OpenAI Claims Navier–Stokes — After an NYU Mathematician Cried Foul
OpenAI claims a full proof of the Navier–Stokes existence and smoothness problem using 300 billion output tokens, following allegations from NYU mathematician Tristan Buckmaster that the effort relied on leaked details of his own unpublished work.
NYU mathematics professor Tristan Buckmaster announced three proofs on Tuesday regarding the Navier–Stokes existence and smoothness problem, a Millennium Prize challenge carrying a $1 million bounty. Working with Anthropic mathematician Levent Alpöge, Buckmaster utilized OpenAI's Codex and Claude models to advance a specific tactical route through smooth force options c and d in Fefferman's statement. Shortly after this announcement, OpenAI published a full proof of the central problem, attributing the discovery to an unreleased next-generation model. The company stated the week-long effort began on September 1, inspired by rumors that two Millennium Prize problems had been solved, and consumed 300 billion output tokens. At current Astra rates, this compute usage equates to $22.5 million.
The core controversy centers on whether OpenAI accessed Buckmaster's research before it became public. Buckmaster asserts that information about his progress was passed to OpenAI, prompting the lab to deploy massive computational resources to replicate and extend his specific approach. He notes that almost no other researchers were pursuing this particular direction, making the simultaneous convergence suspicious. When Buckmaster questioned OpenAI leadership about the timeline and human input involved, he alleges that Sébastien Bubeck suggested removing Alpöge's credit as part of a compromise, warning that publicizing the dispute could ruin Buckmaster's career. Bubeck reportedly stated, "If you don't want me to be nice, then I don't have to be nice."
OpenAI denies accessing specific user data or seeing Buckmaster's work prior to its public release. In their post, researchers claim they did not view the duo's work through any means until it was publicly available, though they acknowledge they cannot rule out that de-identified data from Codex interactions helped improve their models. OpenAI emphasizes that their proofs differ significantly from Buckmaster's, particularly in the Euler case where results vary between forced and unforced scenarios. Despite these denials, the incident highlights risks regarding data privacy when using proprietary models for sensitive pre-publication research, especially given OpenAI's right to train on Codex interactions unless users explicitly opt out.