ψ-Time Traps: Retained Difference at the Edge of Disappearance A Unified Retentive Framework for Horizons, Structural Persistence, and the Collapse of Metric Time

This work develops the formal theory of the ψ-Time Trap, a structural regime in which difference (Δψ) remains finite, invariant, and non-vanishing even as metric time collapses. Within the ψ-Architecture framework (2025–2026), such retentive regimes appear across gravitational, cosmological, quantum, vacuum, and archival domains. The paper constructs the complete retentive Lagrangian, derives the Invariance and Stability Theorem for Δψ under temporal collapse, and introduces the Invariant Floor Condition, proving that whenever a retentive node Ξ survives, the equilibrium Δψ* = β/(2α) · Ξ remains strictly non-zero as t → 0. The work synthesizes a unified topology of horizons—event horizons, late-time cosmological freeze-out, decoherence limits, vacuum minima, and semiotic/archival discontinuities—showing that collapse does not imply disappearance. Instead, collapse becomes a retentive transition. Phenomenological motivations (correlation drift ≈0.6, SNR tail ≈0.1–0.2, late-time structural plateaus) are included as empirical anchors while keeping a rigorous distinction between theoretical formulation and observational interpretation. The ψ-Time Trap is presented as a fourth structural regime of physics, supplementing dynamics, equilibrium, and decay. It provides a retentive mechanism capable of maintaining structural difference when time itself dissolves.