Claude Fable Disproves the Jacobian Conjecture

The Jacobian Conjecture, an 85-year-old mathematical problem, has been falsified by a counterexample produced by the AI model Claude Fable. The discovery demonstrates that a polynomial map with a constant non-zero Jacobian determinant is not necessarily globally invertible, resolving a problem that had resisted human proof and counterexample for decades.

The Counterexample

Claude Fable identified a specific polynomial map from $\mathbb{C}^3$ to $\mathbb{C}^3$ that serves as a counterexample to the conjecture. The map is defined as:

$$( (1+xy)^3 z + y^2 (1+xy) (4+3xy), y + 3x (1+xy)^2 z + 3xy^2 (4+3xy), 2x - 3x^2y - x^3z )$$

This map possesses a constant Jacobian determinant of $-2$. Despite this constant determinant, the map is not injective (and therefore not invertible). It sends three distinct points to the same output:

• $(0, 0, -1/4)$
• $(1, -3/2, 13/2)$
• $(-1, 3/2, 13/2)$

All three of these points are mapped to $(-1/4, 0, 0)$.

Verification and Formalization

Following the announcement on social media by levent (@ extunderscore extunderscore alpoge extunderscore extunderscore), the result was verified through multiple computational and formal methods:

• Computational Verification: The result was verified using Wolfram|Alpha, SymPy, Singular, and PARI/GP. These tools confirmed that the Jacobian determinant is identically $-2$ and that the three points collide at the same output.
• Formal Verification: A Lean 4 / mathlib certificate was created to formally prove the counterexample's validity within a computer-verified proof assistant.
• Open Source Implementation: A verification suite was published to GitHub (by DrAlexHarrison) containing a verify.sh script that reproduces the exact claims regarding the Jacobian determinant and the map's geometry.

Mathematical Implications

Because the Jacobian Conjecture is logically equivalent to several other major conjectures in algebraic geometry, its falsification has a ripple effect across the field:

• Dixmier Conjecture: The Dixmier conjecture for the third Weyl algebra is now considered disproven.
• Poisson Conjecture: The Poisson conjecture is also falsified as a result of this equivalence.

Community Discussion and AI's Role in Mathematics

The discovery has sparked significant debate among mathematicians and AI researchers regarding the nature of mathematical discovery:

The "Crank Graveyard"

Levent noted that the Jacobian Conjecture is a "canonical crank graveyard," meaning it is a problem that attracts many amateur mathematicians who claim to have solved it. The fact that a relatively simple cubic counterexample was found by an AI—something that a computer search in the late 1990s might have been capable of—suggests that the problem was not inherently "difficult" in terms of complexity, but rather lacked the necessary search-based exploration.

Human vs. AI Discovery

Some community members expressed concern that frontier models are now capable of "mopping up" existing open problems that are accessible to current mathematical infrastructure. One mathematician noted:

"Anything that is at all accessible to currently existing methods and mathematical infrastructure is probably going to fall to the frontier models of today... we'll see a world in which a few designated interpreters of the oracle get to 'do' mathematics."

The Process of Discovery

While the specific prompt and reasoning trace used by Claude Fable are not yet public (likely due to NDAs), the result highlights the shift toward "math-vibe coding," where AI generates candidates and humans focus on the rigorous verification and validation of those results. In this case, the discovery was a collaboration between humans (Alex Harrison and others) and the AI model.