Disclaimer: This article is written for both general and specialised audiences.
Huge leap in AI-enabled science, and substantial alarming confidentiality concerns..
Over the past two days, an extraordinary scientific headline has been circulating: AI may have resolved one of the most famous unsolved problems in mathematics — a problem involving the equations that govern the motion of fluids.
Aircraft aerodynamics, weather prediction, blood flow, ocean waves, turbines and pipelines all rely, directly or indirectly, on our ability to describe fluid motion. At the heart of much of this are the Navier–Stokes equations. On 8 September, OpenAI announced that an internal AI system had produced a proposed resolution of the Navier–Stokes Millennium Prize Problem, using roughly 10,000 coordinated AI agents all working together alongside scientists in a calculated collaboration reaching the mathematical construction in about 88 hours and used 130 billion tokens.
There is an immediate paradox here. Engineers and scientists solve the Navier–Stokes equations every day. We use numerical approximations of them to design aircrafts, simulate ocean waves, understand flows and model countless other physical systems.
So how can an equation that we routinely solve still be one of the greatest unsolved problems in mathematics — with a $1 million prize attached to it?
To understand the significance of the claim, we first need to distinguish between solving the equations for a particular engineering problem and proving that the formulation hold mathematically in general.
The Navier–Stokes equations, developed through the nineteenth-century work of Claude-Louis Navier (1822) and George Gabriel Stokes (1845), describe the motion of viscous fluids. In their incompressible form, they can be written as
In simple terms, the first equation is essentially Newton’s second law applied to a moving fluid: the acceleration of the fluid is governed by pressure, viscosity and external forces. The second expresses conservation of mass for an incompressible fluid.
The difficulty is not that we cannot use these equations. We have been doing that successfully for generations. The deeper question is whether, in three dimensions, initially smooth solutions must always remain smooth and well behaved — or whether under certain conditions they can develop a singularity, where some quantity becomes unbounded in finite time. This is the Navier–Stokes Millennium Prize Problem as defined by the Clay organization.
The complete problem statement, however, is more precise than simply asking whether a solution can “blow up”, i.e. a singularity exists. The Clay formulation imposes several mathematical constraints.
The Clay formulation imposes several mathematical constraints. The flow must be incompressible:
It begins from a smooth initial velocity field:
A valid global solution must remain smooth for all time:
while its kinetic energy remains bounded:
The Clay statement then defines four specific cases, A–D: two asking whether smooth solutions exist for all time, and two asking whether there can instead be a breakdown of smoothness. These are considered either in infinite three-dimensional space or in a periodic domain. (The full formulation as published: https://www.claymath.org/wp-content/uploads/2022/06/navierstokes.pdf)
OpenAI claims, with an important caveat, to have established options C and D of Fefferman’s official Clay formulation: that there exist smooth initial data and a smooth external forcing for which no global smooth Navier–Stokes solution exists. Their construction reportedly starts with the fluid at rest, thus avoiding a prescribed existing flow pattern, applies a smooth external force, maintains finite energy, and produces a finite-time singularity. The first caveat (a) is that proposed solution is a deliberately forced singularity using smooth functions of f .
The second caveat (b) is that they did not independently discover this from nothing. OpenAI itself says the effort began after it heard rumours that two Millennium-related problems had been resolved. The relevant concurrent human work was by the unpublished work by Tristan Buckmaster and Levent Alpöge, who obtained finite-time blow-up with smooth forcing for the 3D Euler equations, alongside related Boussinesq and porous-media results.
Therefore, the models were utilised to scale up and further explore the applicability of early results.
Why this is exciting — and unsettling at the same time?
The first concern: we may be moving into an era in which machines can sometimes produce formally correct proofs faster than humans can, but we are still excited by the possibilities. The second crucial aspect is that unpublished research is passing through AI systems, which raises questions about confidentiality, priority, attribution, and what is meant by an independent discovery anymore.