Phoenix 24
A computational breakthrough reveals how artificial intelligence can accelerate discovery without erasing the human ideas behind it.
MADRID, SPAIN
An experimental OpenAI system has produced an analytical proof addressing the Navier–Stokes existence and smoothness problem, one of mathematics’ seven Millennium Prize Problems. According to the company, the proof demonstrates that an initially smooth, three-dimensional fluid subjected to a smooth external force can develop a singularity in finite time—a point at which its calculated velocity becomes unbounded even though its energy remains finite.
The Navier–Stokes equations describe the movement of liquids and gases and underpin research in aeronautics, meteorology, oceanography and blood flow. Mathematicians had been unable to determine whether their three-dimensional solutions must always remain smooth or whether the equations could break down under specific conditions. OpenAI’s result supports the second possibility by constructing a vortex that stretches, accelerates and concentrates until a singularity forms.
Approximately 10,000 coordinated AI agents worked for 88 hours, exchanging 2.7 million messages and generating around 130 billion output tokens. A further 17 hours were required to formalise and verify the argument using Lean, a system that checks whether each logical step follows from explicitly defined mathematical rules. This verification considerably strengthens the result, although it does not replace examination by independent mathematicians.
The conceptual foundation came from research published in 2023 by Spanish mathematicians Diego Córdoba and Luis Martínez-Zoroa, together with Fan Zheng. Their work introduced a singularity mechanism based on infinitely many regions of vorticity separated by vortex-free spaces. The AI system extended that architecture to construct the delicate balance between acceleration, pressure, momentum transfer and viscosity required by the Navier–Stokes formulation.
Córdoba and Martínez-Zoroa have welcomed the achievement while emphasizing that computational speed should not be confused with conceptual origin. Their research required sustained human intuition to identify the mechanism; artificial intelligence then explored, combined and verified possibilities at a scale no individual research group could reproduce. The episode therefore presents scientific discovery as an increasingly hybrid process rather than a simple competition between mathematicians and machines.
Questions concerning attribution have nevertheless accompanied the announcement. Separate work by mathematician Tristan Buckmaster and Anthropic researcher Levent Alpöge explored a related forced-Euler problem. OpenAI says its system did not access their unpublished research or relevant user data and maintains that the resulting proofs are substantially different.
The proof must still withstand extensive independent scrutiny before the mathematical community can regard the Millennium Problem as definitively settled. OpenAI has said it will not claim the associated $1 million prize. Whatever the final judgment, the achievement has already exposed a new research reality: machines may compress years of mathematical exploration into days, but recognition must still distinguish computational execution from the human ideas that made the search possible.
Artificial intelligence can accelerate discovery, but it cannot erase the human origin of knowledge.