OpenAI publishes its Navier-Stokes proof: about 10,000 agents, 88 hours, and an independent math advisory group

Start with the problem itself, otherwise this story reads like a foreign language. Per OpenAI's official page, the Navier-Stokes equations use Newton's second law of motion to describe how fluids move, treating a fluid as a continuous medium rather than tracking individual molecules, and are used for aircraft design, weather forecasting and the study of blood flow. The long-standing open question is whether that continuum approximation can break down even when the motion starts smoothly: specifically, whether a three-dimensional incompressible fluid of constant density can develop a singularity, meaning speeds inside the fluid grow without bound within a finite amount of time. The page stresses that any singularity would have to develop despite the presence of viscosity, which tends to smooth out motion. Because a real fluid cannot move infinitely fast, that would mark a breakdown in how the equations model the fluid, and continuing to model the system would require tracking each particle individually.

The second thread is the scale of the history, which explains why a technical announcement was treated as big news. Per OpenAI's official page, the equations date to the nineteenth-century work of Claude-Louis Navier and George Gabriel Stokes; in 1934 Jean Leray proved that solutions exist in a generalized sense, but whether they always remain smooth became a central unanswered question; and in 2000 the Clay Mathematics Institute named the Navier-Stokes existence and smoothness problem one of seven Millennium Prize Problems. The page also states that the question of whether smooth three-dimensional fluid motion can break down has remained unresolved for roughly 90 years. In other words, the announcement targets not an engineering metric but a mathematical barrier that has stood for nearly a century.

The third angle is the specific shape of the result, because the sentence about proving a singularity rests on a very concrete geometric picture. Per OpenAI's official page, their system produced an analytical proof and a Lean formalization showing that an initially smooth fluid at rest can develop a singularity in a finite time; the fluid has a smooth force applied to it, and its energy remains finite through the entire dynamics, from rest to the formation of the singularity. The page says this resolves the problem by establishing statement C, and also D, in the official Millennium Prize formulation. The solution is a vortex: a spinning swirl of fluid that spirals inward and gets increasingly elongated, which the page likens to spaghetti. The central region shrinks while it speeds up in such a way that its energy still stays finite. The page points to the technical difficulty: the equations must develop the breakdown through the motion of the fluid itself rather than through an infinite force put in by hand, and more mathematically, the terms describing the motion, namely acceleration, pressure gradients, momentum transfer and viscosity, must both become big and cancel in a precise way.

The fourth thread is how the result was produced, because that part is closer to a reproducible engineering practice than the notion of an AI thinking up a proof. Per OpenAI's official page, the company began training a new internal model on August 28 that exhibited unprecedented performance on benchmarks including mathematics; on September 1, after hearing rumors that two Millennium Prize problems had been resolved, it launched an effort to evaluate the model on all open Millennium Prize problems. It used a system of coordinating agents powered by that internal model, with access to tools such as reading from a cached version of the internet and running code, subdivided into groups able to communicate internally; the group that produced the Navier-Stokes resolution involved on the order of 10,000 concurrent agents. The page mentions an unexpected bonus: the company asked the system to try a set of easier problems, one of which was the regularity problem for the Euler equations with the viscosity term removed, and nearly 100 agents worked together for about 50 hours to resolve the unforced version. According to the page's timeline, the agents arrived at their resolution on Saturday, September 5, about 88 hours after the first agents were launched, with Lean formalization and verification taking an additional 17 hours via GPT-6 Astra.

The fifth angle is cost and prioritization, the part most easily overlooked but most useful for judging whether this approach is sustainable. Per OpenAI's official page, across all attempted problems the agents sent 4.9 million messages and used about 300 billion output tokens; in the process of resolving the Navier-Stokes problem specifically, they sent 2.7 million messages and used approximately 130 billion output tokens. The page also describes how compute was reallocated: once they saw the Euler solution, they judged Navier-Stokes the most promising problem, shifted agents away from the other Millennium Prize problems and prompted them with the Euler resolution, later updating the agents to a further-trained version of the internal model when one became available. That pattern of hitting the small problem first and using the result as a ladder is easier for other teams to borrow than raw model capability.

The sixth thread is the controversy attached to the announcement, which touches both credit and data. Per OpenAI's official page, the effort started on September 1 after hearing a rumor that the company later realized related to Levent Alpoge, an Anthropic employee, and Tristan Buckmaster, a math professor at NYU; after completing the project and Lean verification on September 6, OpenAI reached out to them to offer a concurrent release and to recognize their priority in a joint announcement, and learned the pair had produced a resolution of the forced Euler problem using an internal Anthropic model. The page says OpenAI offered them visibility into all the prompts used and later the proof, and recognizes the priority of their work on forced Euler. On the data question, the page says the researchers and the agents did not see any of the pair's work through any means until it was released publicly, that no specific user data was accessed to solve the problem, and that following an investigation it confirmed Buckmaster's Codex prompts over the two months preceding the announcement and the September 8 paper could not have influenced the system in any way, including through training.

Finally, the advisory group thread, because it is OpenAI's institutional answer to the controversy. Per OpenAI's official page, the pace of the internal model's progress in mathematics surprised the mathematicians within OpenAI, and an open letter titled A Severe Misalignment of AI in Mathematics, hosted at mathandai.org, raised concerns about the negative externalities of solving open problems as a benchmark for new AI systems. OpenAI says it is working with mathematicians who have established an independent mathematics advisory group, hosted at the Institute for Advanced Study, which will advise on the review and communication of emerging results and on academic and professional standards of mathematical research. The page sets out several boundaries: the group operates independently, has the freedom to offer advice OpenAI did not request, comment on OpenAI's impact on mathematics and make its advice public; its members are not paid by OpenAI and it can change its own membership; and importantly, it will not be responsible for advising OpenAI on how to pace its internal progress on mathematics. Initial members include Francois Charles (ENS-PSL), Camillo De Lellis (IAS), Timothy Gowers (College de France, Cambridge), Martin Hairer (EPFL, Imperial College London), Nikhil Srivastava (Berkeley), Ulrike Tillmann (Oxford), Ravi Vakil (Stanford), Edward Witten (IAS) and Melanie Matchett Wood (Harvard).

One more thing needs to be stated plainly: accounts of the credit dispute do not fully agree across sources. Per HPCwire, Buckmaster contacted OpenAI on September 3 to say the pair were close to releasing a paper and formal proof, although he did not tell the company which problem they were working on, and according to Buckmaster the discussions later became contentious, including a proposal that would have given him authorship but left Alpoge off the work. The same report raises another uncomfortable question, namely whether OpenAI's AI could have learned anything from the pair's use of Codex; it says OpenAI stated neither its researchers nor its agents saw their work before it became public and no specific user data was accessed to solve the problem, but that the company leaves one possibility open, saying it cannot rule out that de-identified data from the researchers' use of its products helped improve its models. OpenAI's official page, in an update dated September 10, is more definite, stating that an investigation confirmed the relevant Codex prompts could not have influenced the system in any way. Both formulations are presented here without reconciliation so readers can judge for themselves.