ψ-Retentive Existence Criterion: A Two-Mode Stability Condition Linking Cosmological Correlation Plateaus and Laboratory Resonators

This preprint introduces a minimal, falsifiable mathematical criterion for detecting non-dissipative long-time residuals (“retentive plateaus”) in temporal correlation functions. The framework is intentionally model-agnostic and does not modify cosmological theory or laboratory physics. It specifies when a correlation response R(\tau) requires a positive late-time constant term rather than a single-mode exponential decay. The criterion requires the existence of a stationary point \frac{dR}{d\tau}(\tau_0)=0 \quad \text{with} \quad R(\tau_0) > 0, and allows a long-time limit \lim_{\tau\to\infty}R(\tau)=C>0. A two-mode response model R(\tau)\approx A e^{-\gamma_1\tau}+B e^{-\gamma_2\tau} provides a minimal statistical mechanism for empirical plateaus. The hypothesis is rejected if laboratory or survey data converge to single-mode decay or yield B=0. The document outlines laboratory testbeds (optical interferometry, mechanical/acoustic resonators, electrical and microwave cavities) and observational access points (Euclid clustering, Roman weak-lensing correlations, CMB low-ℓ correlations, ISW cross-correlation tails). The purpose is not to propose new physics but to offer a reproducible statistical rule for determining when a non-vanishing residual must be modeled. Failure of the criterion eliminates the hypothesis; success enables terrestrial testing.