CounterexampleAlgebraic geometry19 Jul 2026

The Jacobian conjecture meets a counterexample

A local-to-global intuition overturned by three polynomials.

Levent Alpöge · Claude Fable · Terence Tao · community formalizers

Start with the question

The problem

Suppose a polynomial map over the complex numbers has a Jacobian determinant that is a nonzero constant. Must it have a polynomial inverse? A local invertibility test seems to promise a global conclusion. A counterexample only needs one map that passes the test but sends different inputs to the same output.

det DF = −2, but F(a) = F(b) with a ≠ b

A short announcement with a large consequence

Alpöge’s July announcement credited Fable with finding the map. The example is in three variables. Tao’s subsequent exposition explains that it also gives counterexamples in higher dimensions, while the two-dimensional case remains a separate open question in that account. [1][2]

Two calculations expose the failure

Tao displays the polynomial map, computes its constant Jacobian determinant, and exhibits three distinct rational points with the same image. His article then works backwards to make the geometry understandable: why an object that looks locally invertible can fail to be globally one-to-one. [2]

From a formula to a formal refutation

The independent alpoge-lean project states that it checks the determinant, the collision, and a refutation of a mirrored formal-conjectures statement. This is community formalization of the announced construction, not an official Anthropic proof repository. Its extended structural results should be assessed separately from the central counterexample. [3]

Inspect the evidence

The Lean formalization

Independent community Lean formalization

Alpoge/FormalConjectures.lean is documented as containing isUnit_jacobian_det, no_polynomial_inverse, and jacobian_conjecture_false. Inspect the definitions and pinned dependencies in the repository. We have not rebuilt its proof or certified the repository’s broader claims.

Open the Lean project ↗

Sources & further reading

Reviewed 9 Sept 2026. X permalinks were recovered from indexed posts and linked discussions; direct X pages were not consistently accessible. They document the conversation, not proof correctness.