Xavier's dichotomy for the Keller locus

Xavier's paper appeared a month before the counterexample, and it says nothing about which alternative occurs. The proof uses four facts.^[1]

• The non-injective maps in the parameter space form a constructible set (Chevalley).
• The injective members of $\Omega$ are closed in the Euclidean topology, because a limit of injective local homeomorphisms is injective.
• A constructible set that is Euclidean-closed is Zariski-closed.
• Injective polynomial self-maps of ℂⁿ are automorphisms.

The theorem recasts the Jacobian conjecture as the claim that every component of every bounded-degree Keller locus lies in the automorphism group. After July 2026 the second alternative occurs. Alpöge's map, rescaled to determinant 1, lies in $V_3^7$. So the component through Alpöge's map is one whose general member is a counterexample. The Fong–Fable ledger cited the paper on 20 July as prior art for its family and moduli framing.

1. ↑ F. Xavier, An organizing principle in the study of the Jacobian Conjecture, arXiv:2606.22041 (20 June 2026). Unrefereed.