Volume IV / V ψ-Architecture Research Series) Λψ–Friedmann Control: A Retentive Extension of Late-Time Cosmology

This work develops Λψ–Friedmann Control, a retentive extension of late-time cosmology based on a scalar difference field Δψ and structural retention nodes Ξ. The framework introduces a weakly evolving retentive energy component Λψ that supplements the Friedmann equations without modifying General Relativity or invoking explicit dark-sector interactions. Λψ produces a controlled suppression of linear structure growth, predicts an enhanced growth index γ > 0.55, and identifies a distinct observational redshift corridor (0.5 ≲ z ≲ 1.5) relevant for Euclid DR1 and next-generation surveys. The manuscript provides the full retentive Lagrangian, the resulting effective energy density ρψ, the modified Friedmann equations, and operational observational tests distinguishing Λψ from modified gravity, interacting dark energy, and dynamical dark-energy models. This work constitutes a formal mathematical extension of the ψ-Architecture Research Series and a testable contribution to late-time cosmology.