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PaperclipPerfector  1d ago (text post)   4942 thread views

Culture War Roundup for the week of September 7, 2026

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Navier-Stokes Millenium Problem Solved by GPT

OpenAI says they solved the Navier-Stokes Millenium Prize problem with an internal model that is more powerful than GPT6, so a GPT 7 candidate or something close. OpenAI has been on fire lately really casting a shadow on Anthropic. First the hugging face incident, now the first to crack the Millenium prize problems.

As for the solution to the problem, they are claiming that smoothness is disproved and that the equations do break down.

A fundamental open question for these dynamical equations has been whether the continuum approximation of the fluid can break down. Specifically, can the Navier–Stokes equations for a three-dimensional incompressible fluid with constant density develop a “singularity,” even when the motion starts smoothly? Here, a singularity means the dynamics lead to speeds in the fluid growing without bound within a finite amount of time. The development of a singularity would have to happen despite the presence of viscosity, which tends to smooth out motion. Because a real fluid cannot move infinitely fast, this would mark a breakdown in how the equations model the fluid. To continue modeling the system, one would then need to track the behaviour of each particle individually.

They say that the equation can develop a singularity, which apparently means it is not a perfect model.

This proof, produced by an internal OpenAI system, shows that the dynamics of the Navier-Stokes equations for fluid motion can develop a singularity in finite time.

This is very exciting to me as I believe it to be the first scientific result of an AI model. Previous results have been basically irrelevant to the sciences. However, this result is of course still completely symbolic in nature, which is not surprising because LLMs are not embodied enough to collect data and analyze it autonomously.

While they are generating amazing PR recently, I will say GPT-6 is somewhat disappointing for coding. It is not the same leap that 5.5 to 5.6 sol was. It would appear that the models are getting better at running very long context chains while efficiency improvements and refinements in lower level tasks are lacking. Still, it's great work from OpenAI and it's plausible to me that if there's no singularity in general by 2030, there will be for mathematicians. They will no longer be meaningful for producing math, rather they will only be humans who understand it.

They say that the equation can develop a singularity, which apparently means it is not a perfect model.

It's never been a perfect model; the most obvious problem with the Incompressible Navier-Stokes equations is that everybody knows there's no such thing as an incompressible fluid. (and even if you switch to a version of Navier-Stokes that allows compressibility, you still break down if you have a length scale tiny enough or a gas rarefied enough that atomic mean free paths aren't short enough to ignore, or if you're doing cosmology and your fluid has relativistic effects, etc. etc.)

But IMHO it's still astonishing to see such a singularity possible in these equations. Conservation only allows velocities to grow infinitely large as the singularity gets infinitely thin, but diffusion fights a second derivative of velocity with respect to distances; making the velocity larger while the distances shrink means you've got diffusion fighting you two ways at once. Even with turbulent flows, where there's an energy cascade from the largest down to smaller and smaller length scales, we still expect to hit bottom at a "Kolmogorov microscale" where diffusion is just too strong to overcome and all that energy gets dissipated (lost in the incompressible equations, or turned into heat in the compressible equations and reality). A counterexample where the equations can just go all the way down to a singularity without diffusion ever thwarting us is not something I'd have ever expected to see.

We've had proof of blowup in forced compressible 3D navier stokes even before we got it for the incompressible version. And yes, this is a theoretical result about partial differential equations with minimal relevance to the real world.

Common compressible solutions are literally shocking, which ironically makes finding even less regularity in them metaphorically less shocking. ;-)

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