Paul Mainwood
September 9, 2026 at 11:19 AM GMT+2
Seems like there's two patterns coming out of the Navier-Stokes saga that are particularly worrying. First, worth understanding what the "problem" is. The Navier-Stokes equations are a continuous approximation of the dynamics of fluids - which in the real world are made of discrete molecules. The Millennium Prize problem is to to either show that all initial conditions of the Navier-Stokes equations give smooth solutions, or show some example in which this doesn't happen (basically, do the equations ever break down and give infinities / discontinuities, or does this never happen?)
In the real world, we know that fluids are made of particles, and sometimes this means the continuous approximation is abandoned. This happens in many cases, whether or not the equations break down internally - we handle it numerically. The "smoothness problem" is mainly of mathematical interest. Problem 1: OpenAI taking on the problem was entirely competition - particularly with Anthropic - who they thought had solved it. In fact, a mathematician associated with Anthropic was making progress on it, independent of this affiliation. OpenAI went apeshit and wanted them erased from credit.
Problem 2: The counterexample they found (i.e., initial conditions that break the equations) does not seem very insightful or fruitful for further work. And this seems characteristic of AI counterexamples and proofs so far. They don't advance the field much, at least not in ways humans understand. Both these problems give a bleak vision of the future. Whatever the motivations of individuals within them, the leading labs seem to make major, compute-intense decisions entirely on the basis of short-term competition/IPO value. And this seems to be overruling everything else, including safety.
As they plough through problems, gathering new "high-scores" for the IPO prospectus, we don't seem to be finding the equivalent of Go's "move 37", inspiring human masters to see new horizons within the game. Rather they seem to be picking off problems without suggesting new, productive directions. Two hopes here. First, that leading AI labs can break out from the "beat the other guy" blinkers, whether that other guy is the lab next door, or some more ill-defined guy like "China".
At the moment, they reliably sacrifice even mid-term trust and credibility any time they scent a short-term win. Second, and somewhat reliant on the first, that AI systems can be sent to search not only for solutions to outstanding problems, but also to frame the productive "next problems" beyond them. Historically, this tended to happen - it would be sad if it were an artifact only of human thought.