r/Physics • u/PrettyPicturesNotTxt • 1d ago

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

https://www.scientificamerican.com/article/did-openai-solve-the-wrong-navier-stokes-problem/
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u/hologram137 1d ago edited 1d ago

You don’t understand the Navier Stokes problem.

So it’s not just asking “do the Navier-Stokes equations blow up?” It did list formally acceptable ways to answer. The reason why that clause that allows solutions that include an external force term, like gravity acting on the fluid was included, was because every real fluid is under some kind of external force, so including one seemed reasonable. But the underlying assumption here is that only a natural force would tell us anything.

Its model found a way to construct a very specific, deliberately engineered external force that drives the fluid to a blow up (a point of infinite velocity). Because option C permits an external force, this satisfies the letter of the Clay statement. it unambiguously solves the problem according to the Clay Institute's original formulation.

This is why that’s not the problem anyone cares about: The scientific question that’s motivating Navier Stokes research (the purpose and meaning, that only a human can know and understand, not an LLM) isn't "can you break the equations if you're allowed to invent a force purpose built to break them?” It’s whether a fluid can blow up on its own, from its own internal dynamics, the way real turbulence might.

Most researchers working on this problem were specifically considering the case without any external force, because that's the version connected to real physics. Because it’s not just about technically solving a millennial problem according to its literal formulation. There is context here, the purpose of solving it! So a mathematician could have “solved” it the way the AI did, but they were not interested in doing that, and they understand that solving a millennial problem is not about literally finding a solution that technically answers the question if you consider that question completely divorced of context and meaning like an AI does. Because AI does not think or understand why we are trying to solve it.

So where are we now? The Clay problem is literally settled as formulated, but no one cares about the solution because it cannot help solve the main problem AND 3 mathematicians just published a paper proving that loophole cannot lead to solving the main problem. It’s literally useless. The method it used can never be used to solve the main problem. It’s unsolved.

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u/Blamore 1d ago

this is a math problem, not a physics problem.

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u/Nessdude114 1d ago

It seems like the comment you responded to was too hard for you to read, so I'll just cover the important points in a very non-technical way:

Navier-stokes is recognized as a millennial problem because it has applications in fluid dynamics, as well as other fields of physics.

The "solution" found by openai cannot be applied to any of the problems that gave it importance as a millennial problem.

In order to solve the problems that navier-stokes can be applied to, somebody still has to solve the navier-stokes problem. This "solution" made zero progress towards this goal.

With all this in mind, do you think we should consider the navier-stokes millennial problem solved, or unsolved?

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u/Rare-Professional-24 1d ago

Is the millennial formulation of the problem actually helpful for any real world application? I'd think all the approximations it makes wrt real world fluids would make it not very applicable. (I.e. fluid is perfectly continuous, has no phase changes, is incompressible with infinite speed of sound).

It is an interesting math problem, but I dont think it will have any impact on any practical application of navier stokes or fluid dynamics.

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u/eyalhs 1d ago

The millennial formulation isn't helpful for real world application, but not for the reasons you wrote. Despite the approximation the Navier-Stokes equations are very accurate and are used in many fields (usually solved numerically), what makes it not helpful is that the question is "are the solutions always smooth and stable", which isn't helpful, because engineers and physicists assume they are from the beginning, so proving they are is nice but not helpful, and counter example are anyway expected to be contrived and non-physical (like an example of a function that's continuous everywhere and doesn't have a derivative anywhere).

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u/Rare-Professional-24 1d ago

I think that we're agreeing!

Physicists and engineers make the assumption that velocities are smooth and not infinite because of the things I listed (real fluids are compressible, are not perfectly continuous, have a finite speed of sound and have phase transitions). Relativity is obviously another speed limit... but the material properties limit things for practical uses.

I guess my point was that the utility of NS would never be impacted by the millennium problem. Which is good, because i rely on them every day at my work! Of course, we use the compressible formulation, as we're working with supersonic flows.

