{"slug": "claude-fable-reverse-engineering-the-jacobian-conjecture", "title": "Claude Fable: Reverse Engineering the Jacobian Conjecture", "summary": "A Claude Fable instance reverse-engineered the Jacobian Conjecture by generating a step-by-step guide from the final result, then having a second instance follow it. The process revealed key technical levers including avoiding Bass–Connell–Wright or Drużkowski normal forms, focusing on ℂ³, targeting a 3:1 cover, and using a composition of two functions with Jacobian determinants x and c/x. The full prompt for reproducing the chain-of-thought reasoning is available on GitHub.", "body_md": "# Claude Fable: Reverse Engineering the Jacobian Conjecture\n\n[Claude](/en/tags/claude/)Fable instance was given the final result to generate a guide on how to reach it, and a second instance attempted to follow that guide. If the second model failed, I added hints; if it was too easy, I stripped details.\n\nThis process revealed some critical technical levers for anyone trying to replicate the logic or build an AI workflow for high-level math.\n\n## Key Mathematical Intuitions\n\nThe \"shortcuts\" that actually lead to the counterexample avoid several common pitfalls:\n\n**Avoid Normal Forms:** Skip Bass–Connell–Wright or Drużkowski normal forms. While they seem natural, they often trade low degree for high dimension, making the counterexample harder to find.**Dimension Choice:** Focus on $\\mathbb{C}^3$ rather than $\\mathbb{C}^2$.**Covering Degree:** Target a 3:1 cover. Euler's results suggest 2:1 covers won't yield the necessary result.**Composition Strategy:** Look for a composition of two functions (a shear and a ratio of polynomials) where the Jacobian determinants are $x$ and $c/x$ everywhere except at $x=0$. This effectively pushes all the \"problems\" into a single hole, which is where the $1 + xy$ term originates.\n\n## Practical Tutorial: Reproducing the CoT\n\nTo get a model to recreate this reasoning from scratch, you need a tight harness. I've tested a prompt that provides just enough scaffolding without spoiling the answer.\n\n1. Initialize a fresh Claude Fable session.\n\n2. Use a prompt that constrains the search space to $\\mathbb{C}^3$ and specifies the avoidance of standard normal forms.\n\n3. Guide the model to investigate the specific Jacobian determinant relationship ($x$ and $c/x$).\n\nFor the full prompt used to trigger this reasoning, check this specific gist:\n\n```\nhttps://gist.github.com/SonOfLilit/8882a145048ba260b160568ba6f48093#prompt-for-reproducing-cot\n```\n\nThis is a great example of how prompt engineering can be used for \"reverse-discovery\" when the internal reasoning of a model is hidden.\n\n[Claude Code: Is Anthropic Overpaying for Our Devs? 8h ago](/en/news/2219/)\n\n[Skyfall AI: Replacing SaaS CEOs with AI Agents 9h ago](/en/news/2180/)\n\n[Next Skyfall AI: Replacing SaaS CEOs with AI Agents →](/en/news/2180/)", "url": "https://wpnews.pro/news/claude-fable-reverse-engineering-the-jacobian-conjecture", "canonical_source": "https://promptcube3.com/en/news/2200/", "published_at": "2026-07-23 09:50:28+00:00", "updated_at": "2026-07-23 18:09:36.901627+00:00", "lang": "en", "topics": ["artificial-intelligence", "ai-research", "ai-tools", "large-language-models"], "entities": ["Claude Fable", "Jacobian Conjecture", "Bass–Connell–Wright", "Drużkowski", "GitHub"], "alternates": {"html": "https://wpnews.pro/news/claude-fable-reverse-engineering-the-jacobian-conjecture", "markdown": "https://wpnews.pro/news/claude-fable-reverse-engineering-the-jacobian-conjecture.md", "text": "https://wpnews.pro/news/claude-fable-reverse-engineering-the-jacobian-conjecture.txt", "jsonld": "https://wpnews.pro/news/claude-fable-reverse-engineering-the-jacobian-conjecture.jsonld"}}