Claude Fable: Reverse Engineering the Jacobian Conjecture
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-cotThis is a great example of how prompt engineering can be used for "reverse-discovery" when the internal reasoning of a model is hidden.