Abelian Measure–Defect Equivalence in Divergent Weighted Index Systems

This paper establishes a complete abelian equivalence between additive defect accumulation and multiplicative density decay in divergent weighted index systems, formulated without analytic continuation, Dirichlet series, or arithmetic structure. A minimal framework consisting of a countable index set, divergent cumulative weights, and local admissible densities is introduced to capture global asymptotic behavior. It is shown that the additive defect sum, the logarithmic length of the multiplicative aggregate, and the normalized product profile are asymptotically identical at the level of first–order divergence. All three representations are governed by a single scalar invariant, the defect exponent κ, which admits equivalent formulations as an additive slope, a logarithmic decay rate, and an exponential scaling parameter. The divergence profile of the cumulative weight fully determines the functional form of global scaling laws, with harmonic divergence yielding logarithmic corrections and stronger divergence producing stretched exponential behavior. A cutoff–scale transfer mechanism further connects index-based asymptotics to measure-theoretic scaling laws. The framework demonstrates that additive, multiplicative, and measure formulations are unified as representations of a single structural divergence mechanism.