Shannon-Rényi Control of the Arithmetic Power Geometry Closure Defect

This paper proves a Shannon–Rényi control theorem for the closure-defect functional of Arithmetic Power Geometry (APG). APG studies algebraic closure relations under exponent deformation by fixing positive base coordinates and allowing the exponent to vary as a deformation parameter. We first establish an exact bridge identity showing that the APG local closure defect at the Euclidean target is identical to a normalized power-sum defect associated with the probability vector determined by the squared base weights. This converts the APG defect into an information-theoretic object. Using Rényi entropy monotonicity, we prove two-sided bounds for the APG closure defect in terms of the maximum coordinate weight and Shannon entropy. We then derive explicit integrated-defect bounds for the APG global functional A₀, including the quadratic estimate A₀(a,b;n) ≤ H(W)/4 · (n − 2)². We also prove a Shannon–variance asymptotic law identifying the second-order coefficient of the local APG defect. The paper does not claim consequences for Fermat’s Last Theorem, the abc conjecture, the Szpiro conjecture, or the Birch and Swinnerton-Dyer conjecture; rather, it establishes foundational inequalities and asymptotics showing that APG deformation is quantitatively controlled by Shannon–Rényi information structure.