Tristan Buckmaster and Levent Alpöge, mathematicians at NYU's Courant Institute, published three results: finite-time blowup for the incompressible porous media equation, the Boussinesq equations, and the three-dimensional incompressible Euler equations under smooth forcing. The same day, Buckmaster posted a four-page statement on his personal website — and it wasn't about mathematics.
The Mathematics
The line of work didn't start with the two of them. The statement is explicit about that: the underlying idea is credited to Diego Córdoba and Luis Martínez-Zoroa, who have spent several years constructing blowup solutions with forcing terms, previously achieving this only for rough forcing. What Buckmaster and Alpöge did was push the result further, extending it to smooth forcing and to the Euler equations.
"I believe Luis Martínez-Zoroa deserves a Fields Medal."
The statement lays out a day-by-day timeline. Progress was slow for most of the past year; the smooth-forcing blowup results for Boussinesq and Euler came together on August 15, and verification in Lean was completed on August 22. Buckmaster described the first version of the model-generated proof Alpöge sent him as the worst thing he'd ever read, and said he spent the time since turning it into something a human could follow. The models involved included Anthropic's Claude and OpenAI's Codex (mainly GPT-5.6 Sol), with Astra used only for writing and checking the argument.
Terence Tao commented on the work in a blog post on September 7, saying the demonstration across these three simplified models made further progress look "highly feasible," and that he thought the same set of technique improvements had a good chance of reaching Navier-Stokes. He also noted that, in an era of automated formalization agents, formalizing this kind of work in Lean has become fairly achievable.
The Other Half of the Statement
On September 3, Buckmaster heard a rumor that Anthropic had solved a major open problem, while Alpöge was told that word of their progress had reached OpenAI. He emailed a mathematician at OpenAI to explain the situation; the reply came the same day, saying the person hoped to avoid a collision and offering to provide compute.
On the afternoon of September 6, Buckmaster had two calls with Sébastien Bubeck and others. He was told that an internal OpenAI model had proved finite-time blowup for the Navier-Stokes equations with a forcing term, in a proof roughly 100 pages long. According to the statement, details kept shifting during the calls: the initial account was that the model received only the problem statement with minimal human involvement; this later changed to a whole team having worked on it, the model having first been run on simpler problems including Euler, and even the prompt shown to him having been written using Codex. When he asked when the first prompt had been sent, the answer he eventually got was: in the past few days — after their own work had reached OpenAI.
He also asked whether the model had been trained on, or had access to, the Codex sessions their entire project had been using. The statement says the answer he got was that the model doesn't look at user data; on the question of training, he says he got no answer.
The statement says the other side proposed two arrangements, both involving Alpöge's authorship credit, with reasons tied to his position at Anthropic. Buckmaster says he rejected both, and told them that if they published that way he would make the exchange public.
Bubeck responded on X on September 8, calling the circulating allegations "false and inflammatory," saying he had participated in the discussion following normal academic practice, and promising a fuller account the next day. As of publication, he had not addressed the Codex data question, the authorship proposals, or the call details point by point, and OpenAI has issued no institutional statement.
What's Different This Time
Over the past year, this kind of story has followed a fixed template: a lab announces its model solved some open problem, attaches a benchmark or a proof link, and the discussion centers on the result itself. That was the pattern for the full machine-checked proof of Fermat's Last Theorem and for the Erdős conjecture case.
This time the dispute isn't about whether the proof is correct. It's about who got there first, who gets credited, and whether anyone saw the other side's draft-stage work. Buckmaster draws his own line in the statement: he hasn't seen OpenAI's proof, doesn't know what the model actually did, and isn't accusing anyone of anything — he's only stating what he was told, when, and what was proposed to him.
What he says he actually wanted to write about was something else: what it means for how students are trained, how credit gets assigned, and how papers get reviewed, when a mathematician plus a model can do this much in a month. He calls it a Deep-Blue-versus-Kasparov kind of moment — and then writes that this is the piece he has to write first.
Sources: full statement on Tristan Buckmaster's personal website, Terence Tao's blog What's New, CocoLoop, Sébastien Bubeck's response on X, Analytics India Magazine — timeline and quotes follow the original statement; OpenAI's side is so far represented only by one personal post.