OpenAI admits it cannot rule out training on mathematician's unpublished Navier-Stokes work

NYU math professor Tristan Buckmaster and collaborator Levent Alpöge spent nearly a year working toward a proof of finite-time blowup for fluid equations — a problem related to the Navier-Stokes Millennium Prize — feeding drafts into OpenAI's Codex throughout. In early September, Buckmaster learned that an internal OpenAI model had independently produced a similar proof, using the same smooth-forcing setup he and Alpöge had quietly chosen. He directly asked OpenAI whether his Codex sessions had been used in training, but received no clear answer until the company issued a public statement. OpenAI's statement confirmed that while no user data was directly 'accessed' or 'seen' by its model, it 'cannot rule out' that de-identified data from their product usage helped improve its models. The episode highlights that consumer Codex sessions are used as training data by default, raising broader questions about the use of unpublished professional work submitted through AI coding tools.