The Retentive Threshold Theorem: A Structural Necessity for Late-Time Growth Plateaus in Expanding Universes

This work introduces the Retentive Threshold Theorem, demonstrating that any expanding universe with (i) bounded structure growth, (ii) scale-invariant saturation, and (iii) a non-dissipative stabilizing contribution must exhibit a finite structural threshold separating a growth regime from a retentive plateau. The theorem shows that such a threshold follows directly from structural consistency, without requiring additional dynamics, parameter tuning, or phenomenological adjustments. A minimal realization is provided by the Λψ retentive extension, which predicts a narrow threshold redshift z* ≃ 0.51–0.52, with weak dependence on initial conditions. Rather than creating the plateau, forthcoming observations (e.g., Euclid DR1) act as a recognition of a structure already fixed by the theory’s internal logic. The article includes: • a precise axiomatic formulation of retentive growth; • a formal proof of the Retentive Threshold Theorem; • a retentive Lagrangian showing how the threshold emerges; • a comparison with non-retentive cosmological models; • appendices detailing stability, energy balance, and robustness. This publication is part of the ongoing independent research programme ψ-Architecture / Retention Cosmology and contributes to the formal development of retentive structures in late-time cosmology.