The Jacobian counterexample, validated and extended: structure, an infinite family, exact escape geometry, and characteristic-p certificates

Paper A of a three-paper research record on the Jacobian conjecture after the Alpöge 2026 counterexample in dimension 3. This foundational paper validates the counterexample independently in exact rational arithmetic and builds on it: the structural analysis of the mechanism, an infinite family of new counterexamples, the exact escape geometry and real census, characteristic-p certificates, a rigidity theorem in dimension 2, and a uniqueness result for the mechanism. Every claim is backed by a persisted, hash-manifested machine experiment; refuted predictions remain in the record. The two-variable case remains open. All scripts, artifacts and verdicts: https://github.com/fsantibanezleal/CAOS_RESEARCH (problems/algebraic-geometry/jacobian-conjecture).