And yet it can't even replace a call center employee.
I guess you can't make a career out of being a frog anymore in math.
Claude Fable produced a counterexample to the Jacobian Conjecture
221–230 of 562 posts
Re: Claude Fable produced a counterexample to the Jacobian Conjecture
#222I suspect the LLM was able to synthesize a counterexample because of the availability of a lot of prior work: > The Jacobian conjecture is notorious for the large number of published and unpublished proofs that turned out to contain subtle errors. https://en.wikipedia.org/wiki/Jacobian_conjecture#cite_note-...
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 many scientists refuse to accept that it's truly reasoning; perhaps not in the human sense, but in a very profound and real sense. These powerful results are devastating to their point of view.
I can sympathize, because I too called LLMs "fancy Markov chains" in the GPT 3 era. But there comes a time where you have to update your world view to match reality, or be stranded in fantasy land.
Re: Claude Fable produced a counterexample to the Jacobian Conjecture
#223Earlier quoted context omitted.
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…
In my experience, lean will show that it's correct, but does it not lose the mathematical intuition that led to the result? As far as my experience goes, that's really hard to encode in lean itself. Could we maybe get more information about the problem from the LLM trace itself here?
As proofs become more and more complex, we will need two AI pipelines: one to generate the LEAN proof, and a second one to extract useful lessons for mathematicians from the LEAN proof.
Re: Claude Fable produced a counterexample to the Jacobian Conjecture
#224Earlier quoted context omitted.
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 o…
But actually my feeling is that the final solutions of these last two problems probably is hiding how the AI came up with them! For all we know it used a LOT of theory! As Dolly Parton says "it takes a lot of money to look this cheap" and it takes a lot of intelligence for the proofs to look this dumb. [In high school, I knew of this competition math kid joke where after you derive an inequality with various methods, you use standard results to write the original equations as just a sum-of-squares---like in a "are you stupid, it's >=0 bc it's a sum of squares" sort of way]
Re: Claude Fable produced a counterexample to the Jacobian Conjecture
#225Actual tweet:
>> hello there the jacobian conjecture is false thanx to my close friend akhil for asking about it and my other close friend fable for working during the world cup final
So where does it say that Fable "produced" the counterexample? The tweet says it was a collaboration between two people, using Fable.
Re: Claude Fable produced a counterexample to the Jacobian Conjecture
#226Earlier 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 "…
> that someone named Shestakov had already found a counterexample in 2004. Qwen has the sprit of a grad student
It must mean Chestikoff. He dedicated his life to intellectual pursuits due to chronic ill health.
Re: Claude Fable produced a counterexample to the Jacobian Conjecture
#227Now that is some spicy autocomplete.
"the Riemann Hypothesis is "
Re: Claude Fable produced a counterexample to the Jacobian Conjecture
#228Apparently ChatGPT-verified: https://xcancel.com/__alpoge__/status/2079045382940573896#m
boo on the downvote. The response is pretty funny IMO
Re: Claude Fable produced a counterexample to the Jacobian Conjecture
#229I just fed this to GPT 5.6 Sol: Counterexample to Jacobian conjecture: ((1+xy)^3 z + y^2 (1+xy) (4+3xy), y + 3 x (1+xy)^2 z + 3 x y^2 (4+3xy), 2 x - 3 x^2 y - x^3 z): \C^3\to \C^3, has jacobian determinant -2, and sends (0, 0, -1/4), (1, -3/2, 13/2), and (-1, 3/2, 13/2) to (-1/4, 0, 0) GPT wrote some SymPy code to check it. The response? "As written, this is an explicit counterexample to the Jacobian conjecture. I ch…
...no need for any lean here
Re: Claude Fable produced a counterexample to the Jacobian Conjecture
#230Earlier quoted context omitted.
The interesting thing about using Claude Fable 5 is it's nearly as irritatingly sycophantic as past Claudes while genuinely being smarter than the previous models. So you get a kind of yo-yoing of it glazing you as a creative genius and disappointedly revealing to you that your ideas are bad and dumb.
That is easily fixable with the "Instructions for Claude" in your settings. Just put: "be critical, but correct. I don't want affirmation, i want to actually accomplish things." I don't get any sycophancy from it. The conversations are actually quite good, and it'll push back if it thinks I'm wrong.
gotta try with "think in russian"