ψ–ISW: A Retentive Lagrangian Correction to the Integrated Sachs–Wolfe Effect

This work presents a full retentive Lagrangian formulation of late-time cosmology and derives its structural correction to the Integrated Sachs–Wolfe (ISW) effect. In ψ-architecture, the variable of difference (Δψ) persists even as metric dynamics weaken, contributing a small, slow, and positive derivative term κψΔψ′ to the evolution of the gravitational potential Φ. We analytically derive the Euler–Lagrange equation for Δψ, obtain the linearized late-time regime near the retentive minimum Δψ_min = (β / 2α) Ξ, and compute the modified ISW kernel K_ISW(η) = 2[dΦ/dη + κψΔψ′]. The ψ–ISW mechanism naturally accounts for several well-known late-time anomalies: • flattening of the low-ℓ CMB temperature plateau, • suppression or decorrelation of ISW–LSS cross-correlation, • persistent large-scale structural anisotropies, • weak redshift dependence of the correction, • compatibility with high-ℓ CMB constraints. The model does not modify early-universe physics, introduces no exotic fluids, carries no pressure terms, and remains fully testable with current and forthcoming surveys (Euclid, DESI, LSST, LiteBIRD). This article presents the full derivation, phenomenology, and observational predictions of the ψ–ISW correction.