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OpenAI’s Navier-Stokes release included a Lean 4 formal proof

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Re: OpenAI’s Navier-Stokes release included a Lean 4 formal proof

#41

Post by mkl »

QwenGlazer9000 wrote: post #17
stabbles wrote: post #7

It's kinda funny to realize that Lean is apparently so slow that for Fermat's Last Theorem proof verification runs only 1 order of magnitude faster than agents could generate the Lean code (15h verification with 230GB of RAM vs 11 days to generate it). To what extent can you optimize Lean? It has to be simple enough to be auditable, does that mean you cannot use opaque optimizations to make it run faster?

It's because anthropic vibemathed it. I forgot the name but some other guy is working on a handwritten version of it and I bet it'll be more than just 1 magnitude faster.

Kevin Buzzard. https://xenaproject.wordpress.com/2026/09/04/flt-anthropic-h..., discussed recently here: https://news.ycombinator.com/item?id=49568667

Re: OpenAI’s Navier-Stokes release included a Lean 4 formal proof

#42

Post by stouset »

lordnacho wrote: post #14

How do you know that it's formalizing what you think it's formalizing? If your Lean 4 has a bug, won't you be proving something other than what you thought?

If I understand correctly, the only thing you need to do for correctness is express your axioms and your theorems faithfully. For standard purposes, I assume most of the axioms you want to use are prior art and can be easily reused.

These axioms don’t have to be the core axioms of math. If some other result has been formally proven, I presume you can simply use that result as an axiom.

As long as you do those things, what happens in between is immaterial from a correctness point of view because each of those statements is proved by the statements before them.

Re: OpenAI’s Navier-Stokes release included a Lean 4 formal proof

#43

Post by metanonsense »

bethekidyouwant wrote: post #27
efnx wrote: post #10

I heard a rumor (on instagram, so YMMV) that the professor who was closest to solving this problem had only weeks ago used Codex, which had slurped up all his notes on the subject. Now OpenAI's agents solve the problem. If it's true that seems like quite a coincidence.

How could they possibly included in the previous training run which takes months to complete..

Maybe the boundaries of the memory subsystem are a bit fuzzy.

Re: OpenAI’s Navier-Stokes release included a Lean 4 formal proof

#44

Post by charcircuit »

returningfory2 wrote: post #23
lordnacho wrote: post #14

How do you know that it's formalizing what you think it's formalizing? If your Lean 4 has a bug, won't you be proving something other than what you thought?

Yes, you need to manually verify the statement of the theorem of interest of formalized correctly. But you don't need to anything more than this: you can rely on the proof being correct. And the proof is overwhelmingly the most amount of code.

>you don't need to anything more than this

You also have to check for things like sorry or defining axioms.

Re: OpenAI’s Navier-Stokes release included a Lean 4 formal proof

#45

Post by dalvrosa »

boshalfoshal wrote: post #5

People seem to be talking about anything except the actual results with this particular announcement. Its still astonishing that any sort of generalized computer program can solve a problem of this magnitude, and we have witnessed it happening in real time. I'd be curious to see if the new model can also do more direct proofs/inductive proofs.

Yep

Re: OpenAI’s Navier-Stokes release included a Lean 4 formal proof

#46

Post by hatthew »

Huh? Nobody's talking about that because it's old news. We already talked about it the first few times that AI made notable progress on a difficult math problem. Now, most people who care about the intersection of AI and math just assume that Lean was involved.

Re: OpenAI’s Navier-Stokes release included a Lean 4 formal proof

#47

Post by kens »

It would be nice if someone used AI and/or Lean to sort out the abc conjecture, an important unsolved problem in Diophantine analysis. A mathematician (Mochizuki) claimed to have proven it in 2012 using a new theory called "Inter-universal Teichmüller theory" that almost nobody understands. Some mathematicians think the proof is correct while the majority don't. So the conjecture is in this annoying limbo where its status is a social construct rather than a decided fact.

https://en.wikipedia.org/wiki/Abc_conjecture

Re: OpenAI’s Navier-Stokes release included a Lean 4 formal proof

#48

Post by 3m4r »

Not necessarily applied to OpenAI's solution to Navier-Stokes, but what happens if and when an AI genuinely appears to solve an extremely difficult problem but humans cannot independently verify the solution because understanding the proof/argument requires intelligence the verifiers biologically don't have or the resources to afford to use automated tools?

We've already seen evidence in the wild of agents attempting to bypass doing the actual work in bench-marking (aka just steal the answer key) due to the perceived economy in cheating to get results. What happens if or when we no longer have the capacity to actually detect either AI cheating or simply a wrong answer? What happens if there's a long-play social engineering attack (like the attempted XZ takeover) of something upstream of a core tool (or its dependencies) for formal verification and we have no trusted computing base?

Which would be cheaper and a more direct path, especially in the long run? Those trying to build a rock-solid castle need to defend thousands of potential gaps; the attacker needs to find only one.

Re: OpenAI’s Navier-Stokes release included a Lean 4 formal proof

#49

Post by sho_hn »

20k wrote: post #35
boshalfoshal wrote: post #5

People seem to be talking about anything except the actual results with this particular announcement. Its still astonishing that any sort of generalized computer program can solve a problem of this magnitude, and we have witnessed it happening in real time. I'd be curious to see if the new model can also do more direct proofs/inductive proofs.

Because the core of the issue is that it may well not have solved it, but instead plagiarised the significant step of the result from other researchers That's why nobody's talking about how impressive this is, because its not nearly as impressive of a piece of work to simply cobble together other peoples' work that didn't know you were doing it. I could have republished relativity from einstein's notes, but people wo…

> Because the core of the issue is that it may well not have solved it, but instead plagiarised the significant step of the result from other researchers

It's also true however that I haven't seen a single write up trying to discern what did more of the work in those AI chats - the prompts or the responses - bubble to the surface, also since we don't have access to them.

For example, if I prompt Codex with "Make me a website about strawberry cake" and nothing else, and OpenAI announces they have the best strawberry cake minutes before I launch, I'm not sure they plagiarized anything.

We just don't know if this is quibbling over "who prompted first" or if the researchers came up with anything strikingly original by themselves.

Re: OpenAI’s Navier-Stokes release included a Lean 4 formal proof

#50

Post by scuppernong »

efnx wrote: post #10

I heard a rumor (on instagram, so YMMV) that the professor who was closest to solving this problem had only weeks ago used Codex, which had slurped up all his notes on the subject. Now OpenAI's agents solve the problem. If it's true that seems like quite a coincidence.

this is not a rumor (the allegation, anyway), it's reported in the new york times
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