The Collateral Impact of the Three-Dimensional Jacobian Counterexample: a Source-Verified Cascade

On 2026-07-20, Levent Alpöge announced an explicit counterexample to the Jacobian conjecture in dimension 3, found working with the language model Claude Fable 5 (Anthropic). The credit for every consequence documented here belongs to that single map. This paper documents what else that map decides: through implication chains published over five decades, each re-verified against its primary source (EXP-001, EXP-016, EXP-018, EXP-136), the counterexample resolves or partially resolves the Mathieu conjecture, the Dixmier conjecture, the Poisson conjecture, the Gaussian moments conjecture, Zhao's vanishing conjecture, the Image conjecture, and the symmetric (gradient) form of the Jacobian conjecture. We state each corollary with its chain and its reference, list precisely what remains open, and describe the verification method, which turns a one-line certificate into a status map of a whole neighborhood of problems. The symmetric/gradient and vanishing corollaries are made explicit: a non-invertible gradient Keller map id + grad f on C^48, with f a homogeneous Hessian-nilpotent quartic (382 monomials over Q(i)) and an exact two-point collision, built from Thompson's dimension-24 cubic-homogeneous form by the de Bondt-van den Essen construction and verified end to end in exact arithmetic. Source code, data and computational records: https://github.com/fsantibanezleal/CAOS_RESEARCH (problems/algebraic-geometry/jacobian-conjecture).