Key Highlights
- OpenAI says its internal AI system has produced a solution to the Navier–Stokes existence and smoothness problem.
- The Navier–Stokes problem is one of the seven Millennium Prize Problems and has remained unresolved for roughly 90 years.
- The claimed proof shows that smooth three-dimensional fluid motion can develop a singularity in finite time.
- The proposed solution involves a vortex that spirals inward and becomes increasingly elongated.
- Around 10,000 concurrent AI agents were involved in the Navier–Stokes effort.
- The AI agents reportedly reached the result in approximately 88 hours.
- Lean formalization and verification reportedly required an additional 17 hours.
- The broader project involved about 4.9 million messages and approximately 300 billion output tokens.
- OpenAI says it is not seeking the Millennium Prize for the result.
- Independent mathematical scrutiny will be required to determine whether the claimed proof fully resolves the official problem.
OpenAI has announced that an internal artificial intelligence system has produced what it describes as a solution to the Navier–Stokes existence and smoothness problem, one of the seven Millennium Prize Problems in mathematics.
According to OpenAI, the system generated an analytical proof showing that a three-dimensional incompressible fluid can develop a singularity in finite time. The company has also released a formalization of the result in Lean, a proof-assistant system used to formally verify mathematical arguments.
The claim, if independently validated by mathematicians, would represent a major development in mathematical research. OpenAI said it is not seeking the Millennium Prize for the result.
OpenAI Says AI Model Solved Navier–Stokes Problem
The Navier–Stokes equations are fundamental mathematical models used to describe the movement of fluids. They are relevant to areas including aircraft design, weather forecasting and the study of blood flow.
The central mathematical question is whether a smooth solution to the three-dimensional incompressible Navier–Stokes equations can remain smooth indefinitely or instead develop a singularity in finite time.
A singularity in this context would mean that certain properties of the fluid, such as velocity, become unbounded within a finite period, despite the equations including viscosity, which normally has a smoothing effect.
The problem has remained unresolved since the early 20th century and was formally designated as one of the Millennium Prize Problems by the Clay Mathematics Institute in 2000.
AI-Generated Proof Involves a Developing Fluid Vortex
OpenAI said its internal system produced an analytical construction in which an initially smooth fluid develops a finite-time singularity.
The proposed solution involves a vortex—a rotating region of fluid—that spirals inward while becoming increasingly elongated. As the central region contracts, the fluid accelerates while its total energy remains finite.
According to OpenAI, the construction requires the acceleration, pressure gradients, momentum transfer and viscosity terms in the Navier–Stokes equations to become large while cancelling each other in a precise manner.
The result therefore does not depend on simply introducing an infinite external force. Instead, the claimed breakdown emerges from the dynamics of the fluid itself.
OpenAI described the work as establishing statement C, as well as statement D, in the official formulation of the Millennium Prize problem.
Around 10,000 AI Agents Used in the Effort
OpenAI said the result was produced using a multi-agent system powered by an internal model that the company described as significantly more capable than GPT-6 Astra.
The effort began on September 1 after researchers heard reports that Millennium Prize problems may have been resolved. OpenAI subsequently directed groups of AI agents to investigate the open Millennium Prize Problems and several other mathematical questions.
For the Navier–Stokes investigation, the company said approximately 10,000 concurrent agents participated.
The agents were divided into groups and given different formulations of the mathematical problem. They could communicate within their respective groups and use tools including code execution and access to a cached version of the internet.
OpenAI said researchers encouraged the groups to pursue different approaches before using Codex to consolidate useful intermediate results.
The group working on Navier–Stokes reportedly reached its result on September 5, approximately 88 hours after the initial agents were deployed.
Lean Formalization Took Another 17 Hours
OpenAI said the analytical result was subsequently formalized and verified in Lean, with the formalization process taking an additional 17 hours using GPT-6 Astra.
The company said the formalization provides a machine-checkable representation of the mathematical argument.
Across its broader effort, OpenAI reported that the participating agents generated approximately 4.9 million messages and used around 300 billion output tokens.
For the Navier–Stokes work specifically, the company said agents generated approximately 2.7 million messages and used around 130 billion output tokens.
AI Also Resolved an Euler Equations Problem
Before concentrating resources on Navier–Stokes, OpenAI said its agents investigated a related mathematical question involving the Euler equations, which describe fluid motion without the viscosity term found in the Navier–Stokes equations.
Nearly 100 agents reportedly worked for approximately 50 hours on an unforced Euler regularity problem and produced what OpenAI described as a disproof.
The company said this result encouraged researchers to shift additional computational resources toward the Navier–Stokes problem.
OpenAI Highlights Significant AI Capability Progress
OpenAI said the internal model had been undergoing training since August 28 and had demonstrated substantial improvements on mathematical benchmarks. Training of the model remains ongoing, according to the company.
The company framed the Navier–Stokes result as evidence of a significant increase in the ability of AI systems to perform advanced mathematical research.
OpenAI also said the work involved strict safeguards, including monitoring and isolation, similar to those used during its frontier-model evaluations.
Company Says It Is Not Claiming the Millennium Prize
Despite describing the result as a solution to the Navier–Stokes Millennium Prize Problem, OpenAI said it does not intend to claim the Millennium Prize for the result.
The Clay Mathematics Institute’s Millennium Prize Problems carry a $1 million prize for each problem, but mathematical results must undergo appropriate scrutiny and validation before such a prize can be awarded.
OpenAI’s announcement therefore represents a claimed AI-generated mathematical breakthrough rather than an independently established resolution at this stage.
The company said its objective in releasing the work is to demonstrate the progress of its AI systems and provide researchers with access to the proposed proof and formalization.
The claimed result will now be of particular interest to mathematicians who can examine the analytical proof, its assumptions and the Lean formalization to determine whether the argument fully satisfies the requirements of the original Millennium Prize problem.
