{"slug": "show-hn-how-to-get-a-fable-cot-for-the-jacobian-conjecture-refutation", "title": "Show HN: How to Get a Fable CoT for the Jacobian Conjecture Refutation", "summary": "A Hacker News user posted a method to generate a Chain of Thought for the Jacobian Conjecture refutation using two Claude Fable instances, one to create a writeup and another to follow it, with the goal of providing mathematical intuition for the counterexample. The approach involves looking in C^3 for a 3:1 cover and using a composition of two functions with Jacobian determinants x and c/x, avoiding Bass–Connell–Wright or Drużkowski normal forms.", "body_md": "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.\n\nSince 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.\n\nThis is what a successful run looks like (sadly Anthropic hide Chains of Thought):\n\n[https://claude.ai/share/80526d56-1c23-407d-8f5c-59a704221454](https://claude.ai/share/80526d56-1c23-407d-8f5c-59a704221454)\n\n## Intuition\n\nThis 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):\n\n* 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)\n\n# Prompt for reproducing CoT\n\nHere: [https://gist.github.com/SonOfLilit/8882a145048ba260b160568ba...](https://gist.github.com/SonOfLilit/8882a145048ba260b160568ba6f48093#prompt-for-reproducing-cot)\n\nComments URL: [https://news.ycombinator.com/item?id=48986943](https://news.ycombinator.com/item?id=48986943)\n\nPoints: 1\n\n# Comments: 0", "url": "https://wpnews.pro/news/show-hn-how-to-get-a-fable-cot-for-the-jacobian-conjecture-refutation", "canonical_source": "https://news.ycombinator.com/item?id=48986943", "published_at": "2026-07-21 01:10:03+00:00", "updated_at": "2026-07-21 01:22:59.811218+00:00", "lang": "en", "topics": ["artificial-intelligence", "large-language-models", "ai-research"], "entities": ["Claude Fable", "Anthropic", "Hacker News", "Jacobian Conjecture"], "alternates": {"html": "https://wpnews.pro/news/show-hn-how-to-get-a-fable-cot-for-the-jacobian-conjecture-refutation", "markdown": "https://wpnews.pro/news/show-hn-how-to-get-a-fable-cot-for-the-jacobian-conjecture-refutation.md", "text": "https://wpnews.pro/news/show-hn-how-to-get-a-fable-cot-for-the-jacobian-conjecture-refutation.txt", "jsonld": "https://wpnews.pro/news/show-hn-how-to-get-a-fable-cot-for-the-jacobian-conjecture-refutation.jsonld"}}