Claude Fable produces a checkable counterexample to the Jacobian Conjecture

Mathematician Levent Alpöge announced on X that Anthropic’s Fable model constructed an explicit counterexample to the Jacobian Conjecture, an open problem since 1939 and a fixture on Smale’s list of hard questions. The conjecture claims that any polynomial map from complex n-space to itself whose Jacobian determinant is a nonzero constant must be invertible — and therefore one-to-one. A single well-behaved map that violates injectivity would settle it in the negative.

The proposed counterexample is a polynomial map from ℂ³ to ℂ³ with Jacobian determinant −2, a nonzero constant, that nonetheless sends three distinct points — (0,0,−1/4), (1,−3/2,13/2), and (−1,3/2,13/2) — to the identical image (−1/4, 0, 0). Collapsing three inputs onto one output means the map is not injective, contradicting the conjecture. Unlike a sprawling proof, a claim of this shape is trivially auditable: the whole thing reduces to evaluating a determinant and plugging in three points. Running those computations symbolically confirms every stated value, so the object behaves exactly as advertised.

The significance is twofold. Mathematically, if the community’s scrutiny holds — and the elementary, machine-checkable nature of the artifact makes that likely — an eighty-plus-year problem falls to a concrete example rather than a structural theorem. Culturally, it is a striking data point for AI-assisted mathematics: the result reportedly came together over a weekend, prompted by a colleague’s question, with a language model doing the search for the exact polynomials and points. Standard caveats apply, since this is an informal social-media post rather than a peer-reviewed paper, but the counterexample’s brevity is its own strongest form of verification.