On Tuesday 8 September 2026 OpenAI announced that a system of roughly 10,000 concurrent AI agents had produced a proof that the Navier–Stokes equations — the fluid equations of weather, aircraft design and blood flow — can develop a singularity in finite time.[1][2]
The Navier–Stokes existence and smoothness problem is one of seven Millennium Prize Problems named by the Clay Mathematics Institute in 2000, each carrying a $1m prize, and the question of whether smooth three-dimensional fluid motion can break down had remained unresolved for roughly 90 years.[1][3]
Quanta Magazine called the result, if it holds up to scrutiny, "by a significant margin, the most important mathematical proof to have been arrived at by an artificial-intelligence model to date."[2]
OpenAI says the work used an internal model "significantly more capable than GPT‑6 Astra", with agents subdivided into groups; the group that found the Navier–Stokes resolution involved on the order of 10,000 concurrent agents.[1]
The system produced an analytical proof and a Lean formalization that an initially smooth fluid at rest, with a smooth force applied and finite energy throughout, develops a singularity in finite time — establishing statements "C" and "D" of the official Millennium Prize formulation.[1]
The solution is described as a vortex that spirals inward and grows increasingly elongated, "like spaghetti", while its energy stays finite.[1]
The agents arrived at the resolution on Saturday 5 September, about 88 hours after launch; Lean formalization and verification took an additional 17 hours via GPT‑6 Astra.[1][3]
Across the Navier–Stokes effort the agents sent 2.7 million messages and used approximately 130 billion output tokens.[1]
Sébastien Bubeck, the OpenAI researcher in charge, estimated the computational cost at several million dollars — and others put it far higher, with reports of $22.5 million based on OpenAI's own Astra pricing.[2][4]
A related Euler-equations question (unforced, no external force) was resolved first by nearly 100 agents working about 50 hours, OpenAI says — and OpenAi cedes priority for the 3D Euler result to the other team.[1][2][unverified]
The announcement came twelve hours after NYU professor Tristan Buckmaster and Anthopic researcher Levent Alpöge released their own results on related problems, obtained with a variety of AI models including Anthropic's Claude and OpenAI's Codex.[2][4]
Buckmaster and Alpöge built on an analytic strategy pioneered by Diego Córdoba and Luis Martínez-Zoroa; Buckmaster says the pair made rapid progress from mid-August onward and had a Lean-verified Euler proof by 22 August.[2][4]
OpenAI says its effort began on 1 September after hearing rumors of an imminent Millennium Prize resolution, and that it reached out to Buckmaster and Alpöge on 6 September to offer a concurrent release.[1]
Buckmaster reported a different conversation: repeated questioning on that call, he says, led OpenAI to admit it had only attacked the problem in the past week, and he questions whether the company intentionally accessed his Codex account or trained its unreleased model on his interactions.[4]
He was careful about the accusation: "I do not know what their model did, or how. I do not know whether our data was used. I am not accusing anyone of anything."[3]
Buckmaster went further in a statement, alleging that Bubeck offered publication arrangements that depended on removing Alpöge's name from the paper because OpenAI did not like his Anthropic affiliation — and that when Buckmaster declined, Bubeck asked "Why would you ruin your career?"[4]
Bubeck has denied it all — "We did not use their prompts or proofs to prompt our models or direct our agents" — and OpenAI's post states the researchers and agents "did not see any of their work through any means until they released it publicly."[1][4]
The company also concedes it cannot rule out that de-identified data from the pair's product use "helped improve our models."[1][3]
None of this is an official solution yet: the proof has not been through peer review, and no prize decision has been made — OpenAI says it does not intend to claim the Millennium Prize at all.[1][3][4]
The deeper pattern is the one Terence Tao lamented: AI companies using century-old math challenges as "mere marketing proof points for the prowess of their AI models."[4]
The provenance question — whether a frontier lab's own coding tool quietly fed it a rival's private drafts — is the tail risk that can reshape this story before any referee decides anything.[unverified]
The mathematics, whichever team is credited, will be checked by humans: the crucial step the Lean file cannot certify is whether the formal statement is equivalent to the problem mathematicians set out to solve.[2][unverified]
[1] openai.com/index/navier-stokes-solution — On the Navier–Stokes Millennium Prize Problem (OpenAI, 2026-09-08)"This proof, produced by an internal OpenAI system, shows that the dynamics of the Navier-Stokes equations for fluid motion can develop a singularity in finite time.""In 2000, the Clay Mathematics Institute named the Navier–Stokes existence and smoothness problem one of seven Millennium Prize Problems.""The group that produced the Navier–Stokes resolution involved on the order of 10,000 concurrent agents.""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.""In the process of resolving the Navier–Stokes problem, the agents sent 2.7 million messages and used approximately 130 billion output tokens.""We (the researchers and the agents) did not see any of their work through any means until they released it publicly""While unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models.""Our goal in releasing this result is to report on the substantial progress of our AI models. We do not intend to claim the Millennium Prize for this result."
[2] quantamagazine.org — AI Has Solved One of Math's $1 Million Millennium Prize Problems (Quanta Magazine, 2026-09-08)"On the morning of Tuesday, September 8, mathematicians at OpenAI announced that a group of 10,000 autonomous AI agents under their direction, running on an advanced model not available to the public, had found a “singularity” in the Navier-Stokes equations in three dimensions""Their result has been formally checked in the programming language Lean, giving mathematicians confidence that it is indeed correct.""If the result holds up to further scrutiny, it is, by a significant margin, the most important mathematical proof to have been arrived at by an artificial-intelligence model to date""Both AI-enabled teams relied heavily on work by Diego Córdoba of the Institute for Mathematical Sciences in Madrid and Luis Martínez-Zoroa of CUNEF University""Sébastien Bubeck of OpenAI estimates the computational cost at several million dollars.""OpenAI cedes priority for the 3D Euler result to Buckmaster and Alpöge, while claiming it for the Navier-Stokes result."
[3] theguardian.com — OpenAI claims to have solved maths problem that stumped humans for decades (The Guardian, 2026-09-08)"About 10,000 AI agents – AI systems that carry out tasks autonomously – worked on the problem at once and reached the solution in 88 hours.""Buckmaster had further concerns because the pair’s work in progress was stored in OpenAI’s Codex model, which is used for writing computer programs, potentially making the work visible to the OpenAI team.""“I do not know what their model did, or how. I do not know whether our data was used. I am not accusing anyone of anything.”""At a press briefing on Tuesday, OpenAI researcher Sebastien Bubeck denied that the company had used the pair’s work or accessed material shared with OpenAI’s servers.""It also said it could not rule out that data from the pair’s use of their products “helped improve our models.”"
[4] fortune.com — OpenAI says it cracked Navier-Stokes, one of math's grand challenges (Fortune, 2026-09-08)"Buckmaster says that he and Alpöge took a concept for tackling the Navier-Stokes problem that had been pioneered by two other mathematicians, Diego Cordoba and Luis Martinez-Zoroa, and then used Anthropic’s Claude and OpenAI’s Codex powered by the GPT-5.6 Sol model, to push Cordoba and Martinez-Zoroa’s lines of attack through to completion.""He questions whether OpenAI intentionally accessed his Codex account or whether the unreleased model might have been trained on his interactions with Codex.""“We did not use their prompts or proofs to prompt our models or direct our agents,” Bubeck said in a press conference.""Buckmaster said he declined and said he would go public if OpenAI published in the way it proposed.""Terence Tao, generally considered one of the world’s greatest living mathematicians, lamented on social media AI companies’s use of these longstanding mathematical challenges as mere marketing proof points for the prowess of their AI models."