OpenAI announced that an internal model, which it describes as significantly more capable than GPT‑6 Astra, produced an analytical proof and a Lean formalization asserting that the three-dimensional incompressible Navier–Stokes equations can develop a singularity in finite time. According to OpenAI, this result resolves the Millennium Prize problem statements labeled “C” (and “D”) in the official formulation.
Why this matters
The Navier–Stokes equations describe fluid motion using Newton’s second law (F=ma) within a continuum approximation rather than tracking individual molecules. They are fundamental in applications such as aircraft design, weather forecasting, and the study of blood flow. The long-standing open question has been whether a smooth initial velocity field for an incompressible, constant-density fluid in three dimensions can evolve to a point where velocities blow up to infinity in finite time, despite viscous smoothing. A genuine singularity would indicate a breakdown of the continuum model.
Outline of the claimed proof
OpenAI summarizes the construction as a vortex-like solution: a spinning, inward-spiraling region of fluid that becomes increasingly elongated while its local speed grows, yet the total energy of the flow remains finite throughout. The central technical requirement is that the Navier–Stokes terms describing motion—acceleration, pressure gradients, momentum transport, viscosity—grow large while canceling in a precise way so that the external forcing remains smooth and the system’s energy stays finite even as velocity becomes unbounded.
Model training and project timeline
OpenAI says the internal model was being trained from August 28 and exhibited strong benchmark performance, particularly in mathematics. On September 1 the company launched an effort to evaluate the model on all open Millennium Prize problems and several other high-impact questions after hearing rumors that two Millennium problems had been resolved.
The work was organized as a multiagent system driven by the internal model. Agents had access to tools such as a cached copy of internet content and the ability to run code, and they were divided into communicating groups. The group that produced the Navier–Stokes resolution reportedly comprised on the order of 10,000 concurrent agents. OpenAI states that standard safeguards—monitoring and isolation—were applied throughout the evaluation.
Different groups explored different problem variants
For each Millennium problem, different agent groups were prompted with different variants of the problem statement. For Navier–Stokes, OpenAI says some groups worked on variants labeled “A” and “B” (forms that would yield a proof) while other groups addressed “C” and “D” (forms that would yield a disproof). The multiagent system also tackled related, ostensibly easier tasks: notably a regularity (blowup) question for the Euler equations (Navier–Stokes without viscosity). OpenAI reports their agents produced a disproof for the unforced Euler regularity problem after nearly 100 agents collaborated for about 50 hours.
Resources used and verification
OpenAI reports that the Navier–Stokes resolution was reached on Saturday, September 5, roughly 88 hours after the first agents were launched. Lean formalization and verification required an additional 17 hours via GPT‑6 Astra. Across all attempted problems, the agents exchanged about 4.9 million messages and produced roughly 300 billion output tokens. The Navier–Stokes effort alone involved about 2.7 million messages and approximately 130 billion output tokens.
Contact with other researchers
OpenAI says their effort began after a September 1 rumor later linked to Levent Alpöge (Anthropic) and Tristan Buckmaster (New York University). After completing their project and the Lean verification on September 6, OpenAI contacted them to propose a concurrent release and to acknowledge priority. They discovered Alpöge and Buckmaster had a solution to a forced Euler problem; OpenAI recognizes their priority on that forced Euler result and congratulated them.
Data and attribution notes
OpenAI states that neither the human researchers nor the agents saw Alpöge and Buckmaster’s work before it was made public, and that no specific user data was accessed to solve the problem. The company allows that de-identified data from product usage might have indirectly contributed to model improvements, but emphasizes that the proofs differ substantially and, in the Euler case, concern different variants (forced vs unforced).
Next steps and posture
OpenAI frames the announcement as reporting significant AI progress rather than claiming the Millennium Prize. The company says it will continue to study and understand the model’s behavior and intends to pace capability development deliberately. OpenAI reiterates its stated longer-term goal of building steerable, accountable AI systems connected to people so that artificial general intelligence benefits all of humanity.
OpenAI has published links to the Euler paper and to the Lean formalized proof alongside the announcement.
