Hanners Theorem: A Formalization and Analytical Proof via Harmonic Coherence

Hanners Theorem serves as the mathematical backbone of the formalized **Harmonic Coherence** framework (**Zenodo Record 15337795**). This manuscript presents the formalization and analytical proof of Hanners Theorem, establishing the existence of discrete eigenstates and finite nonzero mass gaps in non-Abelian gauge theories. Using an entropy-minimization principle grounded in statistical mechanics and information theory, the framework derives conditions under which gauge configurations stabilize into discrete equilibrium eigenstates. The companion **Harmonic Coherence Theorem (HCT)** extends these results with topological quantization criteria. Together, Hanners Theorem and the HCT provide an analytical resolution to longstanding open problems in theoretical physics, notably confinement and the mass-gap problem in quantum Yang–Mills theory, with implications for quantum gravity and unified field theories. This record is the formal mathematical treatment of Hanners Theorem within the broader Harmonic Coherence publication ecosystem.