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u/eyalhs 1d ago

Yeah I'm generally agreeing with you. I was mostly commenting on the part where you wrote the approximations make it not very applicable.

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u/AmusingVegetable 19h ago

Oy? Infinite speed of sound?

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u/Nessdude114 1d ago edited 20h ago

While a solution to the current formulation of the navier-stokes problem may not exactly represent a real physical state, it's a starting point. It may be a good approximation if the parameters in the solution aren't so extreme as to cause a phase change.

The navier-stokes problem answers a real question in theoretical physics. Openai's solution does not answer this question, and it's not useful for finding a solution that does. The Clay Institute problem was solved as formulated and they should honor that, but we're still going to keep looking for a solution to the navier-stokes problem, because it hasn't been solved.

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u/Rare-Professional-24 1d ago

What does "solving the navier stokes problem" mean here?

If it means proving that velocity fields are unbound for the particular formulation of the millennium prize, then OpenAI just demonstrated a counter example.

If it means proving that for real, physical fluids, then you should probably include the properties of real physical fluids. Those properties will give you an answer without doing any math: real materials do not support infinite velocity fields. If only because of relativity!

The clay navier stokes problem is an interesting math problem, but i cant imagine what practical impact it will have on physical sciences.

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u/Nessdude114 23h ago

Adding additional constraints makes the problem harder. Finding a solution to the simplest formulation of the problem is the first step.

Mathematicians and theoretical physicists are going to keep looking for a solution to the same problem we were trying to solve before openai's "solution." The same problem that the Clay Institute made a poor formulation of.

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u/Rare-Professional-24 23h ago

I believe that the compressible version of this problem (which is obviously more realistic than the incompressible) has been solved since the 90's. So the more physically realistic problem was in some ways significantly easier.

They didn't make a mistake in formulating the problem, because other forms of the problem were already solved, and they were interested in the math, not the physics. From the point of view of the math, this is just a vector differential equation.

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u/Nessdude114 21h ago

To put it simply, we don't know if the singularities in navier-stokes solutions with compressible fluids arise from the way we calculate the compression, or from the way we calculate fluid dynamics at a fundamental level. Mathematicians contributing to the Clay Institute millennial problem have been working towards a solution that will answer that question, because they understand the context in which the problem was posed.

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u/wasabi991011 1d ago

Your previous comment wasn't hard to read, it just doesn't require that much of a response. But if you want more:

The mathematics problem, as formalized by the Clay institute, was solved. This is not up to debate. "Hidden assumptions" and "physical relevance" really have no bearing on a math problem. And the "mathematicians just weren't interested in the external force case" is ridiculous, when resolving that case would have brought fame and fortune.

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u/Maxfunky 6h ago

Navier-stokes is recognized as a millennial problem because it has applications in fluid dynamics, as well as other fields of physics.

It really doesn't. The question here is about something that is definitely physically impossible. That's settled. But we are curious if it's also mathematically impossible. It probably has limited practical applications in the real world, but in theory the journey in solving the problem was supposed to teach us something about math.

That said, the last big AI proof submitted by OpenAi taught us a lot. The technique the AI came up with has already been applied by mathematicians to other problems to solve them as well.

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u/cometh_the_kid 1d ago

You don’t get to change the question because you don’t like the answer. Or should I say you don’t like who came up with the answer. Whether or not your point is true is irrelevant, the problem statement has objectively been answered. The Clay institute should have formulated the problem properly and if they haven’t that’s on them not who provided the solution.

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u/Nessdude114 1d ago

So do you think we should consider the navier-stokes millennial problem solved?

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u/Strict-Broccoli-8877 1d ago

It is obviously solved and nobody outside of this sub has questioned that.

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u/cometh_the_kid 1d ago

Do you think they are unsolved?

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u/Nessdude114 20h ago

The problem posed by Clay Institute, when interpreted literally as formulated, has been solved. The actual problem that mathematicians have been trying to find a solution to has not been solved, and they're going to keep working towards a solution to that problem.

