OpenAI Can't Rule Out Using Rival's Codex Sessions

OpenAI said Tuesday that an unreleased internal model had resolved part of the Navier-Stokes Millennium Prize problem, and within two days a rival mathematician was asking publicly whether that model had been trained on his own unpublished drafts.

OpenAI published its proof and a formal write-up on Sept. 8, saying the result establishes statements C and D of the Clay Mathematics Institute's official problem description. The company said it will not seek the $1 million prize, because the fluid in its proof required a continuous external force, a condition the original Millennium Prize formulation does not include.

What OpenAI's model proved about Navier-Stokes

The claim concerns a singularity, a point where a smooth fluid develops a mathematical discontinuity in finite time. OpenAI said an internal model, more capable than the publicly released GPT-6 Astra, found such a singularity for the three-dimensional, forced version of the Navier-Stokes equations. Astra was brought in only afterward, to check the work.

The 10,000-agent run behind the proof

OpenAI said the result came from roughly 10,000 AI agents running concurrently for 88 hours between Sept. 1 and Sept. 5, producing about 130 billion output tokens. Astra then spent 17 more hours formalizing the proof in the verification language Lean. Sébastien Bubeck, a member of OpenAI's technical staff who led the effort, and a colleague identified only as Chen said the run cost "millions of dollars." Javier Gómez-Serrano, a mathematician at Brown University who was not involved in the work, said "very few mathematicians will have resources of that scale."

Buckmaster's question about his Codex sessions

Tristan Buckmaster, a mathematics professor at NYU's Courant Institute, and Levent Alpöge, a mathematician at Anthropic, had spent about a year working on a related, unforced version of the same equations using publicly available AI models, including OpenAI's Codex. In a statement posted to his NYU faculty page on Sept. 8, Buckmaster said OpenAI first asked to speak with him on Sept. 3, after rumors spread that he and Alpöge were close to a result, and that a call took place Sept. 6.

Buckmaster wrote that on the call he asked whether OpenAI's model "had been trained on, or had access to, our sessions in Codex, into which we had been putting all our drafts for the whole of this project." He said he was told the model had not looked up user data, but got no direct answer on whether that data had been used in training.

What Bubeck said, and what OpenAI won't rule out

Buckmaster said Bubeck asked him "why would you ruin your career?" during the call, and later told him "if you don't want me to be nice, then I don't have to be nice." Bubeck denied using Buckmaster and Alpöge's work directly. "We did not use their prompts or proofs to prompt our models or direct our agents," he said. OpenAI's own statement went further than a flat denial on training: "While unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models."

Terence Tao, a mathematician at UCLA who has written extensively on AI-assisted proofs, said solving problems purely with AI, without transparency into the process, "can contaminate" the way mathematicians build on each other's work. Neither OpenAI nor Buckmaster has published the Codex session logs that would settle the question, and OpenAI has not said what technical safeguard, if any, would stop a model from training on a paying customer's private coding sessions.