Resolution of Beal's Conjecture

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 A^x + B^y = C^z 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 p_i = 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.

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