Earlier quoted context omitted.
Who knew DOES NOT COMPUTE would be an actual thing?
The unexpected part is it does compute!
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Claude Fable produced a counterexample to the Jacobian Conjecture
241–250 of 562 posts
Re: Claude Fable produced a counterexample to the Jacobian Conjecture
#242Re: Claude Fable produced a counterexample to the Jacobian Conjecture
#243This 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 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.
Re: Claude Fable produced a counterexample to the Jacobian Conjecture
#244Earlier quoted context omitted.
This is nonsensical: Properness of the map is equivalent to its being an isomorphism (quick proof: Jacobian invertible implies that the map is etale, and properness would imply that it is finite etale, but affine space doesn't admit non-trivial finite etale covers), so the lack of properness is just another way of verifying that this is indeed a counterexample.
sure it does? two copies of the affine line? (I guess there's no galois group & no connected finite etale things tho)
I would call that a trivial finite etale cover :)
Re: Claude Fable produced a counterexample to the Jacobian Conjecture
#245A 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.…
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Re: Claude Fable produced a counterexample to the Jacobian Conjecture
#246This 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!
> and yet the counter example is something a grad student in 1997 could have found w a ~3 day computer search
Is that true? Even restricting this to f(x,y,z) and coefficients and powers to 1 ≤ x ≤ 10, there are a lot of polynomials to check, and checking requires checking the Jacobian determinant and, if it’s a non zero constant, finding two points for which the polynomial produces the same value.
Or is there a way to generate all polynomials with a non-zero Jacobian determinant, and does that speed up things? (My intuition say it wouldn’t, because I guess those with zero determinants are rare)
Re: Claude Fable produced a counterexample to the Jacobian Conjecture
#247Earlier quoted context omitted.
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.
Or we just don't use LEAN but something better.
Re: Claude Fable produced a counterexample to the Jacobian Conjecture
#248This 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.
>flabbergasted
phatic mimicry.
Re: Claude Fable produced a counterexample to the Jacobian Conjecture
#249Since 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.
Exactly, yes!
And the conjecture was for a specific class of processes.
And Fable found an example of one concrete* process in that class and three concrete inputs (two were enough of course) giving the same output.
*Concrete here means given by a finite string of characters
Re: Claude Fable produced a counterexample to the Jacobian Conjecture
#250Earlier quoted context omitted.
Same awnser as much of the LLM Proofs - people cared about other things. There isn't a lot of money in academic math, and the ones that love it don't look for low value findings. Proofs like these are, funnily enough, usually the domain of hobbyists - but over the last few years, the "Monetize everything" mentality and struggling first world economy has pushed people away from interesting academic pursuits on their f…
My understanding is that this is in a different league (Smale problem) than a lot of the other results that have been coming out (Erdos), though I could be wrong.
The Erdos problem solutions have been accelerating, including a $250 problem and a $100 problem.
It's been fun watching the cope collapse from day to day. No one told me a slow takeoff Singularity would have so much schadenfreude.