Show HN: How to Get a Fable CoT for the Jacobian Conjecture Refutation

7 pointsby SonOfLilitvor 7 Tagen2 comments

Since the publication of the Jacobian Conjecture refutation, many people have asked for a Chain of Thought to be published so they can better understand the mathematical intuition that leads to the counterexample.

Since none was published, I tried to "clean room reverse engineer" it: give one Claude Fable the result and ask it to generate a writeup of how to arrive at it without any spoilers, then ask a second Fable to follow the write up, if it's too easy remove details, if it's too hard add hints. I could have simplified further, I stopped because it's time to sleep, and I'm sharing because it gave me some intuition for what's going on.

This is what a successful run looks like (sadly Anthropic hide Chains of Thought):

https://claude.ai/share/80526d56-1c23-407d-8f5c-59a704221454

## Intuition

This is my understanding of the interesting ideas (probably too summarized for Fable to consistently get it without a good harness; take with a grain of salt, I'm not a mathematician; later there is a full prompt that was tested and contains everything needed):

* No Bass–Connell–Wright or Drużkowski normal forms (those are reparametrizations that feel natural in this search but they turn low-complexity examples into high-complexity by trading off degree vs dimension, and the counterexample is low complexity)

* Look in C^3, not C^2, C^2 is probably not interesting enough

* Look for a 3:1 cover, not 2:1, there's some result due to Euler (as always) that shows 2:1 will not work

* Look for a composition of two functions (a ratio of polynomials and a shear) that have Jacobian determinants `x` and `c/x` everywhere, except at `x=0` (do the standard trick for not defining at a hole and shoving all the problems into that one hole, that's where the `1 + xy` comes from)

## Prompt for reproducing CoT

Here: https://gist.github.com/SonOfLilit/8882a145048ba260b160568ba...

1 comments

morganriversvor 7 Tagen

cool, i was also hoping for this. However, you are not a mathematician so how did you come up with the prompt? it seems extremely dense

SonOfLilitvor 6 Tagen

I understand basic multivariate calculus (almost every STEM degree teaches it, and I paid attention), which is enough to understand the problem if not every detail about the solution, so I can partially follow what Fable is doing and guess whether it gets too sidetracked.

But the main loop was to show Fable the solution, ask it for a prompt that would get a clean context Fable to find a solution, run it, if it works ask the first Fable to remove details, if it doesn't quote a status report and ask it where the clean Fable went askew and to edit the prompt to prevent that.