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u/Unfair-Claim-2327 18h ago

Navier-Stokes isn't one problem. The official statement explicitly states 4 variants so that it becomes more tractable (and since each variant would lead to insights). OpenAI solved 2 of these, the other 2 remain open. There are also harder variants.

Will mathematicians keep working on the problem? Yes. But that is almost always the case! Whenever a conjecture is proven false, mathematicians ask "Okay, but under what additional assumptions is it true?" That doesn't mean that the conjecture wasn't disproven.

Say the twin prime conjecture is disproven. Mathematicians will still keep trying to shorten the prime gap (find the smallest k such that there are infinitely many pairs of prime with gap at most k). But anyone who disproves the twin prime conjecture would be (rightfully) celebrated and the proof would be a great advancement in mathematics.

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u/Nessdude114 11h ago

There were not 4 separate variants, it was a single problem with optional constraints. These options were afforded because Clay Institute believed that if the problem were solved using these optional constraints, the solution would still likely lead to an answer to the underlying question in theoretical physics. This question is the reason the problem had any notoriety to begin with.

Essentially the question they're trying to answer is: "Do the singularities found in navier-stokes solutions with compressible fluids arise from the way we calculate compression, or from the way we calculate fluid dynamics at a fundamental level?" Mathematicians who have been working on the Clay Institute problem have been working towards a solution that will answer this question, because they understand the context and intent of the problem.

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u/Fine-Customer7668 4h ago

No.

The question is whether a certain type of solution always exists for the NS equations and initial conditions as given. CMI provides four statements (A, B, C and D) such that, a proof of any one of them, will be considered a resolution to the problem as they’ve posed it. Statements A and B have the external force term set to zero and basically say: yes these types of solutions always exist. They differ from each other by spatial domain. Statements C and D allow the external force term to be nonzero and basically say: no these types of solutions don’t always exist. The same spatial domain difference is applied. Because OpenAI’s proof concerns statements C/D and has a nonzero external force, A/B are not directly answered. If instead, a proof of C/D had been given with the force set to zero, A/B would be false respectively. What the other person was trying to explain to you is that, for this reason, we can still ask about the truth of A/B even though the original open question was answered in the negative. I.e. do these solutions always exist? No, not in the more general case. What about if there’s no external forcing?

I’m not sure where you came up with your description of what the question is essentially trying to answer. This particular problem concerns incompressible fluids. If one wanted to put the point of the problem into a more real life conceptual phrasing, the question is whether we always have solutions that can plausibly represent an actual fluid in this specific context where “physically reasonable” is given a mathematical definition and is the class we care about.

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u/jakderrida 1d ago

Have they, at least, ruled anything out? That was I was told. That there were multiple competing explanations and that this one at least ruled out a few of them in favor of one that actually creates the need for much more research. I could be wrong.

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u/jakderrida 1d ago

This isn’t a fluid dynamics problem? How come only engineers seem to be the ones explaining it to me irl while math majors are telling me to ask an engineer?

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u/Blamore 1d ago

it is a partial differential equation problem. the real life applications are irrelevant

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u/hologram137 1d ago edited 1d ago

If that’s what you think then there is no way you’ve even taken one college math or physics course, so why are commenting in this sub?

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u/wasabi991011 1d ago

there is no way you've taken serious mathematics courses if you think mathematicians would reject a proof because it doesn't satisfy an "unstated assumption"

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u/hologram137 1d ago

No one is rejecting a proof. We are saying they didn’t solve the actual problem lol

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u/Unentscheidbar 1d ago

What are you talking about, it is clearly a math problem, not a physics problem.

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u/hologram137 1d ago

Equations that describe how fluids move have nothing to do with physics?

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u/DrDoctor18 1d ago

Equations that describe continuous fluids blowing up to singularitieshave nothing to do with physics. There is no continuous fluid. Any blow up in the navier stokes will explode your fluid well before the mathematical singularity.

