OpenAI has published a solution to the Navier–Stokes existence and smoothness problem. This is one of the seven Millennium Prize Problems, each carrying a $1,000,000 reward since May 24th, 2000.
The accusation
The announcement is complicated by claims from Tristan Buckmaster, a mathematics professor at NYU. He was working on related problems alongside Levent Alpöge, a mathematician currently employed by Anthropic.
Buckmaster released a statement alongside his own findings. He and Alpöge spent nearly a year on the project, relying heavily on Claude and Codex, specifically GPT-5.6 Sol. They achieved a breakthrough on August 15th.
Once the mathematical community began discussing their work, Buckmaster reported that OpenAI had learned of their progress. He contacted the company to discuss their own team, which was using a similar approach.
I asked when the first prompt had been sent by them. This question was not answered directly by OpenAI for some time. Eventually it was agreed that it had been sent in the past few days, after information about our work had reached OpenAI.
I asked whether the model had been trained on, or had access to, our sessions in Codex, into which we had been putting all our drafts for the whole of this project. I was told the model did not look up user data. I asked again, about training, and I did not get an answer.
OpenAI offered to wait for Buckmaster to publish or let him author a paper on their result. They stated clearly that Levent Alpöge would not be invited as a co-author because of his employment relationship with Anthropic.
How the work was done
OpenAI described the timeline in a post on September 1st. Rumours reached them that two Millennium Prize problems had been resolved. They launched an evaluation of their internal model against all open Millennium Prize problems.
The agents arrived at their resolution on Saturday, September 5, about 88 hours after the first agents were launched. Lean formalization and verification took an additional 17 hours via GPT‑6 Astra.
Across all attempted problems, the agents sent 4.9 million messages and used about 300 billion output tokens. In the process of resolving the Navier–Stokes problem, the agents sent 2.7 million messages and used approximately 130 billion output tokens.
The cost of 300 billion output tokens at public API prices for GPT-6 Astra would be $15,000,000.
OpenAI’s view on the competitors
OpenAI acknowledged that their effort began on September 1st after hearing the rumours. They noted the rumours were related to Buckmaster and Alpöge. After completing their project and verification on September 6th, they reached out to offer a concurrent release and recognise priority in a joint announcement.
We (the researchers and the agents) did not see any of their work through any means until they released it publicly — in particular, no specific user data was accessed in order to solve this problem. While unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models.
However, our proofs differ significantly and even the precise results proved are different in the Euler case (forced vs unforced).
OpenAI’s proof differs from the one Buckmaster and Alpöge were developing. The exact results proved are also different in the Euler case, specifically regarding forced versus unforced conditions.
What this means for researchers
This situation highlights a specific risk for people working on hard problems with AI tools. Knowing that a solution is imminent can trigger massive spending to find it first. Buckmaster described the cost of the competing effort as millions of dollars in LLM usage.
The core issue is the ambiguity of how training data is used. When an AI lab states that user data improves model performance, the mechanism is unclear. If a researcher uses a model to partially solve a difficult problem, there is a chance their work influences the training of a later model. That later model could then help someone else solve the problem before the original researcher publishes.
This dynamic mirrors current issues in computer security. A mere rumour of a bug can be enough to find an exploit, as agents are tasked with searching for known vulnerabilities. The same logic now applies to mathematics.
