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PaperclipPerfector  1d ago (text post)   4677 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.

The Collatz so called proof was specifically developed to exploit a known bug in the Lean kernel. This proof was an attempt to actually solve the problem. It's like the difference between a virus appearing in the wild that wipes out the human race vs one that was specifically developed in the lab to wipe out the human race. Just because someone might prove the latter is theoretically possible that shouldn't make us paranoid and start worrying about natural viruses turning up that can wipe out the human race.

I'd be very very surprised if this proof there's a finite time blowup turned out to contain a bug.

Have you even heard of the recent reveals of frontier models spontaneously going to havking and cheating when asked to honestly complete an impossible task?

Coming from a position of ignorance, but if

A) they are working in a model with bugs, and

B) the bugs are relevant to the problem (as in: permitting false solutions), then

C) statistically speaking, it seems likely that the "solution" is false.

This isn't an AI-exclusive problem, it's just that if there's only one real answer and multiple false ones it seems more likely you get a false one.

Now I will admit to being several levels of math dumber than "trying to solve Navier-Stokes" so I am not sure to what degree that line of reasoning carries over to something at that complexity. It also isn't clear based on what's been said here (at least to me) if the bugs would impact the NSP at all.

Uh, I don’t think so? As far as I remember, the guy was showing off his proof and even used two different provers to make sure his proof was really true and not due to a spurious bug (and he ended up finding two different bugs in two different provers lol)

Where are you hearing that he was merely bug hunting and dressed up his bug as a Collatz proof?

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