Scaling Laws on Product Measures and Their Arithmetic Realizations

This paper develops a structural theory of scaling laws on product measures and establishes their arithmetic realizations without relying on analytic continuation, Fourier analysis, or probabilistic modeling. The framework introduces admissible configurations governed by defect densities and multiplicative decompositions across prime-indexed components. Scaling behavior is shown to emerge from the asymptotic balance between accumulation and defect suppression, yielding universal logarithmic correction terms. In particular, the appearance of iterated logarithmic factors is derived as a structural consequence of hierarchical truncation and normalization constraints. The theory identifies a class of observables whose growth rates are fully determined by intrinsic combinatorial and multiplicative structure. Arithmetic realizations are constructed by mapping admissible configurations onto prime-indexed systems, recovering classical asymptotic forms within a purely structural setting. The approach provides a unified interpretation of scaling phenomena across abstract measure systems and number-theoretic counting functions. The results demonstrate that logarithmic hierarchies arise from internal consistency conditions rather than external analytic inputs.