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PaperclipPerfector  2d ago (text post)   8503 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.

For reference, the Lean 4 mathlib library is an attempt to formalise all mathematics known to humankind, and it sits at 15M lines.

yeah 10% of "all mathematics known to man" makes me a bit skeptical. I lean towards the charitable explanation that this is the sort of thing that AI is just good at because no singular human or even team of humans is going to find a proof that is 1.6 million lines long in any bounded time that doesn't span a lifetime of work.

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