AI-Assisted Counterexample Refutes Jacobian Conjecture

The Jacobian conjecture asks whether a polynomial map over the complex numbers must have a polynomial inverse whenever its Jacobian determinant is a non-zero constant. Such a determinant guarantees local invertibility; the unresolved question was whether that local property forces global polynomial invertibility.

The announced counterexample is an explicit degree-seven polynomial map from three-dimensional complex space to itself. Tao verified that its Jacobian determinant is the constant -2 and that three distinct rational input points produce the same output. Because the map is not injective, it cannot have a polynomial inverse.

The Archive of Formal Proofs independently encoded the construction in Isabelle/HOL. Its entry proves that the formal derivatives match the complex analytic partial derivatives, verifies the constant determinant and shared output, and then pads the map with identity coordinates to obtain counterexamples in every dimension of at least three.

The object is compact enough that its defining identities can be checked directly. Discovering it was harder: Tao noted that a generic degree-seven map in three variables has hundreds of coefficients, while forcing its Jacobian to be constant creates more than a thousand cancellation conditions. The concise final expression does not imply that a naive brute-force search would find it easily.

Public accounts attribute the discovery to Alpöge working with Claude Fable 5 after Akhil Mathew suggested the problem. However, the retrieved sources do not include a prompt transcript, search log or detailed division of labor between the mathematician and the model. The strongest supported wording is therefore that the result was AI-assisted, not that a model independently solved the problem without human direction.

For mathematical AI, the important feature is the form of the output: a concrete object whose defining properties can be checked independently. Tao's hand exposition and the Isabelle formalization do not depend on trusting the model's natural-language reasoning.

That makes the result more auditable than a long AI-generated proof with hidden gaps. It also leaves a reproducibility question. Without the model interaction or search procedure, researchers can verify the counterexample but cannot yet reproduce the path that generated it. Future AI-assisted mathematics will benefit from publishing both the compact result and a durable account of the human-model workflow.