This manuscript presents a boundary-level extension of the L(6+7) framework beyond the infinite recursive hierarchy of universes and multiverses. It develops the concepts of admissibility flow, entropic time, the history-closure field Ξ_H, the hyper-closure web X_H, Observer Gate realization, and a bounded residual-manifold program. The work is formulated as a publication-level theoretical candidate within the L(6+7) corpus. It does not claim final empirical proof or a completed discovery of new physics. Instead, it defines a conservative claim boundary and a structured path toward L(6+7) Version 4: action formulation, field equations, reductions, no-signaling constraints, falsifiability gates, and external numerical validation. The manuscript is intended as a theoretical and methodological bridge between the existing L(6+7) field-history framework and a future general theory of admissible realization.