Claude Fable Disproves the Jacobian Conjecture

The Jacobian Conjecture, a problem that remained unsolved for 85 years, has been falsified by a counterexample produced by an AI model known as Claude Fable. The result demonstrates that a polynomial map from $\mathbb{C}^3$ to $\mathbb{C}^3$ can have a constant non-zero Jacobian determinant while failing to be injective (invertible), thereby disproving the conjecture.

The Counterexample

Claude Fable identified a specific polynomial map $F: \mathbb{C}^3 \to \mathbb{C}^3$ that serves as a counterexample. The map is defined by the following three polynomials:

$f_1 = (1+xy)^3 z + y^2 (1+xy) (4+3xy)$
$f_2 = y + 3x (1+xy)^2 z + 3xy^2 (4+3xy)$
$f_3 = 2x - 3x^2 y - x^3 z$

Verification of the Result

To disprove the Jacobian Conjecture, a map must satisfy two conditions: it must have a constant non-zero Jacobian determinant and it must not be globally invertible. The provided map satisfies both:

Constant Jacobian Determinant: The Jacobian determinant of this map is $-2$. This has been verified via Wolfram|Alpha, SymPy, Singular, and PARI/GP.
Non-Invertibility: The map is not injective because it sends three distinct points to the same output. Specifically, the points $(0, 0, -1/4)$, $(1, -3/2, 13/2)$, and $(-1, 3/2, 13/2)$ all map to $(-1/4, 0, 0)$.

Further verification has been provided through a Lean 4 / mathlib certificate and a GitHub repository containing a runnable verification suite (verify.sh).

Implications for Mathematics

Because the Jacobian Conjecture is linked to other major problems in algebraic geometry, its falsification has immediate theoretical consequences.

Related Conjectures

Community discussion highlights that the falsification of the Jacobian Conjecture implies the falsification of several other equivalent conjectures:

The Dixmier Conjecture: Specifically for the third Weyl algebra.
The Poisson Conjecture.

The Role of AI in Mathematical Discovery

This discovery has sparked a debate on the role of Large Language Models (LLMs) in pure mathematics. Some observers note that the counterexample is relatively small (a cubic), suggesting that the result may have been accessible to human mathematicians or computer searches for decades, but was overlooked due to a lack of scaled intelligence applied to the specific search space.

"Given how small the counterexample is... this feels like a a great example of where a lot of interesting results are going to be found: not because they were super difficult, but because intelligence didn't scale, and until computers could do this for us, the number of people who seriously poked at many such things was low."

Community Reception and Methodology

The result was announced informally via X (formerly Twitter) by levent (@alpoge), rather than through traditional academic channels like arXiv. This has led to discussions regarding the outdated nature of the academic journal system and the emergence of "math-vibe coding," where AI is used to generate hypotheses that are then manually verified by humans.

Methodology Questions

While the specific prompt and reasoning trace used by Claude Fable remain undisclosed, there is speculation that the model may have synthesized the counterexample by leveraging vast amounts of prior work and failed proofs published over the last 85 years. Some users suggest the model may have used a sophisticated search technique or constructed the polynomials by guessing families of functions and solving for coefficients.