Hanners Theorem Formalization: Information-Theoretic Foundations of Harmonic Coherence

This manuscript formalizes Hanners Theorem, establishing the information-theoretic foundations of Harmonic Coherence (HC). The argument proceeds in two tracks. The gauge-theoretic track derives discrete spectra and eigenstate stability through variational calculus and elliptic gauge-covariant operators. The information-theoretic track begins with the Contextual Entropy Reduction (CER) identity—Cover & Thomas Theorem 2.6.5 in mixture-model framing—and iterates it under HC regularity conditions R1–R4 to prove fixed-point existence and convergence. Key exports to application papers: R1–R4 regularity conditions (Def. A.7), CER identity (Thm. 2.3), fixed-point existence (Thm. A.9), Condition C (Def. 5.16), phase-locked resonance (Def. 5.2), configuration space and evolution map (Def. A.4, A.6), and Appendix F axioms (Axioms 1–7). Companion documents:• CER Identity (original derivation)• Riemann Hypothesis Resolution (application paper)• Bridge Synthesis (B1–B3)