navier–Stokes existence and smoothness problem—one of the infamous Millennium Prize problems, whose solution carries a $1 million reward from the Clay Mathematics Institute. Their results, achieved using OpenAI’s Codex and Anthropic’s Claude AI models, marked a significant leap in a field that has long stymied the best minds in mathematics.
But almost as soon as Buckmaster made his announcement, OpenAI published its own full proof of the Navier–Stokes problem, igniting a fierce debate over credit, data ethics, and the growing influence of artificial intelligence in mathematical research. According to WIRED and The Decoder, the proof was produced by an unreleased next-generation OpenAI model—described as even more capable than GPT-6 Astra—after a week-long effort involving 10,000 coordinated AI agents and a staggering 300 billion output tokens. The cost? Roughly $22.5 million in compute, or as OpenAI’s Mark Chen put it, “in the millions of dollars.”
The Navier–Stokes equations, central to fluid mechanics, have long been a source of fascination and frustration for mathematicians. Despite their practical utility, the theoretical underpinnings of these equations—particularly questions of existence and smoothness—have remained elusive. A solution to this problem would represent not just a feather in the cap for the mathematicians involved, but a landmark in the history of mathematical physics.
Yet, as Buckmaster revealed in his public statement, the triumph was marred by suspicion and academic rivalry. He alleged that information about his and Alpöge’s progress had been passed to OpenAI before their results were made public. When Buckmaster reached out to OpenAI for clarification, he was told that the company had already achieved a full proof using its internal AI model. The timeline was unsettling: OpenAI confirmed its own research effort began on September 1, 2026, after hearing rumors that two Millennium Prize problems had been solved—a rumor that, it turned out, was linked to Buckmaster and Alpöge’s work.
“It emerged that an entire team had been working on the problem, and that an insane amount of compute had been used,” Buckmaster wrote. He found it suspicious that OpenAI had chosen the same rare approach he and Alpöge had quietly pursued, especially since “almost nobody else I know of was working on it.” According to Buckmaster, this was not a direction “one arrives at in a few days by giving a model the problem statement.”
The situation quickly escalated. Buckmaster alleged that OpenAI’s Sébastien Bubeck pressured him to remove Alpöge from authorship on the grounds that Alpöge was employed by Anthropic, a rival AI lab. “Why would you ruin your career?” Bubeck allegedly asked when Buckmaster refused. When Buckmaster stood his ground, Bubeck reportedly responded, “If you don’t want me to be nice, then I don’t have to be nice.” Bubeck also allegedly texted Alpöge separately, saying, “I don’t know if Tristan is being fully rational right now.”
OpenAI, for its part, has firmly denied any impropriety. In a press briefing cited by WIRED, Bubeck stated, “We, whether it’s the researchers or the agents, did not see any of their work until it was released publicly last night.” OpenAI congratulated Buckmaster and Alpöge on their “monumental achievement” and emphasized that their solution differed fundamentally from the approach taken by Buckmaster and Alpöge, particularly in the Euler case (forced versus unforced). Ven Chandrasekaran, an OpenAI mathematician, underscored that the company’s proof was not derived from their work.
Still, Buckmaster raised pointed ethical questions about whether OpenAI’s models may have been influenced by data from his own Codex sessions. Throughout their project, Buckmaster and Alpöge had uploaded drafts to Codex, and OpenAI reserves the right to train its models on user interactions unless users opt out. “Note that they are openly admitting they used training data from a period after we found our result. Is it ethical to use customer’s data to try to scoop their customer?” Buckmaster wrote on Mastodon. OpenAI, in its official blog post, stated, “While unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models.”
The dispute has thrown a spotlight on the challenges of credit and attribution in the age of AI-assisted research. As The Decoder points out, this is a “Deep Blue–Kasparov moment” for mathematics—a reference to the historic 1997 chess match where IBM’s Deep Blue defeated world champion Garry Kasparov, forever changing the landscape of competitive chess. Today, AI models are not just playing games; they’re tackling some of the most complex unsolved problems in human knowledge.
OpenAI’s own developer, Noam Brown, observed that the cost of deploying advanced AI to solve such problems is dropping rapidly. “I believe that a year from now everyone will have an AI at their fingertips capable of solving problems of this caliber,” Brown wrote. What once required millions in compute and elite teams may soon be accessible to anyone with a $20 ChatGPT subscription. The 2025 Math Olympiad, he noted, required massive compute from both OpenAI and Google DeepMind, but by 2026, everyday users could match that power.
For now, the dust has yet to settle. Buckmaster and Alpöge’s achievement stands as a testament to the potential of human–AI collaboration, but the shadow of controversy lingers. The question of whether AI companies should be able to use customer data for model training—especially when those customers are researchers working on sensitive, high-stakes problems—remains hotly debated. OpenAI maintains that it did not access specific user data before the public release of Buckmaster and Alpöge’s work, but the company admits that de-identified data could have influenced their models.
As the mathematics community digests these events, the debate is sure to intensify: How should credit be assigned when breakthroughs are achieved with the help of AI? What safeguards are needed to ensure that researchers’ intellectual work isn’t quietly absorbed and repurposed by the very tools they rely on? And perhaps most fundamentally, what does it mean for the future of discovery when the boundary between human and machine creativity grows ever blurrier?
Whatever the answers, one thing is clear: the Navier–Stokes breakthrough will be remembered not only for its mathematical significance, but for the ethical and cultural questions it has raised about the role of artificial intelligence in the pursuit of knowledge.