Arithmetic Power Geometry: A Structural Statistical Framework for Distribution Analysis and Learning

Classical statistics provides measures of location, dispersion, uncertainty, inequality, diversity, and concentration, but a single statistic does not describe how the internal organization of a nonnegative multivariate observation changes when dominant coordinates are progressively amplified. We formulate Arithmetic Power Geometry (APG) as a structural statistical framework built from Euclidean squared weights. For a nonzero nonnegative vector X, APG defines wi = xi² / Σj xj², Shannon entropy, maximum and quadratic concentration, effective dimension, the exponent-deformation profile DW(p) = 1 − Σi wi^(p/2), integrated deformation, and an APG Structural Signature. Sixteen theorems and one structural-representation corollary establish normalization, non-negativity, scale and permutation invariance, continuity, differentiability, entropy extrema, perturbation stability, finite-dimensional bounds, local Shannon control, the exact APG–Rényi identity, Herfindahl and Simpson identities, participation-ratio structure, and large-exponent asymptotics. The framework was evaluated on 2,172 usable observations from Iris, Wine, Seeds, Breast Cancer Wisconsin Diagnostic, Pima Indians Diabetes, and Cleveland Heart Disease. Numerical verification confirmed all tested identities to floating-point precision. Raw-scale APG showed the most distributed mean structure for Seeds and extreme concentration for Wine, exposing the importance of feature units. In repeated stratified five-fold classification, APG yielded an improvement over the classical compact representation on Seeds, while the remaining datasets showed marginal, neutral, or negative changes; raw features remained superior throughout. APG is therefore presented as a complementary organizing framework for structural statistics, not as a replacement for established entropy theory, preprocessing, or full-feature learning.