OpenAI Solves the Navier–Stokes Millennium Prize Problem

OpenAI says an internal AI system has produced a solution to the Navier–Stokes existence and smoothness problem, one of the Millennium Prize Problems. The proof argues that the three-dimensional Navier–Stokes equations can develop a singularity in finite time and includes both a written proof and a formalization in Lean.

The problem, unresolved for roughly 90 years, asks whether smooth three-dimensional fluid motion can remain smooth indefinitely. OpenAI says the proof was produced by an internal model significantly more capable than GPT-6 Astra, highlighting the pace of AI progress in advanced mathematical research.

The Problem.

The Navier–Stokes equations apply Newton’s laws to describe fluid motion as a continuous medium rather than tracking individual molecules. They underpin applications ranging from aircraft design and weather forecasting to blood-flow modeling.

The central question is whether a smooth, three-dimensional incompressible fluid can develop a singularity in finite time, where fluid speeds become unbounded despite viscosity, which normally smooths motion. Such a singularity would signal a breakdown of the equations’ continuum model.

Developed in the 19th century by Claude-Louis Navier and George Gabriel Stokes, the equations were shown by Jean Leray in 1934 to have generalized solutions, but whether those solutions always remain smooth remains unresolved. In 2000, the Clay Mathematics Institute named the question one of seven Millennium Prize Problems.

The Result.

The system produced an analytical proof and Lean formalization showing that an initially smooth fluid at rest can develop a singularity in finite time, while its total energy remains finite. The result establishes statements C and D of the official Millennium Prize formulation.

The solution involves a vortex that spirals inward and stretches as its central region shrinks, causing fluid velocity to grow without bound while energy remains finite. The key challenge is that the singularity emerges from the fluid dynamics themselves, not from an artificially infinite force, with acceleration, pressure, momentum transfer, and viscosity becoming large while precisely cancelling to leave a smooth external force.

How the Proof Was Found.

Since August 28, OpenAI has been training a new internal model showing unprecedented benchmark performance, including in mathematics. After rumors on September 1 that two Millennium Prize problems had been solved, an effort was launched to test the model across all open Millennium Prize problems and several other high-impact problems.

The effort used coordinating groups of agents with access to cached internet content and code execution. For Navier–Stokes, roughly 10,000 concurrent agents explored different formulations and proof or disproof strategies, operating under the same safeguards used for frontier model evaluations.

The agents first resolved a related Euler equations regularity problem: nearly 100 agents worked for about 50 hours to produce a disproof of the unforced version. That result led to a greater focus on Navier–Stokes, with agents shifted from other problems, and Codex used to consolidate insights across groups.

The Navier–Stokes resolution was reached on September 5, about 88 hours after the effort began. Lean formalization and verification took another 17 hours via GPT-6 Astra.

Across all problems, the agents sent 4.9 million messages and generated about 300 billion output tokens. The Navier–Stokes effort accounted for 2.7 million messages and approximately 130 billion output tokens.

Concurrent Work.

The effort began on September 1 after a rumor that was later found to involve Levent Alpöge, an Anthropic employee, and Tristan Buckmaster, a mathematics professor at NYU. After completing the Navier–Stokes project and Lean verification on September 6, OpenAI contacted the researchers, believing they had also resolved Navier–Stokes, and offered a concurrent release recognizing their priority. The discussions revealed that their result concerned the forced Euler problem.

OpenAI offered visibility into the prompts used and later the proof itself. The company recognizes the priority of Alpöge and Buckmaster’s work on forced Euler and congratulated them on the result.

OpenAI says neither its researchers nor agents accessed their work before its public release, and no specific user data was used to solve the problem. While the possibility cannot be ruled out that de-identified usage data may have contributed to model improvement, the proofs differ significantly, including the specific Euler results: forced versus unforced.

Progress and Responsibility.

OpenAI says the release is intended to demonstrate substantial progress in AI capabilities, not to claim the Millennium Prize for the result.

The milestone reflects work by mathematicians and AI researchers and represents a snapshot of ongoing AI development rather than a culmination. OpenAI describes the result as evidence of a new phase of AI progress and says it is focused on understanding the model’s capabilities to guide the pace of future development.

The company also emphasizes the goal of building AI systems that are steerable, accountable, and connected to people, with greater consideration for how quickly capabilities advance.

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