GoKawiilMost anyone involved in computing has heard about the P-NP problem, but fluid engineers and mathematicians would love to know if the Navier-Stokes equations have smooth, globally defined solutions. Both questions are part of the Millennium Prize Problems, solutions to which are worth a cool $1 million and eternal renown. OpenAI is claiming that its staff and internal models have solved the conditions of Navier-Stokes solutions set forth in the Millennium Prize. But the company's shouting from the rooftops is being met with a chorus of boos over claims it might have plagiarized the work of a research team that had been toiling on a related, stepping-stone problem for a year.
Tristan Buckmaster (a scientist at NYU) and Levent Alpöge (a member of Anthropic's staff) had been quietly working on proving Euler's equations — another long-standing mathematical problem, and one that is generally acknowledged to be a stepping stone to solving Navier-Stokes.
According to Buckmaster, his work with Alpöge was "a purely personal collaboration, free of any institutional agreements or official involvement by either of our employers." The researchers used Anthropic Claude and OpenAI Codex as assistants, as is apparently now common in the field, to perform busywork (documentation, searching, etc.) as well as running through logic steps. The substantial amount of compute time the project required was paid from Buckmaster's own pockets, too.
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The pair worked for roughly a year until August 15, 2026, when it obtained "the blowup results, with smooth forcing, for both Boussinesq and Euler." Buckmaster says the novel approach was based on previous work by Diego Córdoba and Luis Martínez-Zoroa, and he believes Zoroa should be eligible for a Fields Medal.
Although the team was presumably happy with these achievements, Buckmaster said that the LLM-generated proof was "the most horrendous" he'd seen, calling it "AI slop," and meaning to rewrite it for clarity. Nevertheless, they verified it on August 22 using Lean, a standardized programming language designed specifically to verify mathematical proofs.
Come September 3, Alpöge told Buckmaster of rumors going around that Anthropic had solved an important mathematical problem. This almost certainly alluded to the team's work, and some apparently took it to mean the company itself was working on the problem. The rumor-mongers even theorized that the problem that Anthropic had solved was Navier-Stokes. Alpöge further believed that OpenAI had gotten wind of the news.
This prompted Buckmaster to email an unnamed "prominent mathematician" at OpenAI, clarifying that the effort was a personal collaboration between him and Alpöge and was unrelated to Anthropic. The mathematician replied asking for details, saying "it would be useful to avoid competing," and offering OpenAI compute time. After a few days, on September 6, Buckmaster, the unnamed person, and OpenAI's Sébastien Bubeck talked twice, without Alpöge. He was told that OpenAI had proven a finite-time blowup for the forced Navier-Stokes equations, a subset of the problem.
Alpöge asked by text for the precise statement and was told "existence of forced blowup in R³ and T³", and that "the forcing function is smooth option [C] and [D] in Fefferman," referring to one of the four possible categories established by the Millennium Prize, with any one of them being valid as eligible for the prize, but not constituting a full solution for all scenarios, a distinction remarked on by other scientists.
This is where the story becomes interesting. Buckmaster claims that that idea (forced blowup) was exactly the same one his team had "quietly" chosen, and that nobody else he knew was working on it. Perhaps most importantly, he says that that was "not the direction one arrives at in a few days by giving a model the problem statement," indicating that running the general problem through a bot wouldn't quickly reveal that potential approach.
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