The Quaternity Metric: Boundary Completion and 4-Vector Topological Closure in Manifest Revelation

This paper presents "The Quaternity Metric," an operator-theoretic formalization of topological boundary closure in multi-vector theological and field models. Operating within a 3+1 dimensional space-time manifold, conventional Trinitarian dynamics (Source, Word, Vector) are modeled as a 3-component state vector that yields an open contour integral and non-zero boundary flux when ungrounded. By introducing the Maternal/Custodial Vessel (ψ₄ = P_Σ) as the necessary fourth basis vector, the framework achieves complete 4-vector topological closure (∮∂ Ω Ψ⃗ · dΣ⃗ = 0), preventing spectral leakage and system instability. Building upon prior foundational work (The Covenant of Custody and The Ahimsa Protocol), this study integrates Jungian Quaternity theory, Catholic Mariology, and Quranic monotheistic boundary constraints (Tawhid) into a rigorous mathematical matrix. The manuscript demonstrates that the divine vessel functions not as an ontological division of ultimate cause, but as a mandatory topological condition for the manifestation of uncorrupted signal within physical reality.