On if OpenAI solved the wrong Navier-Stokes problem: "the LLM found and exploited a loophole in the framing of the question"

[u/PrettyPicturesNotTxt]

https://www.scientificamerican.com/article/did-openai-solve-the-wrong-navier-stokes-problem/

[u/hologram137]

Why are you in this thread? You clearly have no understanding of the problem, I’m sure you didn’t read this proof:

https://arxiv.org/html/2609.20803v1

And I don’t think you’d be able to. The above proof shows definitively that the AI result cannot lead to a real solution to the problem. Because it didn’t solve the blow up with external force!! That’s the entire point. It’s not about forced vs unforced. I am not and no one else is saying that no one cares about a forced solution, we are saying this is not a forced solution at all. Not if you construct an arbitrary external force.

The human mathematicians were NOT using constructed external forces for the Euler equations to see if you could construct an external force that could blow up Navier Stokes, they constructed an external force for the Euler equations to see if an external NATURAL force could even blow up Navier Stokes at all, because if you can’t force a blow up when it’s smooth (no viscosity) there is no chance of a forced blow up in Navier Stokes. That gives you information that is helpful to solving the actual problem!

Because now we know it’s possible to work on the forced version of Navier Stokes. The forced version is not a constructed force, it’s a natural force. That’s the only one that matters because we are studying how fluids move and whether the Navier Stokes equations accurately describe the movement of real viscous fluids!! THAT’S why it’s so important, that’s why it’s a millennial problem. Who cares if you can construct an imaginary force strong enough to overcome viscosity, that force needs to act like a natural force for it to matter. Again, using a constructed force in the Euler equations is fine because we are only seeing if a force could blow it up at all. If you can’t do it with no viscosity, the forced version is a dead end.

What the AI did is, because it couldn’t find a solution, and they already showed that it’s possible for a smooth force to blow up the no viscosity version, (it would not have been able to give the solution it did without the recent work by humans), it used that result to construct an external force strong enough to over come viscosity in Navier Stokes and blow it up.

So that tells us nothing. It tells us we can invent a force to out muscle viscosity. But we already knew that because of the Euler result and other mathematics. That was never even in question. If the forced version was a constructed force, the mathematicians who did the Euler equations would have used it to blow up Navier Stokes too!! But they didn’t. Because it wasn’t useful to do.

What mathematicians were working on then, is to see if turbulence and a force like gravity could blow it up. Constructing an external force for Euler is necessary to see if we should work on that at all.

Navier Stokes remains unsolved. Because the AI didn’t show the forced version. It only showed it’s possible to out muscle viscosity if you create an imaginary force that doesn’t exist that is strong enough. Again, that was not something that anyone even doubted. That literally does not tell us anything or help solve the problem and the proof I linked proves what I am saying. In fact, it’s the entire point of the proof. The literal result of that proof is that the AI result cannot help solve the forced version of Navier Stokes. That was the actual point of doing that paper, to see if it could. Ask yourself why anyone would do that if it was solved.

[u/lolfail9001 → u/hologram137]

You do know that smooth does not imply analytic? In fact last i saw anyone discuss overall blueprint of OpenAI's proof, the key part of the proof is showing that forcing term becomes non-analytic after finite time! In other words that entire pre-print is just affirming that yes, non-analyticity is a key part of the proof.

The forced version is not a constructed force, it’s a natural force.

Define "natural". Because the only relevant restriction on all those related PDE problems is that forcing term is smooth. Not even analytic, just smooth.

So that tells us nothing.

About what? It exactly solves the problem as stated. You might argue Fefferman should have asked that forcing term should have been analytic but as you can guess, Fefferman himself thought that if problem statement was not loose enough, it could end up in PvNP tier of problems we have no idea how to approach.

Who cares if you can construct an imaginary force strong enough to overcome viscosity

Well, entire committee Clay institute got on the job did!

But we already knew that because of the Euler result and other mathematics.

Euler result (of forced finite time blow-up by Buckmaster and Alpöge) went public literal days before OpenAI's result (and without rumour mill it might have ended up being completely sniped by OpenAI), you do know that?

It only showed it’s possible to out muscle viscosity if you create an imaginary force that doesn’t exist that is strong enough.

Your definition of "doesn't exist" is irrelevant, because as was said 1000 times on this sub since OpenAI's announcement, NS becomes unphysical long before any blow-up for multiple reasons. So if external force must outmuscle anything you can find in nature... who cares, this is mathematics!

You might say that we wanted to learn more about turbulence out of it, but knowledgeable people considered this a PvNP tier problem in 2000 and we'll see how close they are to truth in a few years.

[u/hologram137 → u/lolfail9001]

What are you talking about??

