Retentive Saturation versus Fata Morgana: Integration-Depth Invariance as a Diagnostic for Late-Time Cosmological Structure

This work introduces a diagnostic framework for distinguishing genuine late-time structural saturation in cosmological observables from observational or methodological artifacts commonly referred to as Fata Morgana effects. Rather than proposing new dynamical mechanisms, the study reframes late-time growth anomalies as a problem of information behavior under increasing integration depth. The manuscript formalizes integration-depth diagnostics as non-violating null tests and establishes a set of invariance criteria—estimator independence, covariance robustness, achromaticity, null-control rejection, and window stability—required for a plateau to be considered physically admissible. Three formal results demonstrate that, when these criteria are satisfied, structural saturation is the only consistent interpretation within the tested regime. The framework preserves early-universe consistency, introduces no additional dynamical degrees of freedom, and is explicitly falsifiable. This work fixes the diagnostic conditions under which claims of late-time structural saturation must be evaluated in future observational and computational studies.