OpenAI has announced that it has generated an analytical solution to the Navier–Stokes existence and smoothness problem, one of the Millennium Prize Problems. Created by OpenAI's internal system, this proof demonstrates that singularities—where velocity diverges to infinity within a finite time—are formed in the equations describing fluid motion.
OpenAI Proves Singularity Formation in Navier–Stokes Equation Solutions Using Internal Model
This article is a translation. Read the Japanese original
An internal model, said to possess significantly higher capabilities than GPT-4 Astra, was used to generate the proof. OpenAI reported that a collaborative multi-agent system involving approximately 10,000 agents was deployed, reaching a solution after about 88 hours of computation. The agents utilized tools such as access to internet cache data and code execution, and formalization using Lean was also performed for verification.
OpenAI stated that this achievement is intended to demonstrate the rapid pace of AI progress to the world and is not aimed at claiming the Millennium Prize reward. It also revealed that during this process, discussions were held to respect the priority of prior research by other scholars, such as studies on the solutions of the Euler equations conducted by Anthropic employees.
Sources
- On the Navier–Stokes Millennium Prize Problem （OpenAI News, 2026-09-08）
- Paper by OpenAI (Description of the proof)