This work presents L(6+7) as a general field theory of admissible realization. It extends the inherited gate-closed L(6+7) backbone by introducing Omega-infinity as the admissibility boundary, admissibility flow as the deformation law of admissible history space, the history-closure field Ξ_H as an action-level bridge between boundary compatibility and local material realization, the hyper-closure web 𝕏_H as a connected domain of compatible closure fields, and cross-scale residual gates. The observable Universe is treated as a locally realized FRW branch. Multiverse and infinite-hierarchy structures are treated only as gate-filtered spaces of admissible histories under normalizability, stability, summability, residual, covariance, and no-signaling constraints. The claim boundary is explicit: this manuscript is a publication-level theorem/action framework and a falsifiable general-field candidate. It is not presented as final empirical proof, not as a replacement of General Relativity or standard quantum theory, and not as a controllable observer-effect or communication channel. Its strength is the unified structure: constrained variational root, lower-sector recovery gates, closure-field action sector, residual-manifold program, no-signaling constraints, and explicit failure modes. Contact: zigangirov@ukr.netORCID: https://orcid.org/0009-0001-8521-8740