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Summary
- Tristan Buckmaster and Levent Alpoge say they proved finite-time blow-up for the Boussinesq and 3D Euler equations with help from tools including Claude and Codex.
- Buckmaster wrote in his statement that during a September 6 call with OpenAI, he was asked to drop Alpoge, who works at Anthropic, from the author list.
- He said he isn't accusing anyone of anything, only recording what he was told and when, and OpenAI's public response has not yet been confirmed.
Tristan Buckmaster, a mathematician at NYU's Courant Institute, published a public statement on September 7 that laid out both an AI-assisted proof of a fluid-dynamics equation and the details of a call with OpenAI that took place just before he announced it. The proof itself is significant news on its own, but roughly half the statement reads as a record of how AI labs compete over credit for mathematical results.
The proof is joint work done in a personal capacity by Buckmaster and Levent Alpoge, a mathematician who works at Anthropic. According to the statement, the two used help from Anthropic's Claude and OpenAI's Codex, GPT-5.6 Sol, and Astra to prove finite-time blow-up for the porous medium equation with a smooth forcing term, the Boussinesq equations, and the 3D incompressible Euler equations. The Boussinesq and Euler results came on August 15, the statement says, and the AI-generated proof was verified using the formal-verification tool Lean on August 22.
In plain terms, finite-time blow-up is when the flow described by an equation shoots to infinity at some point, so the equation itself stops making sense from there on. Whether this actually happens in the Navier-Stokes equations has been an open problem for more than a century.
Buckmaster was careful to credit where the work came from. He wrote that the research builds on a program on forced blow-up developed over several years by Diego Córdoba and Luis Martínez-Zoroa, and that he and Alpoge merely extended that idea, with AI assistance, to smooth forcing terms and the Euler equations. He put a line directly in the statement saying credit for the underlying ideas belongs entirely to those two earlier researchers.
Where the statement turns controversial is what comes next. Buckmaster wrote that on September 3, after hearing a rumor that OpenAI had a similar result, he emailed a mathematician at OpenAI, and on September 6 had a call with people from OpenAI, including Sébastien Bubeck. During that call, he says he was told that an internal OpenAI model had produced a roughly 100-page proof of forced Navier-Stokes blow-up with almost no human involvement — an account he says was later contradicted by other explanations he received.
According to Buckmaster, OpenAI's side made two proposals: to publish the results in a coordinated sequence, and to have the Navier-Stokes paper credit Buckmaster alone as author. He wrote that Bubeck said he wanted Alpoge removed as a co-author because Alpoge works at Anthropic. When Buckmaster said he would decline both proposals and publish anyway, he says he was told, "Why would you want to ruin your career?"
Buckmaster was explicit that he is not accusing anyone of wrongdoing. The statement includes a line saying he is simply recording what he was told, when he was told it, and what was proposed to him. As of this writing, OpenAI has not issued a public response.
What this statement points to is that mathematics has become another arena of competition among AI labs. Just days earlier, Anthropic announced that Claude had produced a computer-verified proof related to Fermat's Last Theorem, while OpenAI said its Astra system had solved an unsolved puzzle. In a climate where who publishes first, with which model, and under whose name has become a scorecard for labs, author lists and publication order are no longer just matters of academic etiquette.
Two questions remain unresolved. One is who verifies, and how, the actual share of human contribution in an AI-generated proof. The other is which company gets to claim ownership of a result produced jointly, in a personal capacity, by researchers who work at different companies. Buckmaster's statement doesn't answer either question — it simply documents a moment when those questions played out in real life.
In the end, this story will likely outlast any single proof. Lean has verified the equations, but no one has yet verified the rules by which labs divide credit for results — and without such rules, the next major proof will likely bring another phone call just like this one.
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