Claude 'Fable' Disproves the Jacobian Conjecture
Original: Claude Fable produced a counterexample to the Jacobian Conjecture
Why This Matters
Disproving the Jacobian Conjecture would resolve a 70-year-old open problem and reshape algebraic geometry and polynomial automorphism theory.
Mathematician Levent Alpoge announced on X that Claude (referred to as 'Fable') produced a counterexample to the Jacobian Conjecture, a major open problem in mathematics. The polynomial map from C^3 to C^3 has a constant Jacobian determinant of -2 yet is not injective, disproving the conjecture.
On July 20, 2026, mathematician Levent Alpoge (@__alpoge__) posted on X that the Jacobian Conjecture — one of mathematics' most prominent unsolved problems — is false, crediting Anthropic's AI model Claude (nicknamed 'Fable') with finding the counterexample during the World Cup final. The counterexample is the polynomial map ((1+xy)^3 z + y^2(1+xy)(4+3xy), y + 3x(1+xy)^2 z + 3xy^2(4+3xy), 2x - 3x^2 y - x^3 z) from C^3 to C^3. Alpoge states this map has Jacobian determinant -2 (constant and nonzero) yet sends three distinct points — (0,0,-1/4), (1,-3/2,13/2), and (-1,3/2,13/2) — all to the same point (-1/4,0,0), violating injectivity. Wolfram Alpha links were provided to verify both the determinant and the point evaluations. Replies note potential implications for related conjectures including the Dixmier Conjecture and the Poisson Conjecture. As of posting, the result had not yet undergone formal peer review, though community members proposed Lean formalization via Mathlib.