# Claude Fable: Reverse Engineering the Jacobian Conjecture

> Source: <https://promptcube3.com/en/news/2200/>
> Published: 2026-07-23 09:50:28+00:00

# 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.

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