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

This manuscript presents a resolution of Beal's Conjecture—a Clay Mathematics Institute Millennium Prize problem—by applying the Harmonic Coherence (HC) framework and Hanners Theorem. Beal's Conjecture (Andrew Beal, 1993) generalizes Fermat's Last Theorem: the exponential Diophantine equation Aˣ + Bʸ = Cᶻ with positive integers A, B, C, x, y, z and x, y, z > 2 has integer solutions only if A, B, C share a common prime factor. We translate entropy-minimization principles from HC and Hanners Theorem into number theory. An entropy functional H (A, B, C) is defined over the normalized terms (Aˣ, Bʸ, Cᶻ) ; equilibrium conditions (gradient zero) yield pᵢ = 1/e. Because e is irrational and the normalized contributions are rational for integer solutions, nontrivial coprime solutions cannot satisfy equilibrium unless a common prime factor is present. The proof is supported by modular arithmetic and congruence arguments (Stewart–Tijdeman, Darmon–Granville) and by extensive computational validation using Python, NumPy, SciPy, and mpmath over large integer domains. No counterexamples to Beal's Conjecture were found. The resolution bridges theoretical physics (entropy minimization, gauge-theoretic stability) and number theory, with implications for cryptography and computational complexity. The document is a formal preprint intended for peer review and is part of the Harmonic Coherence publication ecosystem.