Multi-Variable Shannon-Rényi Control Laws in Arithmetic Power Geometry: Theory, Empirical Validation, and Comparative Analysis on Cache-Like Information Networks

Arithmetic Power Geometry (APG) studies algebraic closure relations under exponent deformation by fixing positive base coordinates and treating the exponent as a continuous deformation parameter. Earlier APG work established that the two-coordinate local closure defect at the Euclidean target is controlled by Shannon and Rényi information structure. This paper develops the finite-dimensional generalization of that result. For a positive vector X = (x₁, ..., xₘ), we define squared Euclidean weights uᵢ = xᵢ² / Σⱼ₌₁ᵐ xⱼ² and the normalized multi-variable APG defect Dₘ(t) = 1 − Σᵢ₌₁ᵐ uᵢ^(t/2) for t ≥ 2. We prove a bridge identity between the APG closure defect and the power-sum defect, establish a Shannon–Rényi upper bound Dₘ(t) ≤ 1 − exp(-(t − 2)/2 H(W)) ≤ (t − 2)/2 H(W), and prove a dimension-stable lower bound Dₘ(t) ≥ 1 − M^(t−2)/2, where M = maxᵢ uᵢ. The theoretical results are evaluated on MovieLens 100K treated as a cache-like request distribution with movies as objects and rating events as requests. The experiment confirms the APG bounds for t ∈ {2.5, 3, 3.5, 4, 5, 6} and compares APG descriptors with Shannon entropy, Rényi entropy, Herfindahl–Hirschman concentration, Gini inequality, and Zipf popularity scaling. The results show that APG descriptors are mathematically valid and most informative at low deformation levels, especially t = 2.5 and t = 3.0, while larger exponents saturate rapidly. The paper positions APG not as a replacement for classical entropy or cache metrics, but as a bounded deformation descriptor for concentration growth in high-dimensional information systems.