ψ-Retention Cosmology: The Structural Fixed Point at γ ≈ 0.64 — A Variational Derivation and Stability Analysis

This work presents a variational formulation of the ψ-Retention Cosmology framework in which the observed late-time growth index γ ≈ 0.64 arises as a structural fixed point rather than a phenomenological adjustment. Starting from a retentive Lagrangian defined over the scale factor, we derive the modified growth equation, establish the autonomous system, obtain the fixed-point solution, and demonstrate its dynamical stability. The analysis explains the k-independent suppression seen in Planck PR4 + DES Y6 + Gold-2026 fσ₈ data, predicts a regime transition near z ≈ 1.7, and identifies the retentive equilibrium Ωψ as the structural origin of the γ anomaly. The result defines a coherent, falsifiable extension of late-time cosmology beyond ΛCDM.