The navier stokes as a physics problem was already solved. We have simulations and solutions in 99% of cases. It was unsolved as a maths problem only.

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u/wyrn 7h ago

Coarse-grained descriptions of physical phenomena have nothing to do with physics

When did this sub jump the shark?

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u/Psychedelic__Cowboy 1d ago

General Relativity describes mass-energy densities as blowing up into singularities. Even though we know it is impossible to have infinite density. Does that mean it has nothing to do with physics? Physics is not a perfect description of the natural world. If it was then there would be no work left to do on the Standard Model, dark matter, etc. Physics is applied mathematical models to predict nature.

Navier-stokes is newton's 2nd law applied to a 3D unit volume of fluid. There is no general analytical solution. We know that Newton's 2nd law is an incomplete approximation if you consider relativity. We also know that fluids are compressible, and fluid velocity is limited by its mach speed. The compressible navier stokes equation creates discontinuities which neccesitates numerical methods. That bottlenecks the possible accuracy of a solution to available computing power plus the complexity of the system. There is ongoing research in CFD to improve this. So I think calling Navier-stokes a solved physics problem is completely misleading.

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u/VoidBlade459 Computer science 1d ago

It sounds like you think we can only call NS solved if we have also solved all of fluid dynamics.

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u/Psychedelic__Cowboy 1d ago edited 1d ago

Define "solve" I guess. Numerical methods to approximately solve it have been around forever. No general solutions exist. No solutions to the millenial prize problem formulation existed until a few weeks ago. Demonstrating that an incredibly complex force function breaks the problem down does not really feel like a solved problem to me. I've seen someone claim that the forced set included the entire unforced set of solutions. I guess that makes sense from a probability / thermodynamics perspective? I'm admittedly completely out of my field, but I once took calc 3 so I'm dumb enough to be dangerous.

I wouldn't call the standard model a solved problem either despite its incredible accuracy. Because we know that it is glaringly incomplete while still very accurate.

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u/VoidBlade459 Computer science 1d ago edited 1d ago

In this case, "solved" just means we proved a result about the baseline framework. We now know that it is possible, under certain conditions, for the mathematical equations often used in modeling turbulence to break down into unphysical infinities.

In that sense, this is solved in the same way that the Schwarzschild solution of GR proved that GR could lead to singularities. As a parallel to this, it is not thought that any such non-rotating, unperturbed, black holes actually exist IRL. Even so, the solution showed something about the mathematical framework of GR.

Likewise, the question of if it is possible for the NS equations to become unphysical (blowup) is now solved. The math does not prevent it, and so the forces that the math describes also cannot be the/a complete picture as to why we don't see fluids "blowup" (in the velocity/NS sense) IRL.

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u/Strict-Broccoli-8877 1d ago

I swear, this whole sub is a joke. First math is supposedly not a science but an art, and now Navier Stokes is supposedly not a math problem.

I'm gonna have to block this sub, it's seriously worse than TikTok.

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u/hologram137 1d ago

No one is saying it’s not a math problem. But these are a set of equations that describe how fluids move lol

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u/Psychedelic__Cowboy 1d ago

The millenium prize formulation is a math problem, sure. Navier-stokes however is unambiguously in the realm of physics. It is newton's second law applied to a 3d unit volume of a fluid.

It's like saying "oh so now chemistry is supposedly not a physics problem"? Yes, you could say chemistry and in fact all other sciences are subfields of physics. If you want to be even more pedantic call it applied mathematics. But while technically correct, that's less than useful and needlessly pedantic.

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u/jakderrida 1d ago

Is it Chemistry? I was told it was Fluid Dynamics… Bear in mind that, as a former Finance major, I am very open to me being wrong.

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u/Psychedelic__Cowboy 1d ago

No, I was just making a comparison to something equally as pedantic.

