Tristan Buckmaster: three blowup results and a dispute with OpenAI

Three results, one still unpublished

Tristan Buckmaster of New York University's Courant Institute has published a statement on three results obtained with Levent Alpöge: finite-time blowup with smooth forcing for incompressible porous media, for the Boussinesq equations and for three-dimensional incompressible Euler. A fourth result — blowup for hypo-dissipative Navier–Stokes — is being withheld until its Lean verification finishes.

Buckmaster stresses that the program was not theirs and was not proposed by an LLM. The idea belongs to Diego Córdoba and Luis Martínez-Zoroa, who have spent years exploring forced blowups; Buckmaster and Alpöge, helped a great deal by language models, pushed their rough-forcing result to smooth forcing and to Euler.

August 15th and the role of the models

Progress was slow for most of the past year; the turn came on August 15th, when the pair obtained the blowup results for Boussinesq and Euler. The first model-generated proof, Buckmaster writes, was the most horrendous he has ever read; it was verified in Lean on August 22nd. They used Anthropic's Claude and OpenAI's Codex, especially GPT-5.6 Sol and, more recently, Astra — the latter only for writeups and for auditing their arguments.

Buckmaster is openly unhappy with the presentation: the Boussinesq and Euler write-ups are much closer to what models produce under human direction than to a paper written by a person, and the Euler text can only be described as "AI slop". He apologises, and says he believes Luis Martínez-Zoroa deserves a Fields Medal.

The dispute with OpenAI

On September 3rd, as a rumour spread that Anthropic had resolved a major open problem, Buckmaster wrote to a prominent mathematician at OpenAI. He was later told that an internal OpenAI model had produced a proof of finite-time blowup for the forced Navier–Stokes equations — "existence of forced blowup in R3 and T3", with a smooth forcing function, "option c and d in Fefferman". He has not seen the roughly 100-page proof. The word "forced", he writes, was a red flag: that is the route Córdoba and Martínez-Zoroa opened, and the one he and Alpöge had quietly chosen.

He was told the model had simply been given the problem statement, but it emerged that a whole team had worked on the problem, that even the prompt he was shown had been written by prompting Codex, and that an enormous amount of compute was used. The first prompt, it was eventually agreed, went out in the past few days, after their work reached OpenAI. Asked whether the model had been trained on their Codex sessions, he was told it does not look up user data; he asked again and got no answer.

Two proposals were put to him: OpenAI posts its result the day after they post Euler, or Buckmaster alone writes a paper presenting the Navier–Stokes result. He declined both; when he said he would go public, the reply was: "Why would you ruin your career?" He has not seen the proof and is not accusing anyone: if an OpenAI model really did close the gap, that is remarkable and should be said loudly, with the history intact.