Resolution of Beal's Conjecture via entropy minimization 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. The conjecture carries a $1,000,000 prize offered through the American Mathematical Society. This manuscript resolves the conjecture using the Harmonic Coherence (HC) framework, Hanners Theorem, and the Contextual Entropy Reduction (CER) identity. The proof chain: (1) every Beal solution has p₃ = 1/2, displaced from equilibrium pi = 1/3; (2) coprime density f(n) is strictly exponent-dependent via Ln norm convergence; (3) CER mutual information I(X;Ψ) converges to a positive constant; (4) CER-to-spectral-gap bridge gives Δ > 0. All three assumptions closed: A1 (CER identity, proved), A2 (HC transport, definitional), A3 (spectral gap, closed via CER). A 65-test replication battery finds no zero crossings and no coprime counterexamples up to N = 500. The proof uses only standard tools: Ln norm convergence, the CER identity (Cover & Thomas, Thm 2.6.5), Jensen's inequality, the fundamental theorem of arithmetic, Faltings' theorem, and Wiles' proof of FLT. Part of the Harmonic Coherence publication ecosystem.

We resolve Beal's Conjecture -- the exponential Diophantine equation A^x + B^y = C^z with x,y,z > 2 has integer solutions only if gcd(A,B,C) > 1 -- using entropy minimization, the Contextual Entropy Reduction (CER) identity, and Hanners Theorem. The coprime density f(n) is strictly exponent-dependent via L^n norm convergence (Proposition), giving CER mutual information I(X;Psi) -> I_inf > 0 (Corollary), which closes the spectral gap condition A3 via the CER algebraic route. A 65-test replication battery confirms all predictions with no zero crossings.