On 20 July 2026 Levent Alpöge announced that Anthropic’s AI model Claude Fable 5 produced a counterexample that disproves the Jacobian Conjecture.
Jacobian Conjecture — precise statement
- Conjecture (Keller, 1939): For a polynomial map F: C^n → C^n, if det(Jacobian(F)) is a non‑zero constant then F has a polynomial inverse.
- Local vs global: Non‑zero constant Jacobian gives a local analytic inverse by the Inverse Function Theorem but does not a priori guarantee a global polynomial inverse.
AI-assisted counterexample
- Construction: Claude Fable 5 produced a 216‑character polynomial formula satisfying the constant non‑zero Jacobian determinant condition yet lacking a polynomial inverse.
- Injectivity failure: The map has distinct inputs mapping to the same output, demonstrating non‑invertibility despite the Jacobian condition.
- Agents involved: Result announced by Levent Alpöge (Harvard/Anthropic) and attributed to the AI model Claude Fable 5.
Verification and mathematical context
- Formal checks: Symbolic verification used SymPy; formal proof‑assistant checks used Lean.
- Peer scrutiny: Mathematicians examined the construction; the finding engages long‑standing research on polynomial automorphisms.
- Relation to Smale: The conjecture appears among problems noted in Stephen Smale’s list of 21st‑century mathematical challenges.
IASPOINT Booster Facts
- Known reductions: Bass–Connell–Wright reduction shows it suffices to prove the conjecture for cubic homogeneous maps.
- Trivial case: The conjecture is true for n = 1 (one variable).
- Tools: SymPy is a computer algebra system; Lean is an interactive theorem prover.