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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

#441

Post by zamadatix »

ethbr1 wrote: post #311
siddboots wrote: post #243

Earlier quoted context omitted.

I fed ChatGPT the map with no other context, just “tell me about this function”. It did a bit of work finding the Jacobean etc and eventually worked out the implications of what it was seeing. It then proceeded to check the arithmetic 4 times, and then decided to do a manual verification using an ad hoc symbolic checker in case its SymPy had been tampered with.

Skepticism is the flipside of knowledge. If an LLM has knowledge encoded inside it (and it's hard to argue it doesn't), then cognitive dissonance can be experienced. And once experienced, must be dealt with, especially in longer-running agentic loops. A friend was joking the other day about sending some messages under a previously-used Slack identity for an agent (since turned off), then asking the agent about the me…

Another way to look at it is the LLM is, by definition, what's expected to be probable based on the training data and this, by the same definition, is extremely unlikely data to run across. With high uncertainty comes the need to verify until it can level out as "really surprising" instead of "plausible sounding error".

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#442

Post by evenhash »

rao-v wrote: post #141

This is so unreasonable! As @__alpoge__ himself notes this is classic crank graveyard territory and yet the counter example is something a grad student in 1997 could have found w a ~3 day computer search. Wild!

The search space for a naive brute force of three polynomials of degree <= 7 with integer coefficients <= 12 is roughly 10^500. I think it would take a little longer than that.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#443

Post by Rzor »

eru wrote: post #292
aizk wrote: post #112

This is a rare instance where feeding this groundbreaking information into an LLM gives _them_ psychosis. I fed this to claude code and watched it verify the result in 7 different ways to be 100% certain, and it was just flabbergasted. Quite remarkable.

Can confirm, Claude is flabbergasted. Gemini just checks the web first it seems, and already references the news. Kimi doesn't quite believe it.

Deepseek Pro got stuck after munching it for a minute or so and couldn't quite believe it either.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#444

Post by UltraSane »

0x1ceb00da wrote: post #245
koito17 wrote: post #176

A little over 10 years ago I remember meeting a postdoc who believed he had something close to a counterexample to the Jacobian Conjecture. He and another person was bruteforcing polynomials in about 16 variables, something like 80 - 700 terms each, using binary trees for mapping coefficients. They were guessing, at the time, that the lower bound of a counterexample (P, Q) for max(deg(P), deg(Q)) would go up to 200.…

[flagged]

Finding a counterexample that humans have failed to find for 85 years is a marketing stunt?

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#445

Post by hgoel »

aurareturn wrote: post #97
cyphar wrote: post #28

Earlier quoted context omitted.

I don't think the (fairly factual) description of these systems as stochastic parrots means that they will never do useful work, just that they are not intelligent in the way we believe animals to be (to "push back" on your anecdata, I've also heard fewer people claiming that LLMs are actually conscious in the past year -- maybe we're reaching the happy medium?). That was the point the stochastic parrots paper and Ch…

I don't think LLM labs ever claimed that their LLMs are intelligent the way we believe animals are.

The labs do tend to kind of anthropomorphise their models, as do us users, but admittedly part of the issue is that, for example, "thinking" is a decent layman description of the process even though it implies more human-like intelligence than there really is.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#446

Post by danyim »

glimshe wrote: post #276
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 told Gemini Pro I woke up after having dreamt that polynomial and asked if it was related to the Jacobian conjecture. It spent some time thinking and referenced this tweet announcement, saying: "If you truly dreamt about that specific polynomial, you might be mathematically clairvoyant." In the rest of the answer, it maintained a cautious skepticism about my claim, saying: "Here is exactly why the math world is cur…

Must be that LLMs are just as suspicious of humans as some us are of them.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#447

Post by YeGoblynQueenne »

antirez wrote: post #254
0x1ceb00da wrote: post #245

Earlier quoted context omitted.

[flagged]

So many mathematicians over the years tried hard and failed, but now Anthropic just for some PR magically did it? And this after LLMs obtaining different math wins? What is your logic here really escapes my understanding.

I mean we have no idea what happened exactly, how Fable was used, how many times it was run, whether earlier models were also tried, what was the prompt, how long it run for, etc etc. All we have to go by is a tweet.

Why not be skeptical about that?

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#448

Post by mswphd »

nxtfari wrote: post #424
kelseyfrog wrote: post #131

Earlier quoted context omitted.

Same result. Public share https://claude.ai/share/19fd1a34-d63b-4a16-8d83-60d5b79e7747 It did the multiple verification sequence before expanding to internet search where it found this thread.

Everyone using Claude Fable to verify this proof is so funny. If you read the definition of the Jacobian Conjecture and (I am not exaggerating this) have passed a college Calc 3 class, you can just verify the proof yourself in 30 seconds. The problem was very hard to solve but the counterexample is very easy to verify! --- edit, adding an explanation: To summarize it, the conjecture says if you have any multi-variabl…

verification also requires computing the jacobian. an undergraduate-level exercise, but harder than verifying p(pt1) == p(pt2).

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#449

Post by YeGoblynQueenne »

hgoel wrote: post #345
fn-mote wrote: post #299

Earlier quoted context omitted.

The concern is that there could have been expert knowledge input, whose importance/worth we are unable to evaluate. I don’t believe a mathematician produced the counter example secretly, but how much did they contribute to the result? AI isn’t magic, so to evaluate the value delta, you need to know the value of the input.

While I agree that we need the inputs to properly evaluate what this means for LLM capabilities, I don't really believe that the amount of knowledge input matters much for the overall significance of the result. These kinds of results are interesting for LLMs because mathematicians have been working on them for decades. If the result doesn't already exist, there's no way it's in the training data, and if mathematicia…

Yes, but the question is the extent to which the new tool was guided by the mathematician.

Are we talking a bicycle, powered by a human stepping on the pedals; or a rocket that will fly to the moon on its own with people inside?

Don't you want to know? I mean, doesn't everyone want to know?

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#450

Post by YeGoblynQueenne »

igravious wrote: post #333
William_BB wrote: post #318

Earlier quoted context omitted.

AI is great at reducing the search space and using human-like reasoning (in a brute-force way) to carry out the brute-force search. I'm not surprised by this result. This is exactly what AI should excel at, with human guidance.

"using human-like reasoning (in a brute-force way)" that's self-contradictory -- what brute force means is doing an exhaustive search of a search space (brute forcing it) 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

LLMs don't search trees. They generate plausible proofs and a human has to check it's true. Repeat until.
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