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Claude Fable produced a counterexample to the Jacobian Conjecture

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Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#531

Post by tptacek »

YeGoblynQueenne wrote: post #502
tptacek wrote: post #460

Earlier quoted context omitted.

Why are you asking them which basic computer science graph traversal algorithms a frontier model used?

OP: >> using human-like(?) reasoning means cutting down the search space by having some sort of insight or intuition which allows you to prune branches from the entire tree So according to the OP there's a search of a tree and it also uses pruning btw, so I want to know what search they mean. Why are you asking?

Because your question doesn't make sense; this isn't how implicit search works.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#532

Post by tptacek »

YeGoblynQueenne wrote: post #505
tptacek wrote: post #459

Earlier quoted context omitted.

That's not what happened here. This isn't a proof; it's a counterexample. The model was perfectly capable of verifying its correctness. You could have verified it by hand if you wanted; the verification is trivial. Finding it was the hard part.

>> The model was perfectly capable of verifying its correctness. It's an LLM. It can't do that.

Are you doing the "LLMs don't know how many R's are in 'Raspberry'" thing here?

A bunch of people on the original thread about the Conjecture were like "we need to wait and see if real mathematicians verify this proof it's probably just LLM psychosis", because they don't understand that the counterexample is a trivial calculation. Checking it isn't hard; any AP calc student can do it quickly. It's finding the counterexample that's the challenge.

It's as if someone presented a SHA-2 collision, which anyone could just feed to `openssl sha256` to see, and then naysayers were like "we need to wait for independent verification because the LLM can't know if that collision was real".

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#533

Post by alerad »

jlev1 wrote: post #368
Davidzheng wrote: post #324

Earlier quoted context omitted.

here's another version directly from the horse's mouth : "Consider the natural map π: P¹ × Sym²(P¹) → Sym³(P¹), (p, {q,r}) ↦ {p,q,r}. Let R be its ramification divisor and let H ⊂ Sym³(P¹) ≅ P³ be a hyperplane tangent but not osculating to the small diagonal; identify X := (P¹ × Sym²(P¹)) \ (R ∪ π⁻¹(H)) ≅ A³ and Y := Sym³(P¹) \ H ≅ A³. Take π|X: X → Y." This is in fact so simple if correct that someone should have fo…

My Claude found a similar description (it phrased it in terms of the natural map from "cubics with a choice of root" to "cubics"). The part that seems not at all simple or obvious is the fact that X is isomorphic to A^3. In your presentation (and more or less similarly in the one my Claude found), X is given as P1 x P2 minus a reducible hypersurface, also I think R itself is reducible since it contains points of the…

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Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#534

Post by orangecat »

aureate wrote: post #421
beering wrote: post #45

Earlier quoted context omitted.

I think parent’s point is that every false conjecture can cost a lot of time to be spent on futile affirmative proofs. So if we “clean up” a bunch of false conjectures, then more effort can be spent on interesting proofs of the others. (Probably a rather naive view of the value of conjectures but I’m just offering an alternative interpretation of the comment.)

The Collatz conjecture is a question about positive integers, so enumerating and checking all the possible counterexamples is trivial, albeit requiring infinite time. It has been verified up to 2.36×10^21. It could turn out to be false, but nobody's going to find a counterexample as surprisingly simple as the one Claude found for the Jacobian conjecture, which would be like finding a Collatz counterexample in the fir…

I haven't seen any thorough analysis of how "hard" it would have been to find by brute force

Pretty hard. I asked Fable and it gave an estimate of 10^46 candidates in the counterexample's "reference class", and that's assuming you know how many distinct terms there are (as opposed to searching all polynomials of degree 7/6/4 for the three coordinates, which it estimates at 10^334).

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#535

Post by orangecat »

icepush wrote: post #344
meowface wrote: post #198

Earlier quoted context omitted.

Hilariously absurd cope. They're still not very good at writing, but you've flipped the takeaway. The correct conclusion is writing style isn't very relevant to intelligence.

I think the real tragedy is that they are good at writing, just by default are tuned to have a kind of bland corporate tone. If you give the LLM a few pages of writing you like and tell it "Continue, but using this style" it will do a pretty good job of it. Most people just .... don't bother to do that.

Yeah. And really their default tone isn't that bad objectively, it's more that it becomes annoying when it's everywhere. There are a bunch of writers with distinctive styles that I enjoy, but would quickly get sick of if half of everything I read was like that.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#536

Post by CamperBob2 »

YeGoblynQueenne wrote: post #504
CamperBob2 wrote: post #457

Earlier quoted context omitted.

