Anthropic's Claude Fable AI model has generated a verified counterexample disproving the 87-year-old Jacobian Conjecture.
On July 19, 2026, an Anthropic researcher using the newly released Claude Fable model announced a concrete counterexample to the Jacobian Conjecture, a famous open problem in algebraic geometry since 1939. The model constructed a polynomial map from three-dimensional space to itself that has a constant Jacobian determinant of -2 but is not injective, mapping three distinct points—(0, 0, -1/4), (1, -3/2, 13/2), and (-1, 3/2, 13/2)—to the same output, (-1/4, 0, 0). Because the map fails to be one-to-one, it cannot have a polynomial inverse, disproving the conjecture for all dimensions n >= 3.
While historical claims of solving the Jacobian Conjecture have been notorious for containing subtle flaws, this AI-generated counterexample is exceptionally significant because it is computationally trivial to verify.
* **Instant Verification:** Unlike typical mathematical proofs that require months of peer review, this counterexample is checkable in seconds using basic polynomial calculus and exact rational arithmetic.
* **AI Capability Leap:** This discovery marks a shift from AI assisting in proofs or search spaces to directly generating complex, valid mathematical structures that have eluded human mathematicians for nearly nine decades.
* **Remaining Frontiers:** While the conjecture is now disproven for dimensions n >= 3, the two-dimensional case (n = 2) remains open, though the structure of this counterexample may provide new tools to attack or resolve it.
DISCOVERED
19h ago
2026-07-20
PUBLISHED
21h ago
2026-07-20
RELEVANCE
AUTHOR
loubbrad