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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

#231

Post by 7373737373 »

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 "…

Current LLMs behave very counterproductively around unsolved problems, especially if they learned that humans consider them difficult. This has many straight up preventing themselves from attempting anything...

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#232

Post by FacelessJim »

minimaxir wrote: post #134
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.

> So the conjecture that survived Keller, Abhyankar, Moh's degree-100 verification, and five-plus published wrong proofs appears to have died via tweet during the World Cup final.

I lol’d. I didn’t know Fable was this sassy

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#233

Post by elbear »

danbruc wrote: post #163
k1t wrote: post #150

Since I actually don't know math, maybe my ELI5 understanding can be helpful (or corrected). The conjecture says that you can always reverse (a process) to determine the original inputs. But this proof shows multiple inputs creating the same output - which obviously cannot be reversed to determine the input - thus falsifying the conjecture.

[...] you can always reverse (a process) to determine [...] There is a precondition - constant non-zero Jacobian - to the inverse existing and the inverse is claimed to be of a specific kind - polynomial. The counter example satisfies the precondition and by mapping two different inputs to the same output makes any inverse impossible, including polynomial ones. But maybe that is already ELI7.

This helps because the ELI5 was too general. To me at least.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#234

Post by inglor_cz »

laichzeit0 wrote: post #32

The great thing about these mathematical mopping up type operations is that no person will waste their time trying to prove it to be true anymore. If anything that’s a win. It would be great if an LLM could settle the Collatz conjecture next, god knows how many man-years have been burned on that by unsuspecting victims.

I used to be a young mathematician sometime at the turn of the century...

"waste their time trying to prove it" is the MBA approach, where you should spit out results and articles.

Outside the MBA-thinking box, attacking hard problems, even unsuccessfully, is the way to gain deeper insight into various results and tools that you can later apply to other problems, i.e. no waste of time at all, unless you go to the extremes (like spending years on a single problem and nothing else).

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#235

Post by [deleted] »

ibarrajo wrote: post #203

I’ve been math-vibe coding a few months now. It’s surprisingly easy to do with AI. The hard part has been manually verifying and validating the results. I took one of the smaller findings (disproving a conjecture) and wrote a paper as my first endeavor into publishing. Because the next few findings i have in the pipeline are substantial in the field of quantum topology and physics im taking some time to publish them…

[deleted]

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#236

Post by SonOfLilit »

krackers wrote: post #137
tyre wrote: post #87

Earlier quoted context omitted.

Reading the chain of thought for Fable, it is incredulous as well. It keeps thinking that it must be missing something or that this is a trick, because it can't have just found a counterexample to a famous conjecture. It's just like us! https://cdn.xcancel.com/pic/orig/EA1E99C3DAE99/media%2FHNpQX... https://cdn.xcancel.com/pic/orig/3B5C0B867A7C3/media%2FHNpQU...

Which tweet has the chain of thought? I was curious how it discovered this, did it use some symbolic brute force or some clever trick? Also is that the raw or summarized COT, it seems to have the hallmarks of raw COT (the frequent interjections) but I thought Anthropic always used summarizers in between

By the CoT content, this is a screenshot of someone giving Fable the result and asking if it's true, not the CoT of discovery.

e.g. "maybe the user's example is DESIGNED to be "correct in the stated facts" &c"

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#237

Post by somenameforme »

SubiculumCode wrote: post #113

For all thehubbub, as far as I know, all the math breakthroughs via AI that I've heard about have come from Anthropic and OpenAI, not Chinese models. I could have missed those announcements, but one might think that between close to frontier performance plus cheap tokens, that they'd be leading the way on these things.

All (unless I've missed one?) the math breakthroughs are also coming from the relatively small number of mathematicians working at these companies, as opposed to the many orders of magnitude more mathematicians using LLMs for mathematics outside of the companies. I assume the missing link everywhere is being able to casually burn a few rainforests worth of tokens in pursuit of something publishable.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#238

Post by frotaur »

slopinthebag wrote: post #68
mathisfun123 wrote: post #44

The funniest thing about LLMs is the cognitive dissonance they cause people. People clearly recognize (and bemoan) the fact that LLMs produce derivative breathless prose ie they fundamentally fail at "unstructured creativity" (something the might accurately labeled intelligence ) but are then shocked that the same LLMs can do math. It's reasoning from a flawed premise that math universally requires intelligence and c…

I would think it's more that some aspects of math (like anything really, including programming) don't require creativity and can be solved through "brute force" or whatever you want to call what LLMs do. But it's pretty obvious that LLMs are not capable of solving the vast majority of problems in math (or programming) at this time. Eg. Google went 9/353 on Erdos problems. If a more powerful LLM is capable of solving…

Didn't humanity also score very low on 353, namely 0 since they were open? Probably collectively we could have gotten a slightly better score if all of Math started trying to solve those, but not by that much, I think, since they are precisely still open.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#239

Post by Lerc »

slopinthebag wrote: post #206
Lerc wrote: post #183

Earlier quoted context omitted.

> I really hate the idea that AI can replace artists, or humans as a whole. I think it’s anti-human to its core. AI can no more replace artists, than other artists can. A sunset can be nice to look at yet be created without meaning. Any non-conscious process that produces images can produce beauty that has worth without intentionality. Any conscious process that produces artworks is just another artist. Most artists…

Well, I think an AI model could replace artists in a meaningful way without being conscious. So you end up with less artists. Art is about human expression, it's not about the reflection of light off a surface. But economically I could see a world in which human expression is replaced by a facsimile and the world would be worse off for it. And that is what scares me. Brave new world indeed. edit: whoever downvoted th…

I think the fundamental problem is that art is not economical. Banksy once made a comment about the advertising industry, more or less to the effect that for all the corrupting influence of Advertising, the thing he truly hates about it is that it consumes the most creative people to do so.

People will make things they don't care about to survive. This is how we get reality television.

Irrespective of what the short term future of AI is, sooner or later there will be technical means of sustaining yourself without needing to work. If we aim for a society that values working on something because they believe in it, then we could have a better world. Right now it seems the only thing a lot of people are aiming for is to undermine whatever their adversaries want.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#240

Post by c32c33429009ed6 »

nommynommynom wrote: post #182
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 opposing argument there is that the hope is that solving these problems reveals other interesting maths knowledge along the way. Finding a counter example all but ensures that won't ever happen.

No, it opens up a whole new suite of questions. Now we can ask, for example: what conditions do we need for the result to hold? What dimensions does it hold in? And many more...
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