OpenAI announced on September 8, 2026 that an internal system produced a solution to the Navier–Stokes existence and smoothness problem, one of seven Millennium Prize Problems named by the Clay Mathematics Institute in 2000. According to OpenAI, the proof establishes that a three-dimensional incompressible fluid starting smoothly at rest, under a smooth applied force and with finite energy, can develop a singularity in finite time — resolving statements 'C' and 'D' of the official formulation. OpenAI said it published both a writeup and a Lean formalization, and that the work used an internal model 'significantly more capable than GPT-6 Astra,' with a group of roughly 10,000 concurrent agents. OpenAI states the agents reached the resolution on September 5, about 88 hours after launch, with Lean verification taking a further 17 hours. Across all attempted problems the agents sent 4.9 million messages and used about 300 billion output tokens.
Writing September 8, Simon Willison reported the result is 'overshadowed by accusations' from NYU professor Tristan Buckmaster and mathematician Levent Alpöge, who say they worked on the problem for nearly a year using Claude and Codex before a August 15 breakthrough, and allege OpenAI began its effort after rumors of their work reached the company. OpenAI states it did not access any specific user data but 'cannot rule out' that de-identified data helped improve its models.
- OpenAI says its internal model, described as more capable than GPT-6 Astra, produced the proof plus a Lean formalization
- The resolution establishes statements 'C' and 'D': a smooth, finite-energy fluid can develop a singularity in finite time
- Agents used ~300 billion output tokens across all problems; ~130 billion on Navier–Stokes
- Tristan Buckmaster and Levent Alpöge allege OpenAI began work only after rumors of their nearly year-long effort spread
- OpenAI says it accessed no specific user data but cannot rule out de-identified usage data aided its models
What it means for you
A major, decades-old math problem was reportedly cracked by AI agents — genuinely notable for the field, but it changes nothing about running a business or using AI day to day. The more relevant part for ordinary users is the dispute buried underneath: a rival team alleges OpenAI moved in on a problem they'd worked on for a year using OpenAI's and Anthropic's own tools, raising unsettled questions about what 'we use your data to improve our models' actually means.
Try this
If you put sensitive or proprietary work into an AI tool, spend ten minutes checking the data-retention and training settings for your plan — many business tiers let you opt out of your inputs being used for training, and consumer tiers often default to opt-in.
Who should care
Researchers and technical readers following AI's role in serious math or science, plus anyone who feeds confidential or original work into AI chat tools and hasn't checked how that data is retained.
Skip this if
You use AI for everyday drafting, summarizing, or coding help and don't put confidential unpublished work into it — then this is interesting reading, not something you need to act on.
Sources: OpenAI, Simon Willison — read the original