OpenAI’s breakthrough solution for the elusive Navier-Stokes problem overshadowed by plagiarism controversy — researcher says OpenAI scraped Codex session and issued career threats
Most 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.
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.
In fact, Buckmaster claims that over the calls, Bubeck ultimately revealed that instead of just AI models and agents with a couple of handlers, there was an entire team of live humans working on Navier-Stokes. The OpenAI team first had the models try to work through easier paths, and the text prompt that generated the Navier-Stokes proof had itself been generated by prompting Codex, with an “insane” amount of computing needed.
Buckmaster then asked when the initial prompt was issued, and OpenAI’s response of “in the past few days” did not arrive until “some time” passed. He proceeded to ask if the model “had been trained on, or had access to, our sessions in Codex,” and was told by OpenAI that Codex does not access user data. Finally, he asked if the data was used for model training more generally and, crucially, apparently did not get an answer.
OpenAI allegedly offered Buckmaster two options: one, that Buckmaster and Alpöge publish their Euler proof first. The following day, OpenAI would post its Navier-Stokes proof, giving the two priority. The second option was that Buckmaster alone, without Levant, was to write a paper with the Navier-Stokes proof, acknowledging that an internal OpenAI model resolved it. Bubeck was apparently adamant about Levant’s removal from the Euler proof, as his employment at Anthropic was “annoying.” Buckmaster opted for neither, and told OpenAI that if it chose the first option, he’d go public with his findings, as has since occurred.
This prompted what Buckmaster interpreted as a threat from Bubeck, who asked him “why [he] would ruin [his] career.” After Buckmaster asked why that would happen, Bubeck told him, “If you don’t want me to be nice, then I don’t have to be nice.” Bubeck then allegedly reached out to Alpöge, questioning Buckmaster’s sanity, to which Alpöge responded with a refusal, pointing inquiries back to his colleague.
The entire story raises pointed questions about what OpenAI (and others) are actually doing with user data collected via its LLMs, despite the toggle switches that are supposed to disable it. Not only has OpenAI neglected to tell Buckmaster whether it used his team’s data for training, in its PR about Navier-Stokes, the company says while it “no specific user data was accessed in order to solve this problem,“ it “cannot rule out that de-identified data derived from their usage of our products helped improve [its] models.”
OpenAI’s proof still needs to undergo a likely years-long peer review before any party can take the Millennium Prize home. The firm has stated it does not intend to claim it. As for Bubeck, he predictably paints the story in a very different light, but insists that his pushing away of Alpöge is justified on the basis that “it would be inappropriate for an Anthropic employee to author OpenAI’s work,” a puzzling statement that some could take as meaning a double standard regarding scientific authorship, based solely on corporate rivalry.
For his part, OpenAI CEO Sam Altman claims his team was well-intentioned and cooperative, and supported Bubeck, saying “it was challenging to offer [the same publication options] to Levent.” Neither person opted to discuss the matter of whether OpenAI used the research of Buckmaster and Alpöge as training data, or offered any further explanation of why Alpöge didn’t deserve credit for his work as an equal to Buckmaster.
Given the groundbreaking nature of this apparent discovery and the ensuing fight for priority that these competing accounts have sparked, it’ll likely take quite some time and review before we know whether and how OpenAI or Buckmaster and Alpöge will be credited with this discovery. But given the inter-lab rancor already on display, the process will surely be ugly.