What you're asking for is exactly the sort of thing that belongs in, and will appear in, a journal article. There will likely be a preprint on arxiv, so you might keep an eye out for that. In any case, the fact that it was found by a commercial model means that the unfiltered reasoning trace isn't available even to the original author. So there are aspects of the problem-solving process we'll never see. Even if we di…

No, I think it works the opposite way. Until there is an article somewhere that describes what happened, if there is one, all we have to go by is that some guy posted a counter-example for the Jacobian on X, with a vague allusion to using Fable and without any further information. Assuming and guessing anything about e.g. the method used at this point is just raising the noise level.

No one can figure out where you're coming from here. If the human solved a significant open problem without relying on AI, don't you think they'd have claimed the credit for themselves?

The suggestion that a human, working at Anthropic or elsewhere, did the hard work needed to disprove the Jacobian Conjecture yet chose to claim falsely that their AI did it, amounts to an extraordinary accusation that requires extraordinary proof.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#537

Post by CamperBob2 »

nullc wrote: post #461
CamperBob2 wrote: post #129

Earlier quoted context omitted.

Interestingly, even Qwen 3.6 27B was able to verify the solution, but I didn't get any glazing for discovering it. Instead, it thought that someone named Shestakov had already found a counterexample in 2004. GLM 5.2 whiffed, it insisted the counterexample wasn't valid. VibeThinker 3B also recognized that the counterexample was valid. But it kept trying to convince itself that it wasn't, over and over, since it's an "…

I think vibethinker is heavily overtrained on not attempting to solve open problems. I had a fun time taking some open problems and disguising them algebraically so that vibethinker 3b would work on them. It managed to prove some interesting things that I didn't know and would be publishable, but for the fact that they already have been. :) (though hard to know if this was because it had been exposed to that knowledg…

IMO VibeThinker is the most interesting open model since the OG DeepSeek R1. The conventional wisdom has always been that specialization is not very helpful for LLMs, yet it outperforms models hundreds of times larger in its specialized area. It shows that there is a lot of fruit left to be picked, still out of reach but hanging low enough to be worth going back to the barn to fetch a ladder.

If I were a young Turk in this business, I'd drop everything else and figure out how VT3B is so ridiculously good at math.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#538

Post by YeGoblynQueenne »

tptacek wrote: post #531
YeGoblynQueenne wrote: post #502

Earlier quoted context omitted.

OP: >> using human-like(?) reasoning means cutting down the search space by having some sort of insight or intuition which allows you to prune branches from the entire tree So according to the OP there's a search of a tree and it also uses pruning btw, so I want to know what search they mean. Why are you asking?

Because your question doesn't make sense; this isn't how implicit search works.

"Implicit search" with pruning?

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#539

Post by YeGoblynQueenne »

CamperBob2 wrote: post #536
YeGoblynQueenne wrote: post #504

Earlier quoted context omitted.

No, I think it works the opposite way. Until there is an article somewhere that describes what happened, if there is one, all we have to go by is that some guy posted a counter-example for the Jacobian on X, with a vague allusion to using Fable and without any further information. Assuming and guessing anything about e.g. the method used at this point is just raising the noise level.

No one can figure out where you're coming from here. If the human solved a significant open problem without relying on AI, don't you think they'd have claimed the credit for themselves? The suggestion that a human, working at Anthropic or elsewhere, did the hard work needed to disprove the Jacobian Conjecture yet chose to claim falsely that their AI did it, amounts to an extraordinary accusation that requires extraor…

I guess? But I didn't say anything like that.

>> No one can figure out where you're coming from here.

No, I think it's just you and I think that's because you have preconceived ideas about the only possible positions that people can assume in this debate. I'm sorry, of course, because that's not conducive to productive dialogue, but it's not my fault. I think what I wrote so far is very simple, no technical jargon, no tortured metaphors: all we know is a guy posted a Jacobian conjecture counterexample on X thanking another guy and Fable for it but without explaining what that means, and there's no reason to assume anything else besides that very limited amount of information at all.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#540

Post by [deleted] »

tptacek wrote: post #532
YeGoblynQueenne wrote: post #505

Earlier quoted context omitted.

>> The model was perfectly capable of verifying its correctness. It's an LLM. It can't do that.

Are you doing the "LLMs don't know how many R's are in 'Raspberry'" thing here? A bunch of people on the original thread about the Conjecture were like "we need to wait and see if real mathematicians verify this proof it's probably just LLM psychosis", because they don't understand that the counterexample is a trivial calculation. Checking it isn't hard; any AP calc student can do it quickly. It's finding the counter…

[deleted]
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