On the Navier-Stokes Millennium Prize Problem
B · Useful, limited OpenAI announcement post for its claimed Navier-Stokes proof with figures on agents, hours and tokens; a primary vendor release whose process numbers are self-reported and unaudited.
Summary based on the full text·Original: about 8 minutes to read
In short
OpenAI's research post of 8 September 2026 (updated 10 September) announces that an internal system produced a proof, with Lean formalization, that the three-dimensional Navier-Stokes equations can develop a singularity in finite time, resolving a Clay Millennium Prize Problem. The work used an internal model OpenAI describes as significantly more capable than GPT-6 Astra, still in training since 28 August. The post explains the problem for a general audience, describes the multi-agent search, and addresses a priority dispute with concurrent work by Anthropic's Levent Alpöge and NYU's Tristan Buckmaster.
Key numbers
- ~10,000concurrent agents in the group that produced the Navier-Stokes resolution
- 88 hoursfrom first agent launch to resolution on 5 September; Lean formalization took 17 more hours
- 130 billionoutput tokens used on Navier-Stokes (2.7 million messages); about 300 billion across all problems
- ~100 agents, ~50 hourseffort behind the unforced Euler regularity disproof
As reported by the source.
English·Web page·opens openai.com
What it reports
- The fluid starts at rest, a smooth force is applied, energy stays finite, and speed grows without bound: statements "C" and "D" of the official problem.
- The search began on 1 September after rumours that two Millennium problems had been solved; agents were run on all open Millennium problems and some easier ones.
- Agents had a cached copy of the internet and code execution, worked in communicating groups, and were given proof and disproof variants of each problem.
- About 100 agents disproved regularity for the unforced Euler equations in about 50 hours; resources were then moved to Navier-Stokes, seeded with the Euler result.
- Codex was used to consolidate insights across agent groups; the model was upgraded mid-run.
- Resolution after about 88 hours with on the order of 10,000 concurrent agents; Lean verification 17 hours via GPT-6 Astra.
- Totals: 4.9 million messages and about 300 billion output tokens across all problems; 2.7 million messages and about 130 billion tokens for Navier-Stokes.
- OpenAI credits Alpöge and Buckmaster with priority on the forced Euler result, states its investigation found Buckmaster's Codex prompts could not have influenced the system, and says it will not claim the prize.
How they got it
This is the company's own account. Figures on agents, hours and tokens are internal logs reported without external audit; the mathematics is in the separate proof paper and Lean files.
Limits
- No independent review of the proof or the process at the time of posting.
- The model is unreleased, so outside researchers cannot reproduce the run.
- Announced amid a priority race with a competitor, which makes the timeline part of the message.
Why it matters for companies
The post gives rare concrete numbers on what a frontier-scale agent run looks like: thousands of agents, days of wall-clock time and hundreds of billions of tokens for one result. That spend is far beyond normal business budgets today, but it shows where verified, many-agent work is going. OpenAI also says it may need "more deliberate choices about the pace of progress", which signals faster capability releases than most corporate AI plans assume.
Summary based on the full text. Read the original for the full argument and data.Read the original on openai.com
- Tags
- AI capability