ψ-Atlas II — Law of Retentive Invariance: Field-Theoretic Extension of Memory Curvature

This work presents the verified revision of ψ-Atlas II within the ψ-Architecture framework and introduces the Law of Retentive Invariance. The central claim is that curvature may arise not from matter or energy but from the persistence of differentiation itself. Formally, the theory defines a field of difference Δψ sustained by retentive nodes Ξ on a pseudo-metric manifold parameterised by intrinsic ψ-time τ. The Lagrangian \mathcal{L}{ψAtlasII} = \tfrac{1}{2}(\partial{τ}Δψ)^2 - V(Δψ,Ξ) + Λ(Ξ)\nabla Δψ encodes the stabilising tension between transient differentiation and structural retention. From this structure follow the ψ-field equations and a curvature tensor R_{ψ} whose source is the persistence tensor T_{ψ μν}. The ψ-Einstein extension R_{ψ μν} - \tfrac{1}{2}R_{ψ} g_{ψ μν} = κ T_{ψ μν} + λ g_{ψ μν} introduces a retentive cosmological constant λ, modelling baseline persistence even in isolation. The Law of Retentive Invariance is stated as \delta(Δψ\,Ξ)=0 \Rightarrow ψ-curvature is conserved, establishing a conservation principle for memory geometry. Topologically, retention replaces spatial continuity with continuity of differentiation; existence is defined by the non-vanishing of Δψ for at least one Ξ. Applications sketch analogues across cosmology (information survival beyond event horizons), cognition (distributed, self-sustaining recollection), ethics (non-violent stability via positive \langle Δψ,Ξ\rangle), and aesthetics (form as finite embodiment of ψ-curvature). Overall, ψ-Atlas II consolidates a field-theoretic model in which existence = continuity of difference under retention, offering a coherent basis for simulations and comparative formal analysis in post-metric regimes. This study contributes to the emerging field of memory geometry — the investigation of curvature and retention as fundamental forms of intelligibility.