2607.02851 Resolution of the Swallowtail Branch Locus of the Alpöge–Mathew–Fable Jacobian Counterexample: A Proof via Hironaka's Theorem in All Dimensions
On July 19–20, 2026, Levent Alpöge announced that Fable (Anthropic) had found a polynomial map F : C^3 → C^3 with det J_F ≡ −2 that is not injective, refuting the Jacobian conjecture. The map was found by sweeping a C*-equivariant ansatz F = (h_1(u,v)/x^2, h_2(u,v)/x, x h_3(u,v)) with u = xy, v = x^2 z, which forces det J_F to depend only on (u,v) and reduces the search to a linear equation in h_1.