ψ-Retentive Tail Law: A Statistical Criterion for Non-Vanishing Late-Time Response Tails

This manuscript introduces a falsifiable statistical decision rule for determining whether late-time response signals fully dissipate or require modeling as non-zero constant offsets. The ψ-Retentive Tail Law formalizes three measurable acceptance conditions—late-time decomposition, plateau-variance stability relative to a quantified noise floor, and model-selection preference using AIC/BIC with a Gaussian threshold. The framework is compatible with standard laboratory protocols for acoustic, optical, mechanical, and electrical ring-down experiments, and it includes a synthetic benchmark demonstrating reproducibility. No new dynamical assumptions are proposed; the method operates strictly as a statistical boundary for rejecting or accepting residual structure. A minimal cosmological example illustrates how the same criterion applies to angular correlation plateaus without implying new physical parameters. The law is fully falsifiable: a single failure of variance, significance, or model-selection thresholds obligates rejection of retentive claims. The manuscript is intended as a transparent methodological contribution supporting independent verification and critical data analysis.