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PaperclipPerfector  2d ago (text post)   8502 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.

May be tossing egg on my face, but I'll post this anyway:

There's a decent chance this proof is not valid. It's formalised in Lean 4, yes, but we also had a Lean-verified proof of the Collatz conjecture a couple months ago that turned out to be spurious (they didn't actually find a proof; they found a bug in Lean).

That proof was 320 lines. This Navier-Stokes proof is 1.6 million lines, and according to their README, requires 100GB to even check. Thanks to somebody eating all the ram, my PC does not presently have 100GB, so I can't run Lean myself to check it. But I will point out that the README explicitly mentions Lean 4.32.2, and the release notes for Lean 4.33.1 mention two soundness bugs fixed, so... it sure looks like there are exploitable soundness bugs in the version they used for this proof.

And if an LLM can find a soundness bug in 320 lines, surely it can sneak one into 1.6 million lines.

This Navier-Stokes proof is 1.6 million lines

How big are human written proofs in this space? This Euler proof that Levent and Co wrote, how big was it? 1.6 million lines feels like a brute force solution. Mathematics at this level is sufficiently arcane for me to never begin to understand it, but I am skeptical that ANY human has ever written a proof of more than 10-20k lines.

The classification of Finite Simple Groups is the largest I recall hearing of, off the top of my head. Looking it up now, it added up to over 10k pages, so in the same ballpark as 1.6M lines, and probably a significantly larger proof when you consider how much more verbose formal proofs have to be. I wouldn't call the classification something any human, singular, has written, though; it's essentially the combination of several hundred proofs of intermediate steps written over decades by around 100 mathematicians.

Looks like there's work in progress to simplify it, but it's only getting cut down to several thousand pages?

I guess the follow on to the other comment here is what is the length of the FSG in lean, in this Lean 4 Mathlib library? I think the FSG being the compilation of hundreds of proofs over decades by 100 mathematicians, and it be comparable to this Navier-Stokes proof in length/complexity(?) is reinforcement to my belief that LLM-AIs are very good at the sort of thing that is just too large in scale for human's to perform at. Assuming this is the solution to the NS, then it would have taken 100s of mathematicians decades to solve this, just by the shear scope of knowledge and effort required.

what is the length of the FSG in lean, in this Lean 4 Mathlib library?

Good question! Right now the answer is "mu"; it's not in there. We might try to extrapolate from the English proof - if around 20 pages of English turns into 250K lines of Lean and around 1000 pages of English turns into ~13M lines of Lean, I'd guess we'd be in the ballpark of a billion lines in total.

Assuming this is the solution to the NS, then it would have taken 100s of mathematicians decades to solve this, just by the shear scope of knowledge and effort required.

Well, the trouble is that we might not yet know what the scope of knowledge and effort required was, only what the scope that was sufficient was. The same theorem can admit scores of different proofs, of greatly varying difficulty levels and lengths. Human mathematicians tend to find proofs uglier the longer they are, counteracted by the extent to which they can be broken into intermediate lemmas (or in the FSG classification, whole-paper-worthy theorems) that look interesting on their own. But AI mathematicians so far appear to just be trained to Do The Task and get to any proof. My wild-ass guess is that without any AI assistance it'd have taken us decades to get a solution here, but it wouldn't have been a thousand man-years of effort on this problem, it would have been hundreds of man-years of effort on other related problems that eventually made this one look more tractable.

Nobody will ever try for a better non-AI-assisted solution, though. At this point the fastest way to get to a nicer proven counterexample will be to train (or at this point maybe just task) AIs with finding shorter+simpler+more-interesting proofs.

reinforcement to my belief that LLM-AIs are very good at the sort of thing that is just too large in scale for human's to perform at.

I think "too large in scale" has always been computers' strong suit vs humans, but the scope where we can actually employ that scale has greatly changed. It used to be that computers were good for tasks where describing how to get a solution was simple but actually executing that process was incredibly tedious. It feels like we've cracked the next level, tasks where describing how to verify a proposed solution is simple but actually figuring out how to get to that solution is incredibly tedious. There's still one level left, that of problems where we can't cheaply verify/score a proposed solution so we have to actually get AI to learn efficiently rather than just self-playing with a billion artificial problems before tackling a real one ... and then that's pretty much it.

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