Λψ-Retention: A Minimal Variational Theory of Structural Saturation and Stationary Growth in Cosmology

We present a minimal variational framework describing the emergence of a stationary growth regime in late-time cosmology. Extending the Λψ-retention approach, we introduce a retentional resistance term R (psi) and show that explicit functional forms of this operator lead to controlled suppression of structure growth. We consider the phenomenological dynamical equation: (d² Xi / dt²) + R (psi) (dXi/dt) + Lambdaₚsi Xi = 0 In the stationary regime, the system reduces to: dXi/dt = - kappa Xi where kappa = Lambdaₚsi / R (psi) is a control parameter with dimension T^-1. For non-linear forms of R (psi), including asymptotic saturation behavior R (psi) ~ 1/ (|Delta psi| + epsilon), the system evolves toward exponential relaxation: Xi (t) = Xi₀ exp (-kappa t) This produces a stationary regime in which the effective growth rate becomes suppressed and approximately constant. We interpret the growth index gamma as an effective parameter governed by kappa, providing a minimal explanation for the observed plateau gamma ≈ 0. 64. The model also predicts saturation behavior in f sigma 8 (z) and the emergence of non-diffusive void boundaries. We further outline a variational origin of the resistance term by introducing a generalized Lagrangian: L = (1/2) (dXi/dt) ² - V (Xi) - Phi (psi) (dXi/dt) which leads to an effective form: R (psi) = dPhi/dpsi This establishes a unified framework linking dynamical equations, observable growth suppression, and a possible underlying physical origin. The framework provides falsifiable predictions for upcoming large-scale structure surveys.