OpenAI announced on Tuesday, September 8, that approximately 10,000 coordinating AI agents, powered by an internal model the company describes as significantly more capable than GPT-6 Astra, produced a proof that the three-dimensional Navier-Stokes equations can develop a singularity in finite time4,6,14. The agents arrived at their resolution on Saturday, September 5, roughly 88 hours after the first agents were launched, at a cost OpenAI described as millions of dollars15.
The Navier-Stokes existence and smoothness problem is one of seven Millennium Prize Problems selected by the Clay Mathematics Institute in 2000, each carrying a $1 million prize for a verified solution. Only one other Millennium Prize Problem had previously been solved. OpenAI said it does not intend to claim the $1 million prize.
The technical claim is substantial: OpenAI said the system found a finite-time singularity in the Navier-Stokes equations, which govern fluid motion. The agents had access to tools including a cached version of the internet and code execution, and were subdivided into communicating groups. Ethan Knight, an OpenAI-affiliated figure, said the company had spent the past year training models to collaborate via multi-agent reinforcement learning5.
The announcement was immediately overshadowed by an authorship dispute. NYU mathematics professor Tristan Buckmaster, a tenured faculty member, announced three proofs on Tuesday with a preliminary finding on the same problem, developed in collaboration with Anthropic mathematician Levent Alpöge using both OpenAI's Codex and Anthropic's Claude2. Buckmaster said information about his and Alpöge's progress had been passed to OpenAI. When he contacted OpenAI, the company told him it had already achieved a full proof, but Buckmaster said answers about when OpenAI began its research and how much human input was involved "became more evasive".
Buckmaster said it eventually emerged that an entire team had been working on the problem and that substantial compute had been used, and that the first prompt had been sent "in the past few days, after information about our work had reached OpenAI". Both the Buckmaster-Alpöge proof and the OpenAI proof use an approach to the Navier-Stokes problem pioneered by Diego Córdoba and Luis Martínez-Zoroa. Buckmaster said almost nobody else he knew of was working in that direction.
According to Buckmaster, OpenAI employees presented him with two options: he and Alpöge could post their work and OpenAI would post its solution the following day, or he could collaborate with OpenAI on a paper that excluded Alpöge from authorship. When Buckmaster declined the latter, he said OpenAI asked him why he wanted to ruin his career10. Buckmaster also said OpenAI employees offered no response when asked whether OpenAI's models had been trained on transcripts of the work he and Alpöge had done with OpenAI models. OpenAI reserves the right to train models on Codex interactions, though users can opt out.
OpenAI's Sébastien Bubeck said the team was inspired to pursue the problem after hearing a rumor about Buckmaster and Alpöge's efforts. Bubeck called Buckmaster's claims "false and inflammatory".
Acceptance of the proof by the mathematics community remained unresolved at the time of reporting. Mathematician Javier Gómez-Serrano said the Córdoba-Martínez-Zoroa approach was one of several thought to hold promise for solving the problem. Terence Tao wrote a Mastodon thread describing how important mistakes, wrong directions, and incomplete solutions are for pure mathematics.
ANALYSIS The dispute raises a concrete question for AI-assisted research: whether frontier labs' ability to deploy thousands of agents and millions of dollars of compute on a problem, potentially informed by users' interactions with their own tools, creates an asymmetry that academic researchers cannot match or defend against. The episode also marks the first time a multi-agent system, rather than a single model, has been central to a claimed mathematical breakthrough of this magnitude.