The Mathematics of the AMEM Model: A Self-Contained Derivation Guide with Exercises

[CRITICAL REASSESSMENT — 2026-06-02] On adversarial consistency review (gauge invariance, CPT, and anomalies), the central mechanism of this model appears to be forbidden: the charge-sign-dependent coupling requires a non-conserved current (b†b + d†d is not conserved in QED), and CPT pins the matter/antimatter coupling magnitudes equal. The claims of renormalizability and anomaly-freedom are withdrawn; the negative-frequency framing is retired; the model is flagged as an open consistency problem, not an established result. See the attached file AMEM-CRITICAL-REASSESSMENT.pdf for the full statement.

A self-contained mathematics companion to 'Charge-Sign-Dependent Dark Gauge Coupling' (Kilgore 2026). Nine chapters covering every derivation in the primary preprint, with exercises and worked examples.

Contents: (0) Notation and conventions; (1) Standard dark photon and kinetic mixing; (2) Construction of the C-odd current J^μ_C with proofs of C-oddness, CPT-evenness, and conservation; (3) AMEM Lagrangian — renormalizability and anomaly cancellation; (4) System-dependent coupling table; (5) Hydrogen vs. antihydrogen frequency shift derivation (factor-of-8 traced); (6) Both experimental constraint derivations (Baker 2025 and Van Dyck 1987) with full arithmetic; (7) UV completion mechanics; (8) N_eff cosmological bound; (9) Parameter space summary. Appendices: two worked numerical cross-checks, further reading, and complete symbol reference table.