I have a question I'm surprised people are not asking: How did Fable find this? Was it like guessing a bunch of families and then solving for possible solutions in those families? Was it clever search? something else?
Claude Fable produced a counterexample to the Jacobian Conjecture
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Re: Claude Fable produced a counterexample to the Jacobian Conjecture
#202[dead]
Re: Claude Fable produced a counterexample to the Jacobian Conjecture
#203I’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 with a ton of scrutiny. And verification has taken more time than it did to make the discoveries.
Here’s my first piece if anybody is interested in number theory: https://arxiv.org/abs/2607.09793
Re: Claude Fable produced a counterexample to the Jacobian Conjecture
#204Re: Claude Fable produced a counterexample to the Jacobian Conjecture
#205Somebody said this on the xcancel thread but this also means that the Dixmier conjecture for the third Weyl algebra is disproven
Re: Claude Fable produced a counterexample to the Jacobian Conjecture
#206Earlier quoted context omitted.
I initially wanted to downvote you but I can’t because you’re right. I’d probably be called an anti-ai luddite on this site despite using it all the time for programming. I naturally gravitate to anti-ai viewpoints, but only because the breathless hype of the AI boosters is so nauseating. I have no moral qualms about AI (although I’d like nothing more than Anthropic/OAI/Google dismantled), but I really hate the idea…
> 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 the person replying to me, shame on you. they make good points and makes for good discussion.
Re: Claude Fable produced a counterexample to the Jacobian Conjecture
#207This 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!
So IIUC no one really worked on that in „professional” or rather university world since then?
Re: Claude Fable produced a counterexample to the Jacobian Conjecture
#208I have a question I'm surprised people are not asking: How did Fable find this? Was it like guessing a bunch of families and then solving for possible solutions in those families? Was it clever search? something else?
I’m doing this by working all logical steps into lean (formal verification) the quick feedback loop between the AI prose and the Lean verification errors and warnings ensures that its logically consistent.
The issue that remains are two things, ensuring the idea of the proof is actually the thing you want to prove and the interpretation of the results you get. But besides that, everything inside of the kernel checked code is logically consistent
Re: Claude Fable produced a counterexample to the Jacobian Conjecture
#209Earlier quoted context omitted.
> This example, you could have given an undergraduate good at programming and computer algebra and told them to come up with a counterexample please try go try it. There's no way someone didn't do massive computer algebra searches before today. > All three of the big conjectures solved the answers were at the level where if you had given a grad student the questions and the right background reading there's a good cha…
Why would I try it to win an argument on HN? That's a bizarre suggestion. Just look at the degree. If it were degree 47 in 17 variables then it wouldn't be surprising, but here it's surprising. Of course people tried hard to solve them all, which is why it's so surprising that they were open. If anything, the solutions have gotten easier. The unit distance graph solution relied on a famous theorem remote from graph t…
Sorry last comment was a bit emotional from me, but I do not think it's findable like you say--during my PhD I tried to find some ideals I knew existed in char 2 in 5 variables and low degree and I didn't think I ever got close. 3 variables, 7 degree, coefficients up to 6 is like 6^100 possibilities. You've got to narrow it down somewhat no? Even sparse is intractable I would guess.
I think the solutions which rely on the least amount of theory are the most telling of the AIs being higher in intelligence than humans today already. There's almost no theory to teach someone to understand the cycle double cover conjecture as you say, yet no one finds it. I don't think the conclusion is that it was "easy", but that it was in fact irreducibly difficult in a way that proofs developed with theory are not. Theory gives the human brain abstractions to simplify complex proofs to be understandable at our capacity--I think there are many proofs which probably are not of this form.
But I think our differences hinge on how hard we perceive these solutions to be--I think they are very hard to find!
Re: Claude Fable produced a counterexample to the Jacobian Conjecture
#210Earlier quoted context omitted.
Smarter in some ways at least. Probably still not quite as smart at understanding human emotions (and things that aren't well enough catalogued on the internet or amenable to Reinforcement Learning on virtual environments).
Probably not as smart as highest EQ humans but smarter than me, I use LLMs to help pin down what emotion I'm feeling.
Same here