OpenAI Says It Solved the Navier-Stokes Millennium Problem, Mathematicians Say Credit Was Stolen

OpenAI has announced a solution to the existence and smoothness problem for the Navier-Stokes equations, one of the seven Millennium Prize Problems set by the Clay Mathematics Institute and unsolved for nearly 90 years. The announcement comes with a priority dispute attached: mathematicians Tristan Buckmaster and Levent Alpöge say OpenAI's model used the exact same, not-yet-published method they had been developing for months.

The Navier-Stokes problem asks whether smooth, three-dimensional fluid motion described by the classical equations of hydrodynamics can, in finite time, collapse into a singularity, a point where the solution stops being well defined. The answer to that question, worth a $1 million prize from the Clay Institute, had been unknown since the 1930s and was considered one of the hardest open problems in mathematical physics.

How the process worked

According to OpenAI's account, the agents worked in parallel on multiple variants of the problem, including a related question for the Euler equations, where roughly a hundred additional agents were deployed. The company published a full writeup of the proof along with a formalization in Lean, a tool mathematicians use to machine-verify whether every logical step of a proof actually holds.

OpenAI says the model proved the existence of a singularity under specific boundary conditions, answering the question as officially posed by the Clay Institute. Company spokespeople stress that the result passed formal mathematical verification rather than just peer assessment.

Buckmaster's allegations

The dispute broke out when Tristan Buckmaster revealed that, starting September 3, OpenAI had been pushing for a phone call with him. During a call on September 6 with several OpenAI researchers, he learned the company was preparing to announce that its unpublished model had solved Navier-Stokes via exactly the same route he and Alpöge had been developing for months. Buckmaster says he asked directly whether OpenAI's model had been trained on, or had access to, their sessions in the Codex tool, and did not get a satisfactory answer.

Buckmaster also described OpenAI researcher Sebastien Bubeck presenting him with conditional options: publish with limited credit, remove Alpöge's name from the paper, or accept a simultaneous publication by OpenAI. When Buckmaster refused, he says Bubeck asked why he wanted to ruin his own career.

Why would you ruin your career? If you don't want me to be nice, then I don't have to be nice. - Sebastien Bubeck, OpenAI researcher, as recounted by Tristan Buckmaster

OpenAI's response

Bubeck denied any wrongdoing, saying the team did not use Buckmaster and Alpöge's prompts or proofs to steer its agents and had not seen their work before the public release. He later added that OpenAI acknowledges both mathematicians' priority and congratulated them on the result. OpenAI's head of research, Mark Chen, said neither people nor AI systems had searched through user data to solve the problem.

OpenAI also admitted it only took up the Navier-Stokes problem after hearing rumors that rival Anthropic was preparing its own announcement on the matter. That admission reinforces suspicions that the rush was driven by competition between labs rather than a planned research agenda.

Pushback from mathematicians

Terence Tao, a Fields Medal winner, voiced broader doubts about how large AI models approach hard mathematical problems. In his view, these systems produce answers without deeper insight, rarely explain their reasoning or the connections between different fields, and AI companies almost never disclose failed attempts, which are crucial to progress in mathematics.

Using excavators to loot an archaeological site, destroying the context needed to give treasures any historical meaning. - Terence Tao, Fields Medal winner

Fortune's report also describes a broader sense of disorientation in the math community, where at one weekend gathering academics reportedly found themselves questioning the point of mathematical research now that AI models can solve problems considered unsolvable for decades.

For Polish researchers and tech companies, the case shows how hard it will be to set rules for crediting authorship in an era when AI models are trained on ever-broader datasets, potentially including researchers' working sessions in tools like Codex. The question of access to unpublished user data from commercial development and research tools could become the subject of legal disputes outside the United States as well.

The Navier-Stokes case also lands amid intensifying rivalry between OpenAI and Anthropic over prestigious scientific results, after both companies announced fresh achievements by their models in mathematics and the natural sciences in recent weeks. How the priority dispute between Buckmaster, Alpöge, and OpenAI is resolved could shape how institutions like the Clay Mathematics Institute verify and award prizes for Millennium Problem solutions in the future, now that AI systems are in the mix.