ψ-Retentive Angular Tail Law: A Statistical Criterion for Non-Vanishing Large-Angle Correlation Plateaus

This preprint introduces a reproducible statistical criterion for evaluating whether large-angle two-point correlation measurements require a constant positive offset instead of a pure decay-to-zero model. The method is model-agnostic: it does not introduce new cosmological parameters, does not modify ΛCDM, and does not interpret large-angle anomalies. It specifies a falsifiable decision boundary for angular correlation analysis. Given an estimator \xi(\theta), two nested models are compared: M_0 = A f(\theta), \qquad M_1 = A f(\theta) + C_{\mathrm{ang}}. A dataset satisfies the angular-tail criterion only if three independent conditions hold: (1) late-angle decomposition between decay and plateau components; (2) variance stability on a selected interval compared against a noise estimate \sigma_{\mathrm{noise}}; (3) model-selection advantage \Delta\mathrm{AIC} \ge 10 or \Delta\mathrm{BIC} \ge 10 together with Gaussian significance Z \ge 3 for C_{\mathrm{ang}}. A single failure rejects the claim. The protocol includes a synthetic demonstration and a minimal workflow for ISW and weak-lensing products using mock-based covariance. The result is strictly a statistical screening tool. Failure of the criterion eliminates the claim; success enables further data-driven testing.