Beal's Conjecture: Toward an Entropy-Minimization Resolution via Harmonic Coherence and Hanners Theorem

A conditional structural analysis of Beal's Conjecture within the Harmonic Coherence (HC) framework. We define an entropy functional over the normalized terms of the exponential Diophantine equation Ax + By = Cz and show that every solution is permanently displaced from the entropy equilibrium, with a quantifiable lower bound for coprime bases. Under Assumption A3 (spectral gap), this displacement implies non-existence of coprime solutions. The result is supported by 34 computational tests (all PASS) over large integer domains. Companion documents: • Contextual Entropy Reduction Theorem • Canonical Reconciliation • HC Bridge Note • Fixed-Point Convergence Theorem • Paper A: Transformer Distillation • Paper B: GW Kerr Ringdown • Paper C: HC Bridge Synthesis