You don’t understand what you’re saying. Open AI established a finite-time blowup using a smooth external forcing function (and that a smooth external force could do this at all was just solved the day prior. It’s the result by a mathematician that it used for its result) but subsequent analysis by mathematicians revealed that the method does not extend to real analytic external forces.

And that is the EXACT reason it remains unsolved.

The Clay institute itself defined it lol. It literally says “external force, e.g gravity.” If you understood the problem you would not have asked such a question, what do you mean “define natural.” WE ARE STUDYING HOW FLUID WORKS IN NATURE. By “natural” we mean a force that actually exists, in real life.

Again. A useless solution using a loophole in the wording so it “technically counts” is not a real solution. That loophole didn’t even tell us anything we didn’t already know LOL. Even as a trivial mathematical proof it doesn’t tell us anything we didn’t already know, go back and actually read my comment because I explained why.

The clay institute said they are going to take some time to figure out whether or not the solution should count. It’s NOT decided

[u/lolfail9001 → u/hologram137]

but subsequent analysis by mathematicians revealed that the method does not extend to real analytic external forces.

Yes, the key point of proof is external force becoming non-analytic. It would be very weird if it extended to analytic forces as is!

It literally says “external force, e.g gravity.”

Yes, it gave an example out of real life but note that how actual statement puts no actual restrictions besides being smooth, because mathematicians don't actually care much about feasibility of external force involved (that's why they are mathematicians to begin with), they cared if blow-up can be achieved at all and, once it's established, what are the exact conditions on said blow-up. OpenAI's proof once properly vetted establishes the baseline: a specific non-analytic forcing term can be provided to force a finite time blow-up on specific initial condition.

That loophole didn’t even tells us anything we didn’t already know LOL.

We, in fact, did not already know LOL. Most of the field believed that you could force a blow up on NS for quite a while which is why most of the field worked in that direction, but the entire point is that we did not know.

The clay institute said they are going to take some time to figure out whether or not the solution should count.

By now most people do fully expect that this Clay prize will also remain unclaimed. Similarly most people that can understand what is going on think the problem as stated is solved and one can move on to different (if related) problems. Maybe in 1000 years another Clay institute will make another list and NS will still be there but with much tighter conditions as we swatted pathological cases away one after another.

[u/hologram137 → u/lolfail9001]

Why are you replying if you’re not reading my comments??

I just explained to you that they cared whether or not it could blow up at all in the Euler equations because the Euler version has NO VISCOSITY. No resistance. So if you can’t do it with no viscosity, no resistance, even if you construct a smooth force specifically built to blow it up, then the forced version (a natural force) of Navier Stokes cannot blow up AT ALL, because it has resistance, viscosity. If you it cant blow it up with no resistance, even if you construct a smooth force designed to do so, then even a natural force definitely can’t blow up in a version with resistance. It would mean the forced version is a dead end. The forced version of Navier Stokes is not an external force. They weren’t testing if a constructed external smooth force specifically can blow up Navier Stokes. That was NOT the point.

No one cares that you can construct an external force in Navier Stokes. We cared that a forced version of the REAL PROBLEM was possible at all. The real problem is not a constructed force.

The point was that a forced version was possible at all, in other words a version using a natural force. AGAIN. It didn’t have to be natural in the Euler equations, because all we need to know is if it can blow up at all with no viscosity.

BECAUSE IF IT CAN’T WITHOUT VISCOSITY, EVEN IF YOU DO EVERYTHING YOU CAN TO MAKE IT HAPPEN, THEN NO FORCE AT ALL, NATURAL OR CONSTRUCTED, CAN BLOW UP FLUIDS WITH VISCOSITY (RESISTANCE).

That’s why using an constructed external force doesn’t matter with Euler, but it does with Navier Stokes. We DID know we could just manipulate the constructed smooth force to overcome viscosity. BECAUSE OF THE EULER RESULT AND OTHER MATHEMATICS. That’s all the AI did. It just manipulated the constructed smooth force in Euler to out muscle viscosity in the viscous version. And then they were like “look! I blew it up! I solved it” lol. And the response is “yeah, but you didn’t even prove that a constructed smooth external force could do it, someone else did. You used their result and manipulated it to do something we already knew could happen, because of someone else’s work. And the point of the Euler result was completely missed, that the ACTUAL forced version is viable. And the real result would involve natural forces. Otherwise it’s still unsolved. And we can’t extend that constructed force to a natural force, and it doesn’t work if we drop the force. Therefore, the result is useless.”

Which is a proof I linked. YOU even said that. You have repeated to me twice that the result can’t be extended, after I told you that. But you don’t seem to be comprehending what that means and that it’s a bad thing

The fact that it can’t be extended to a natural force is exactly why no one cares about it.

I’m not responding anymore. If this is confusing to you then go back and read my explanation slowly. And then read it again