2) We are witnessing the intelligence explosion from the first row, wherever this takes us

3) I'm still processing the drama, just found out about it after reading the blog post. If that happened based on private data, that's horrible. If that happened based on public tweets, then it's still abuse of power as OA employees access to compute (launching 10k agents) is quite heavy weight in boxing terms.

But apart from AI and drama now that we have working solution to Navier-Stokes, what improvements can we expect in engineering?

▲

20k 7 hours ago | parent | next [-]

Drama aside, this solution would be a counterexample disproving the smoothness postulate, which means that it leads to nothing new unfortunately. We already had working solutions to navier stokes, the only thing we didn't know is if the equations possessed a technical property

Its a bit like solving p = np with a negative result. Its an incredibly difficult problem, but it doesn't lead to anything at all on its own. This is why people are talking about the fact that the solution methodology is much more interesting than the solution - the tools used to crack something like this may lead to solving more useful problems

▲

inkysigma 7 hours ago | parent | prev | next [-]

To be quite clear, the solution to the _Navier Stokes problem_ is one in which you get a finite time blow up (i.e. infinite pressure). This is more meant to suggest that Navier Stokes is unphysical in some way which is not necessarily unexpected.

There's unlikely to be any engineering applications since even if the solution can be approximated, you still need to set up the initial conditions but at that point you can also drive pressure in other ways.

▲

thangalin 6 hours ago | parent [-]

The proof of finite-time singularity may impact both fluid dynamics models (CFD) and AI reasoning models. Under specific conditions, Navier–Stokes equations allow velocity to grow infinitely, causing the continuum fluid assumption to break down. Knowing the exact mathematical breakdown mechanisms helps developers improve adaptive mesh refinement and sub-grid scale models around high-vorticity regions (like vortex stretching and turbulent shear layers). While aerodynamic simulations for vehicles operate far from singularity thresholds, their stability at extreme boundaries could improve?

Proving out the combination of scaling inference-time compute and agent collaboration to solve previously intractable mathematical problems is WOW. By pairing creative candidate generation with automated proof checkers (like Lean) we are leaning into a repeatable framework for AI-driven scientific discovery.

	▲	semi-extrinsic 4 hours ago \| parent [-]
		> Knowing the exact mathematical breakdown mechanisms helps developers improve adaptive mesh refinement and sub-grid scale models around high-vorticity regions (like vortex stretching and turbulent shear layers). This is 100% wrong and reads like copy paste of AI slop. Any simulation which uses sub-grid scale models is already solving a different PDE than the actual Navier-Stokes considered in the Millenium problem, and that PDE is guaranteed to have different properties. Full stop. And to claim this is somehow connected to AMR methods is an example of the kind of pseudoscientific statement Wolfgang Pauli would have called "not even wrong".

▲

xyzsparetimexyz 4 hours ago | parent | prev | next [-]

> But apart from AI and drama now that we have working solution to Navier-Stokes, what improvements can we expect in engineering?

Minor productivity boost in mathematics as people are no longer nerdsniped by the problem

▲

cyberax 7 hours ago | parent | prev [-]

> But apart from AI and drama now that we have working solution to Navier-Stokes, what improvements can we expect in engineering?

Nothing, really. This mirrors other examples of blowups from the classical physics. It's possible to create a system with just gravitating bodies that exhibits a blowup to infinite speeds in a finite time. The root cause is that, in classical physics, the speed of gravity is instant.

In the case of Navier-Stokes, the fluid is incompressible. So technically any force that you apply to it is supposed to instantly affect everything else. This can be exploited to create these blowups. In reality, no fluid is incompressible, and it takes time for any action to affect the material.

It's just that Navier-Stokes equations are so slippery that it's hard to pin their behavior down. They basically just restate the momentum conservation law for a continuous medium.

▲

mcfry 2 hours ago | parent [-]

Great comment! I've been trying to understand it more and was hoping to find more people discussing the result, or the implications of the result, and this was the missing piece for me after watching a few videos.

	▲	cyberax an hour ago \| parent [-]
		Here's the paper about the blowup for the gravitating bodies: https://www.jstor.org/stable/2946572?origin=crossref There's a Wiki article about it: https://en.wikipedia.org/wiki/Painlev%C3%A9_conjecture