The Jacobian Conjecture after the Three-Dimensional Counterexample: Exact Validation, a General Seed Family, the Geometry of Non-Properness, and Two-Dimensional Equivariant Rigidity

On 2026-07-20, Levent Alpöge announced (crediting the language model Claude Fable 5 and a question posed by Akhil) an explicit polynomial map F from C^3 to C^3 with constant Jacobian determinant -2 and a repeated value, refuting the Jacobian conjecture for every N ≥ 3. We report an exact-arithmetic study of this map. First, we validate the counterexample independently and compute the full fiber over the announced target: it consists of exactly the three announced points. Second, we reverse-engineer the structure of F: it is a weighted skew-product over the invariants v = xy, t = x^2 z, its Keller property reduces to a two-variable identity, and its fibers are governed by a cubic in a single composite variable. Third, we derive and verify a general constructor: every admissible triple (seed polynomial p, scale k, section q) yields a Keller map with determinant -k p(1)^2 and generic fiber degree deg p + 1; we exhibit new explicit counterexamples with rational collision certificates in fiber degrees 4, 5 and 6, plus a section-tail instance beyond the announced family shape, prove the degree law (5d - 3, 5d - 4, 4) within the family, and show that the announced map is its smallest member. Fourth, we make the failure geometry exact: a preimage escapes to infinity precisely when its fiber root is a multiple root, so the asymptotic variety of F is the plane C = 0 together with the explicit discriminant surface 27A^2C^2 - 18ABC + 16A + B^3C - B^2 = 0; over the reals this wall separates a three-sheet from a one-sheet region, and F restricts to a surjective, non-injective real Keller map. The counterexamples reduce at suitable primes to non-injective Keller maps over finite fields F_l of degree below the characteristic. Fifth, we prove that the mechanism cannot reach the still-open two-variable case: for opposite-sign or one-zero weights, every G_m-equivariant Keller map of C^2 is linear (a classification with a positive-factor kill at the top degree), so a planar counterexample, if any, must be non-equivariant. Shaska's contemporaneous independent all-signature classification is reconciled explicitly: same-sign actions may give nonlinear triangular automorphisms, while equivariant planar Keller maps remain automorphisms in every signature. All computations are exact (over Q, over finite fields, or symbolic) and reproducible from the repository; every claim is labeled as machine-verified, derived, or conjectural. Source code, data and computational records: https://github.com/fsantibanezleal/CAOS_RESEARCH (problems/algebraic-geometry/jacobian-conjecture).