Mathematical Foundations of Retentive Physics and the ψ-Quant Formalism

This work develops the mathematical foundations of Retentive Physics within the ψ-Architecture framework. The article introduces a minimal Lagrangian formulation in which structural retention, rather than dynamical flux, governs late-time cosmological behavior. A quadratic stability potential U(\Delta\psi,\Xi) defines the retentive minimum and gives rise to the structural remainder C_\psi\,\Xi^2, interpreted as a non-dissipative equilibrium contribution. Building on this foundation, the ψ-Quant formalism is introduced as a scale-free analytical structure describing the smallest unit of retentive potential, Q_\psi = C_\psi \Xi^2. We then define the retentive density \rho_{\text{ret}}, derive the supplementary term \Lambda_\psi = 8\pi G\,\kappa_\psi\,C_\psi\,\langle\Xi^2\rangle, and embed it into the Friedmann equations without altering the structure of General Relativity. A structural amplification mechanism is identified through the Ω–Ξ–Qψ coupling, showing that even modest variations in ψ-quanta propagate nonlinearly into the retentive density due to the late-time behavior of H(t). This leads to clear, falsifiable signatures in the Euclid DR1 sensitivity window 0.5 < z < 1.5, including correlation-time plateaus and suppressed structure growth. The ψ-retentive sector is presented with conceptual modesty: as a mathematically coherent and empirically testable supplement to standard cosmology, grounded in stability, topology, and internal ψ-time. The article aims to provide a rigorous foundation for future theoretical and observational work on structural retention in the late universe.