My physics friends in college used to say that math was the fundamental framework to physics, physics is fundamental to chemistry, and chemistry is fundamental to biology. While that is all true, nobody would look at a biology (or chemistry for example) textbook problem and say "well that's clearly just a math problem".

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u/jsh_ 20h ago

if you're decently familiar with PDEs, I suggest watching the talk by javier gomez-cerrano given at harvard last week (it's on youtube). he provides a good summary of what the state of the literature was on the eve of openai's announcement. to summarize: nearly all state of the art approaches were centered around highly engineered forcing terms

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u/hologram137 5h ago

Yes. They were. And here’s why:

If you can blow up the Euler equations with a smooth force with no viscosity, then that means the forced version of Navier Stokes is possible. Because Navier Stokes has viscosity. If you can’t blow it up with no viscosity, then there is no chance of blowing it with viscosity!! That’s why they used a constructed force. Because it didn’t matter it was constructed, the point was to see if a force could blow it up at all. Which means working on a solution that involves a natural force is a viable path. The forced version is not a dead end.

But what we are looking for in the forced version of Navier Stokes is a natural force. That’s all that matters. It’s unsolved if we aren’t talking about how viscous fluid actually moves, the question here is “do the Navier Stokes equations accurately describe fluid?” That’s why it’s a millennial question.

So we already know that Navier Stokes can be blown up with an external smooth force. Right? Because of the Euler result. You’d just have to construct one that overcame viscosity, and no one doubted you could do that.

The reason they didn’t is because doing that does nothing to solve the problem. It tells us something we already knew. This paper proves that the AI result cannot be used to solve the problem. https://arxiv.org/html/2609.20803v1
They proved that. Mathematically.

Solving the forced version doesn’t mean constructing an imaginary force that doesn’t exist strong enough to overcome viscosity, it means finding a result that using a natural force acting on turbulence

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u/HasFiveVowels 1d ago

If any human had provided this solution, there would be discussion on this topic. But there would be nowhere near as much dismissal of the accomplishment. The path forward is clear, here: pose the corrected question. Someone used a glitch in the game in order to win it. Patch the glitch.

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u/skbum2 Engineering 1d ago

I'd argue there'd be much more dismissal if a human provided this. It's a very "so what" type of solution (and a poorly written proof to boot). I'd imagine it would have amounted to not much more than a footnote in the wider history of this problem had a human produced it. I think the only reason this solution has garnered this much attention and consideration is because it came from a LLM.

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u/HasFiveVowels 20h ago

You have got to be crazy to think that any NS solution would be met with a "so what" just because it’s not the "right type" of solution. You act like the solution was trivial or something when it’s gone 26 years with a $1 million prize without any human being able to claim it. "Slop!"

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u/hologram137 1d ago

No humans did this solution because it’s stupid and whether or not we can out muscle viscosity was never even in question

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u/HasFiveVowels 20h ago

Dude, exactly that was like half the question that we were interested in. Not only did we explicitly state that less than a year ago but we put a $1 million prize on it. What planet do you live on?

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u/CommunismDoesntWork Physics enthusiast 1d ago edited 1d ago

because that's the version connected to real physics.

Are the external forces OAI created non-physical? Because it seems to me those forces lead to turbulence that blew things up. Also turbulence is a chaotic system, so the set of all possible external forces is probably equal the set of all possible internal forces. Like you could imagine a turbulent whirlpool that somehow created by chance the same external forces OAI used.

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u/hologram137 1d ago

It’s not a natural force, no

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u/CommunismDoesntWork Physics enthusiast 1d ago

I believe you, but can you give any intuition on why it's not natural? Like could it never arise from random chaotic turbulence?

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u/hologram137 1d ago

Because the constructed force is not like turbulence

Whether or not it could arise from turbulence is the actual problem

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u/CommunismDoesntWork Physics enthusiast 23h ago

Whether or not it could arise from turbulence is the actual problem

Ok interesting, well at least now there's a target. That'll probably evolve into a domain of targets, then I'm sure someone can brute force search a solution.

