OpenAI's AI Proof of Navier-Stokes Draws Doubts
An unreleased model reportedly cracked the problem in about 88 hours; two researchers who worked on it for a year are asking where its knowledge came from.
Symbolic image: in a night-time office, someone sketches vortex curves on a whiteboard while a nearby monitor shows an abstract proof structure and a small server blinks.
OpenAI says an unreleased model produced a solution to the Navier-Stokes problem, along with a formal proof in Lean, in roughly 88 hours, but where that reasoning came from is now disputed.
At a glance
- OpenAI dates the agent run to September 1–5, 2026, about 88 hours of work.
- For this single problem OpenAI cites 2.7 million messages and 130 billion output tokens.
- Across all problems attempted: roughly 300 billion output tokens, about $15 million at public API rates.
- The Lean proof was machine-checked on September 6, 2026, according to the published timeline.
- Tristan Buckmaster (NYU) and Levent Alpöge (Anthropic) reached a related breakthrough on August 15.
OpenAI says an unreleased model produced a solution to the Navier-Stokes existence and smoothness problem, published alongside a writeup and a machine-checkable proof in Lean. Within hours the story had stopped being about fluid dynamics and turned into a question about provenance.
The numbers OpenAI put on the record
The agent run is dated September 1 to September 5, 2026, roughly 88 hours. For this one problem the company reports 2.7 million messages and 130 billion output tokens. Across every problem attempted in the same effort, the total comes to about 300 billion output tokens, which works out to roughly $15 million at public API rates. The Lean proof was machine-checked on September 6, 2026.
The target is not new. The Millennium Prize Problems were posted on May 24, 2000, and Navier-Stokes has resisted every attempt since.
Who got there first
Tristan Buckmaster of NYU and Levent Alpöge, who works at Anthropic, had spent close to a year on related problems using Claude and GPT-5.6 Sol, and reached a breakthrough on August 15. They have asked whether OpenAI's model had access to their work, or to training data derived from it. OpenAI denies reading their sessions but concedes it cannot rule out that “de-identified data” from their use of its products helped improve its models.
The uncomfortable part
Simon Willison reads the release as a display of capability made without much thought for how it would look, given that OpenAI could reasonably tell a solution was within reach. He also names a failure mode with no clean answer yet: working through a hard problem with a commercial chatbot may quietly improve the model a competitor then uses to finish that same problem first.
What is still unverified
Only Willison's account was reachable for this article; OpenAI's own announcement page refused the request. Several things therefore go unchecked here: the exact wording and scope of the announcement, whether the Lean proof covers the full problem statement or a restricted version, and how the Clay Mathematics Institute treats an AI-generated submission. Until independent experts have worked through the argument, “solved” remains OpenAI's claim rather than an established result.
FAQ
Did OpenAI actually solve the Navier-Stokes problem?
OpenAI says so. Nothing in the sources available for this article shows an independent expert review, and whether the Lean proof covers the full problem statement could not be verified here.
What does a formal proof in Lean actually prove?
Lean is a proof assistant that checks every logical step by machine. That largely rules out reasoning errors, but it says nothing about whether the formalized statement matches the problem people care about.
Why are researchers questioning where the result came from?
Two researchers who spent close to a year on related problems and reached a breakthrough on August 15 are asking whether their use of commercial chatbots fed indirectly into OpenAI's model.
