Buckmaster and Alpöge Post AI Fluid Blowup Proofs, Dispute OpenAI Contact

NYU mathematician Tristan Buckmaster and Anthropic researcher Levent Alpöge have publicly posted three preprints establishing finite-time blowup with smooth forcing for the incompressible porous medium equation, the two-dimensional Boussinesq system, and the three-dimensional incompressible Euler equations, alongside Lean formalizations of the proofs. In a statement posted with the preprints, Buckmaster also recounts contacts with OpenAI in which, he writes, the company told him an internal model had produced a roughly 100-page proof of finite-time blowup for the forced Navier–Stokes equations — the direction of one of the Clay Mathematics Institute’s Millennium Prize Problems. Clay’s official problem page continues to list the Navier–Stokes existence and smoothness problem as unsolved. OpenAI had published no announcement of such a result on its news page as of September 8, 2026, and Buckmaster states he has not seen the proof.

The Posted Results

The preprints carry the blowup program of Diego Córdoba and Luis Martínez-Zoroa to smooth forcing across three model equations. The Euler paper constructs a finite-time singularity for the incompressible Euler equations on three-dimensional space with a force that stays smooth in space and time up to and including the blowup time. The initial velocity is smooth, axisymmetric with nonzero swirl and zero meridional component, and supported in a fixed solid torus; the vorticity and circulation-gradient norms tend to infinity at the terminal time, and the solution remains smooth and unique in a stated finite-energy class on every earlier closed interval. The paper describes the construction as a continuation of the Córdoba–Martínez-Zoroa program on singularity formation through interacting layers at successively smaller scales.

The Boussinesq paper proves finite-time blowup for the inviscid Boussinesq system on the plane with smooth forcing in both equations, from smooth compactly supported initial temperature and zero initial velocity. The temperature remains bounded while its gradient norm tends to infinity and the vorticity norm has infinite limsup, with both forces smooth in every mixed derivative on the closed time interval and supported in one fixed ball. The IPM paper, additionally coauthored with Matei P. Coiculescu, proves finite-time blowup for the incompressible porous media equation on the two-dimensional torus with a uniformly space-time smooth force, extending an earlier Córdoba–Martínez-Zoroa result whose forcing was smooth only in space.

All three works were verified in the Lean proof assistant, and the authors posted the formalizations to a public repository containing separate Euler and Boussinesq blowup modules. In his statement, Buckmaster writes that he and Alpöge also believe they have blowup for hypo-dissipative Navier–Stokes but are not releasing that paper because its Lean verification has not finished and no presentable writeup exists.

LLMs in the Proofs

The authors disclose heavy use of large language models across the project. The IPM paper states that the authors used Anthropic’s Claude to identify key elements of the prior Córdoba–Martínez-Zoroa proof and reproduce its argument, and used Claude and OpenAI’s Codex to write the main body of the text, delegating bookkeeping of inductive orders and constants to the models under the authors’ direction; the introduction was written by the listed authors. The Boussinesq paper’s AI statement reports that the pair obtained their first blowup solution on August 15, 2026, and Lean-verified it on August 22, 2026, and that they iterated on the writeups with Claude and Codex, later using models designated 5.6 Sol and Astra, the latter only for writeups and auditing of arguments. Buckmaster’s statement says the first LLM-generated proof Alpöge sent him was “the most horrendous I have ever read,” and the Boussinesq AI statement calls the first model-produced writeup the worst the authors had ever seen in mathematics. Buckmaster describes the released Euler writeup as closer to model output under human direction than to a human-written paper, apologizing for its presentation and citing outside pressure for the accelerated release.

Buckmaster writes that the collaboration was purely personal, with no institutional agreements or official involvement by either employer, and that he paid for tools, including a large OpenAI bill, from his own research funds. He assigns credit for the program’s basic ideas to Córdoba and Martínez-Zoroa, writing that the pair had explored forced blowup constructions for several years and that, in his view, Martínez-Zoroa deserves a Fields Medal.

The Disputed Contacts

The latter half of Buckmaster’s statement is his account of events beginning September 3, 2026, when, with a rumor circulating that Anthropic had resolved a major open problem, he emailed a mathematician at OpenAI to clarify that the work was a personal collaboration and would be posted shortly. He writes that after requesting a later meeting, he was asked to meet on September 6, and that he and the mathematician, joined by OpenAI’s Sébastien Bubeck, spoke twice that afternoon without Alpöge present.

On those calls, Buckmaster writes, he was told an internal OpenAI model had produced a proof of finite-time blowup for the forced Navier–Stokes equations, with existence on both three-dimensional Euclidean space and the three-torus and the forcing smooth under two of the options in Charles Fefferman’s official statement of the Clay problem. He recounts being told the proof was about 100 pages and that “very little human input” had been used — a characterization he says unraveled over the call as it emerged that a team had worked on the problem, had first set the model on easier problems including Euler, and had written even the displayed prompt by prompting Codex. Buckmaster writes that it was eventually agreed the first prompt had been sent in the preceding days, after information about the pair’s work had reached OpenAI. When he asked whether the model had been trained on or had access to their Codex sessions, into which they had placed all of the project’s drafts, he was told the model did not look up user data; on the training question, he writes, “I did not get an answer.”

Buckmaster describes two proposals he says were put to him: that the pair post their Euler result with OpenAI posting its Navier–Stokes result the next day, or that Buckmaster alone write a paper presenting the Navier–Stokes result while acknowledging that an internal OpenAI model had resolved it. He writes that Bubeck twice asserted he wanted Alpöge removed from authorship, citing Alpöge’s employment at Anthropic, and that it was said OpenAI would, if posting after the pair, credit them as deserving the Clay Prize and as the “closest humans to the problem.” Buckmaster writes that he declined both offers and said he would go public if OpenAI released the result as proposed; he recounts the reply, “Why would you ruin your career?” followed, after he asked why going public would ruin his career, by “If you don’t want me to be nice, then I don’t have to be nice.”

Buckmaster states plainly what he is not claiming: he has not seen OpenAI’s proof, does not know what the model did or how, and does not know whether the pair’s data was used. “I am not accusing anyone of anything,” he writes. “I am stating what I was told, when, and what was proposed to me.” He adds that if an OpenAI model did close the gap to Navier–Stokes, “that is a remarkable thing and it should be said loudly, by them, with the history intact.”

The Clay Institute’s million-dollar Millennium Prize asks, in Fefferman’s formulation, for a proof either that smooth solutions to the three-dimensional Navier–Stokes equations exist for all time or that they break down in finite time; the posted preprints address the forced Euler, Boussinesq, and IPM equations, which lack the viscosity term that distinguishes Navier–Stokes.