ψ⁸: Shadows of the Past — A Minimal Retentive Equation for Late-Time Cosmology

This paper introduces the Retentive Shadow Equation, a minimal dynamical law governing the persistence of structural difference in the late-time universe. We formalize the condition ddt| (t) | 0 for all >0, showing that once a retentive node is formed, structural difference ceases to decay and approaches a stable equilibrium — the Δψ-floor. This regime explains several late-time anomalies observed in Euclid weak-lensing data, including correlation-time plateaus (0. 2 < z < 0. 5), void ghosting, and extended topological persistence. The proposed formalism provides a parsimonious alternative to ΛCDM and defines a falsifiable criterion for identifying retentive behavior in cosmic structures. As part of the ψ-Architecture Research Series, this work consolidates the mathematical and observational basis of retentive cosmology.