Hanners Theorem: Formalization and an Analytical Framework via Harmonic Coherence

Preprint submitted to *Communications in Mathematical Physics*. This work presents the formalization and an analytical framework for **Hanners Theorem** in non-Abelian gauge theory, with application to the mass-gap problem (Clay Mathematics Institute Millennium Prize). It builds on the prior Zenodo preprint **Harmonic Coherence** (https://zenodo.org/records/15337795), which reframes gauge field dynamics via entropy minimization. Here, Hanners Theorem is formalized and conditions are established under which discrete eigenstates and finite nonzero mass gaps arise in non-Abelian gauge theories; the main results (Hanners Theorem and the Harmonic Coherence Theorem, HCT) are developed with detailed derivations in the main text and appendices. The framework yields predictions for a mass gap, confinement, and gauge field regularity under entropy-minimization and harmonic coherence conditions. An illustrative numerical comparison (Appendix C) and replication package are documented in the manuscript and supplementary material; numerical outcomes may be implementation-dependent and do not constitute proof of the theorem. This record provides the manuscript and supplementary materials as submitted to CMP.