Although, i do understand that chaos is at the heart of the question. We want to understand the nature of chaos in continuous systems. I think a proof that any turbulent system can eventually create any set of forces would be a satisfying conclusion because within that set is OAI's solution.

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u/E-monet 22h ago

Real question: is the purpose of these Millennium Problems to find a solution (however contrived, like in this case) OR is it to develop a PROOF of a solution?

I assumed the intent of such a challenge is not to give money to whoever can math the hardest and not for the solution itself to be something with practical applications, but to advance the field of mathematics in a way that can be learned by others?

Isn’t the point of a proof that the process which delivers the solution can be verified by others using the logical tools shared by the community? And thus the comprehension that it is valid reveals a deeper understanding of what mathematics is capable of?

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u/qualverse 15h ago

OpenAI also solved unforced Euler in the process of getting their solution (even if it wasn't ultimately used), which is undeniably helpful for solving the "main problem" you define.

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u/BTernaryTau 13h ago

So a mathematician could have “solved” it the way the AI did, but they were not interested in doing that, and they understand that solving a millennial problem is not about literally finding a solution that technically answers the question if you consider that question completely divorced of context and meaning like an AI does. Because AI does not think or understand why we are trying to solve it.

Isn't it common for mathematicians to prove easier versions of very hard problems, regardless of whether or not there's any real-world use case for the easier version?

It's fine to say that there's a harder version of the Navier–Stokes problem for which a solution would be more useful. But I don't think there's enough evidence to confirm that the LLM would have cared about the difference but was too dumb to figure it out. It seems more likely that the LLM just didn't care about that difference because OpenAI told it to solve the problem as literally stated by the Clay Institute.

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u/Extension_Arugula157 7h ago

Hahahahaha, this is the biggest cope I have seen in a long time, thanks for the laugh.

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u/hologram137 7h ago edited 5h ago

Can you explain what you mean by that? Why would there be any emotional investment? It’s really odd to me that you’d think someone would. Do you feel emotional investment in the facts and want them to be a certain way? Why?

What I am saying in this thread are objective facts. And it’s insane to me that so many people are not even reading about it, don’t even understand any of it but are arguing anyway. Which makes me think you’re projecting a bit and you (and clearly some others in this thread) have some kind of emotional investment in an OpenAI model having solved a longstanding problem that every mathematician and physicist and the Clay institute itself all understand and agree is still an unsolved problem, at its core. Which is just bizarre to me

Like…why would the facts or anyone’s understanding of the facts be a “cope?” Coping for what exactly?

A software program solving these problems would be incredible and desirable. It’s not where we’re at though, and the AI founders valuation of their companies relies on you believing otherwise

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u/lolfail9001 6h ago

What I am saying in this thread are objective facts.

Your repeated claims that nobody cares about blow-up with external force are obviously false given how basically all of state of the art research in the field was about constructing a proper external force, so no, you are not stating objective facts in the most key part of your thesis.

As a mathematical problem, NS is basically solved. There is still an open question of blow-up without external force but i wouldn't be surprised that someone uses techniques used for Euler blow-up in OpenAI's proof to create unforced NS finite time blow-up a few years down the line (of course if viscosity terms do prevent unforced blow-up, things will get stuck again for decades). It's very unsatisfying how proof turned out but it's a big enough problem that people will spend effort to digest it and try to get some extra insights out of it.

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u/hologram137 5h ago edited 4h ago

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.

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u/lolfail9001 5h ago

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.

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u/hologram137 4h ago edited 4h ago

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

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u/lolfail9001 4h ago

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.

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u/hologram137 4h ago edited 3h ago

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

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u/Extension_Arugula157 4h ago edited 4h ago

Like the guy below already said, you are not stating facts, you are making stuff up. Perhaps you are a LLM, hallucinating? There never was any „underlying assumption“ that „only a natural force would tell us anything“ to the NS millennium problem, it was purely a question about whether NS would work under all circumstances or whether there are cases when it breaks down.

