Claude Fable: Reverse Engineering the Jacobian Conjecture

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.

Claude Fable: Reverse Engineering the Jacobian Conjecture 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. This process revealed some critical technical levers for anyone trying to replicate the logic or build an AI workflow for high-level math. Key Mathematical Intuitions The "shortcuts" that actually lead to the counterexample avoid several common pitfalls: 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. Practical Tutorial: Reproducing the CoT To 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. 1. Initialize a fresh Claude Fable session. 2. Use a prompt that constrains the search space to $\mathbb{C}^3$ and specifies the avoidance of standard normal forms. 3. Guide the model to investigate the specific Jacobian determinant relationship $x$ and $c/x$ . For the full prompt used to trigger this reasoning, check this specific gist: https://gist.github.com/SonOfLilit/8882a145048ba260b160568ba6f48093 prompt-for-reproducing-cot This is a great example of how prompt engineering can be used for "reverse-discovery" when the internal reasoning of a model is hidden. Claude Code: Is Anthropic Overpaying for Our Devs? 8h ago /en/news/2219/ Skyfall AI: Replacing SaaS CEOs with AI Agents 9h ago /en/news/2180/ Next Skyfall AI: Replacing SaaS CEOs with AI Agents → /en/news/2180/