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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

#271

Post by inigyou »

baq wrote: post #185
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.

sol medium can't believe its own input and output tokens either despite computing everything itself; this is what it gave me: > Taken literally, these two facts would make this map a counterexample to the complex Jacobian conjecture in dimension 3: scaling one output coordinate would normalize the determinant to 1 without restoring injectivity. Since the complex Jacobian conjecture is still treated as an open problem…

Just take it as more confirmation that LLMs are unintelligent pattern-matchers.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#272

Post by jibal »

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.

They gave their "logic", such as it is ... and it's utterly irrational.

Note that the "they" who published the counterexample on X is some rando mathematician (Levent Alpöge) working for Anthropic, not Anthropic the organization. He posted the counterexample in a tweet -- reason enough for "not disclosing the LLM chat session". There's no reason to think that it won't provided if asked for, but it hardly seems relevant.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#274

Post by baq »

inigyou wrote: post #271
baq wrote: post #185

Earlier quoted context omitted.

sol medium can't believe its own input and output tokens either despite computing everything itself; this is what it gave me: > Taken literally, these two facts would make this map a counterexample to the complex Jacobian conjecture in dimension 3: scaling one output coordinate would normalize the determinant to 1 without restoring injectivity. Since the complex Jacobian conjecture is still treated as an open problem…

Just take it as more confirmation that LLMs are unintelligent pattern-matchers.

statistical parrot indeed, just like the one which built the counterexample. maybe.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#275

Post by abernard1 »

cornholio wrote: post #222
rowbin wrote: post #24

Earlier quoted context omitted.

Maybe, maybe not. The "proofs" may not have helped at all with finding a counterexample. Either way, it doesn't matter. A counterexample was found, no one found one before even though clearly a lot of people have tried who also had access to the prior "proofs".

I think it's becoming harder and harder to argue that LLMs don't really reason and just mimicry human speech. This counterexample is clearly the result of a sequence of steps that build on previous knowledge in context and logically combine it to reach other true statements - to a degree and complexity that rivals the best human minds. For someone that use Claude Code every day, this is obvious, but for some reason m…

I think this is actually a perfect case of an impressive thing being done exactly by mimicry.

Why do LLMs still have trouble on floating point math without forking out to a tool, but they can perform symbolic manipulation just fine?

Because symbolic manipulation is just rote work and textual stepping through symbols. A side poster commented on the number of prior attempts on this problem which were close, but not quite.

Starting from a known "close" solution (which this did), and using exploration to search around the space is exactly something an LLM would and could be good at (clearly).

The "transformer LLMs are next-token predictors with some in-GPU processing of bounded complexity with respect to token count" remains undefeated. Both because that is mathematically what they are, and also because we don't have counterexamples to that effect that don't require some higher-order tooling wrapping the systems.

Symbol manipulation is what LLMs are good at.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#276

Post by glimshe »

CamperBob2 wrote: post #129
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.

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 currently scrambling to verify the polynomial you "dreamt" about."

I love how it put "dreamt" in quotes.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#277

Post by Xmd5a »

monocasa wrote: post #139
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.

I've heard about mathematicians going through kind of the same thing when they get a weird proof that ends up being right from some weird source or themselves. Which is fair, they get inundated with kooky proofs from amateurs all the time and odds are incredibly good that there's some major fatal flaw that the amateur doesn't see. Or in the case of themselves, there's a certain blindness that makes it a little more d…

> they get inundated with kooky proofs from amateurs

Source?

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#278

Post by gilgoomesh »

tacomonstrous wrote: post #138

Speaking as a mathematician, it does seem like we're a bit fucked as a community. Anything that is at all accessible to currently existing methods and mathematical infrastructure is probably going to fall to the frontier models of today, and at this rate of progress it's likely that, already by next year, we'll see new infrastructure being put into place by AI, giving us a world in which a few designated interpreters…

Goldbach's Conjecture is very accessible.
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