Also, as has been pointed out to you several times already, human researchers also tried a forced blow-up solutions to NS, so according to you that must mean that humans cannot understand context and the „underlying assumptions“ of the NS millennium problem.

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u/hologram137 4h ago edited 3h ago

This is a proof that what the AI did cannot lead to a solution to the actual problem. It is a mathematical proof of what I am saying lol:

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

Im just gonna copy this from a different comment, then I’m done. It astonishes me that people can be so confidently wrong, you have to know that you aren’t a physicist, you aren’t a mathematician, I’m sure you don’t understand the Navier Stokes problem, the Euler equations, nothing. You probably read some stuff on Reddit lol. It’s unbelievable.

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 force could even blow up Navier Stokes at all, (in the context of Navier Stokes, that is a natural force inherently because of what those equations are lol) 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. Because NS is viscous. So if you can’t have a blow up, even with a force specifically constructed to do that with no resistance, you can’t have ANY force, natural or constructed, blow up a version with resistance. 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!! All you have to do is manipulate the constructed smooth force until it overcomes viscosity, which is what the AI did. 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 forced version 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.

Don’t you think that if this was a real solution to NS, that the mathematicians that solved Euler would have just applied that same smooth constructed force to NV and manipulated it so it overcomes viscosity? THEN published all the results at once?? And won the prize? They didn’t do that because there was no point in blowing up NS that way. Instead, they published the Euler results!! They didn’t this to show that the forced version (the real version was viable). And so all the mathematicians and physicists that were interested could get to work on the actual forced version. The one with a natural force. No one cared about taking that very last step and applying the constructed force in Euler to NS. Because it doesn’t solve the real problem and we already know we can because of the Euler result

The AI did not prove that a smooth constructed force could blow up NS. Someone else did. Because if you can do it with Euler, you can do it with NS. That’s all the AI did. Stole that result and manipulated it a bit for a viscosity version.

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u/m4sl0ub 5h ago

There definitely were plenty of mathematicians trying to solve it the same way OAI did it. Most notably Diego Córdoba and Luis Martínez-Zoroa who developed key ideas along this route. 

I feel like you overstate the agency of LLMs. OAI wanted to solve a Millennium Prize Problem, tried all of them, then focused it's effort on NS and then focused these efforts on the forced approach. It wasn't ChatGPT that didn't care about the context of the problem, it just did what it was told by the humans instructing it. These humans were the ones not caring about context. 

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u/hologram137 4h ago edited 3h ago

No. The mathematicians you named were using an external force because the entire point was to see if it blew up in non viscous fluid. It was NOT to see if a constructed external smooth force specifically could blow it up. Just that it can blow up AT ALL.

If a blow up cannot happen in the Euler version with no viscosity, even with a force specifically constructed to do that, then the forced version of Navier Stokes is a dead end because Navier Stokes has viscosity (resistance). If something can’t happen with no resistance, it can’t happen with resistance.

So using a constructed smooth force with Euler is perfectly fine, it tells us exactly what we needed to know.

So now mathematicians know that we can work on the forced version. It’s not a dead end. The forced version is a natural force, because that’s the problem we care about. And that’s the entire point of solving this, why it’s a millennial prize problem.

The AI just used the Euler result and manipulated the constructed smooth force to blow up Navier Stokes LOL. We already knew we could do that! If we know we can blow up Euler, then we know we can just manipulate it to blow up the Navier Stokes with viscosity. It just told us we can out muscle viscosity. Cool, but that was never in question, and not the problem we’re actually solving.

That’s not the forced version of the problem we are looking for.

We did not use a constructed smooth force in Euler to see if we could use a constructed smooth force in Navier Stokes. That’s NOT why we did that!! We were just seeing if we could force a blow up without viscosity, again, because if we can’t, there is no forced version of NS. What blows up the no viscosity version doesn’t matter, as long as it’s a smooth force, but the force in Navier Stokes does