OpenAI says internal model proved finite-time Navier-Stokes singularity
- OpenAI says an internal model produced an analytical proof, formalized in Lean, that a smooth three-dimensional incompressible Navier-Stokes flow can develop a finite-time singularity, claiming resolution of the Millennium problem's statements C and D.
- The proposed construction starts with fluid at rest under a smooth external force and keeps total energy finite while an inward-spiraling, axially stretched vortex drives fluid velocity without bound in finite time.
- OpenAI says it began the effort on September 1 after hearing rumors of other Millennium-problem results, then used coordinating groups of internal-model agents with internet-cache access and code execution; the Navier-Stokes group had about 10,000 concurrent agents.
- Across all attempted problems, the agents sent 4.9 million messages and generated about 300 billion output tokens, according to a quoted figure in the discussion.
- OpenAI says the internal model is substantially more capable than GPT-6 Astra and remains in training; it says its proof differs from concurrent work by Levent Alpoge and Tristan Buckmaster on the forced Euler problem.
Hacker News opinions
I do not even want to calculate the API bill for 300 billion output tokens, especially when the internal model may cost more than Astra.
At Astra's listed output price of $50 per million tokens, 300 billion output tokens alone comes to roughly $15 million, before input tokens.
I would read the thread linked from the article's Concurrent Work section too. It gives a much less filtered account and alleges aggressive conduct by particular OpenAI employees.
If agents can produce a mathematical proof with massive prompting and compute, calling the output 'slop' has stopped meaning much. This looks like a real AGI-era use case to me.
I do not think that is the full story. Buckmaster's statement raises questions about where the approach came from and how OpenAI used related work.
I work at OpenAI but not on this team. My understanding is that we started after rumors of Millennium-problem results and tested many problems, did not read private chats, and produced a substantially different proof; we also tried to arrange a joint announcement.
The central issue is that Alpoge and Buckmaster appear to have proved a forced-Euler result rather than Navier-Stokes itself. The dispute is whether their route gave OpenAI the strategic idea for its Navier-Stokes work.
Buckmaster says he asked whether the model had trained on or accessed their Codex drafts and did not get an answer about training. OpenAI says it cannot rule out de-identified product data helping its models, which leaves an uncomfortable gap.
I do not buy a story that removes people from this result. Buckmaster says OpenAI mathematicians corrected and guided the process, and that the team tried easier Euler problems before the unforced case using enormous compute.
Tao says the other authors had significant AI input but are reworking the proofs into acceptable form. AI was involved on both sides, whatever the final publication record is.
The contamination allegation is not well evidenced, but Buckmaster's account of being told 'Why would you ruin your career?' and then being offered $1 million to separate from collaborators is alarming if accurate.