How Claude Fable 5 Disproved a 142‑Year‑Old Jacobian Conjecture with Three Simple Formulas

Yesterday, mathematician Levent Alpöge posted on X that he had used Claude Fable 5 to discover a three‑dimensional counterexample to the Jacobian Conjecture, a problem that had remained open for 142 years. In his post he presented only three lines of formulas and thanked Fable 5 for its continuous work during the World Cup final.

Initial reactions dismissed the claim as a typical amateur‑scientist stunt, especially after a previous incident where an OpenAI executive falsely claimed GPT‑5 solved ten unsolved math problems. Some also suspected Anthropic marketing because Alpöge is currently an Anthropic researcher.

Verification proved the counterexample straightforward: using Mathematica the Jacobian determinant of the proposed map was computed to be a constant –2, and the three points given in the post all mapped to the same image, confirming the violation of the conjecture’s condition. Within an hour, the English Wikipedia entry for the Jacobian Conjecture was updated to include this AI‑assisted counterexample.

Alpöge holds a Ph.D. in mathematics from Princeton under Fields Medalist Manjul Bhargava and was formerly a Harvard scholar, specializing in number theory and arithmetic geometry—precisely the fields dealing with polynomial equations and algebraic curves.

The Jacobian Conjecture asks whether a polynomial map with a non‑zero constant Jacobian determinant is globally invertible with a polynomial inverse. Historically, it ranks alongside the Riemann and P/NP problems, having been listed by Fields Medalist Smale as one of the “21st‑century mathematical problems.” Over more than a century, mathematicians narrowed the search space but never found a full counterexample; the first viable three‑dimensional example appeared only now.

Earlier attempts include Ludwig Kraus’s 1884 two‑dimensional formulation, Keller’s 1939 generalization, and later work by Stuart Wang (1998) ruling out quadratic counterexamples, as well as Bass, Connell, and Wright’s reduction to cubic maps. In 1994, Sergei Pinchuk found a real‑field counterexample that failed to meet the constant‑determinant condition, leaving the complex case unresolved until Alpöge’s discovery.

According to the post, Fable 5 likely assisted in the most difficult phase: searching for a formula that satisfies the algebraic constraints (constant non‑zero Jacobian and coincident images for distinct inputs). The AI presumably transformed the problem into a searchable set of algebraic conditions, iteratively adjusted polynomial structures, and filtered candidates until the successful three‑dimensional map emerged, while Alpöge performed the final mathematical verification.

Although the full dialogue and search process remain undisclosed, the episode illustrates a broader trend: AI is moving beyond data retrieval and computation to generate original, verifiable mathematical insights. Recent examples include GPT‑5.4 Pro’s contribution to Erdős problems, OpenAI’s internal model overturning a core conjecture in the unit‑distance problem, and GPT‑5.6 Sol Ultra’s proof of the “double‑covering conjecture” in graph theory, all accompanied by formal verification via Lean.

These developments suggest that AI can shoulder the labor‑intensive search and construction phases of research, freeing mathematicians to focus on problem formulation, result validation, and value judgment, potentially accelerating breakthroughs in fields that rely on massive trial‑and‑error such as drug discovery, material design, and chip layout.

Jacobian Conjecture

Let F = (f₁, …, fₙ) : ℂⁿ → ℂⁿ

be a polynomial map.

If det(∂fᵢ/∂xⱼ) = c ≠ 0,
then F has a polynomial inverse.