The Operator Geometry of Absolute Custodianship: Triadic Field Projection, the Marian Boundary Condition, and Structural Admissibility

Abstract

This manuscript constructs a rigorous operator-theoretic framework that unifies triadic theological field projections with the mechanics of structural admissibility and spectral stability. Classical dogmatic models often treat triadic structures (Source/Father, Word/Son, Spirit/Breath) as closed, self-sustaining loops. However, within continuous-time operator dynamics and functional analysis, an unconstrained, non-self-adjoint field projection (M) acting on a Hilbert space (H) induces severe transient pseudospectral spikes and spectral leakage unless constrained by a localized zero-point boundary condition. We formalize the uncreated ground state generator (L_0), the manifest perturbation (M), and the continuous transport medium (S(t)), proving that the triadic field projection remains unmanifested or catastrophically unstable without an explicit containment vessel. We define the Marian Boundary Operator (P_Sigma) and the Magdalene Custodial Chain (Q_Magdalene) as non-negotiable projection operators that enforce det_2(I - K) != 0, absorb non-normal transient spikes, and prevent state-space collapse. Moving beyond linear models of succession, this work establishes Spiritual Stewardship as a distributed matrix of custodians holding and transmitting divine custody across time. Ultimately, the framework demonstrates that the Divine Feminine in prophetic and esoteric traditions functions not as an arbitrary theological addition, but as the essential, non-linear grounding geometry (Sigma) required for absolute physical and structural admissibility.