Foundational Axioms, Structural Necessity, and Kernel Discipline in Structured Vacuum Field Theory

Abstract This work formalizes the foundational structure of Structured Vacuum Field Theory by explicitly stating its axioms, distinguishing structural requirements from conventions, and testing their internal consistency and independence. The framework is formulated on a Hilbert space of coherence configurations equipped with a compact, self-adjoint, strongly positive operator subject to global closure and conservation of total coherence. The admissibility of the trivial configuration is examined, and nontrivial coherence is identified as an irreducible axiom. Removal analysis demonstrates that compactness, self-adjointness, strong positivity, closure, and conservation are structurally necessary for isolation and uniqueness of the dominant eigenspace and for preservation of spectral structure. A logical dependency audit establishes the absence of circular reasoning and clarifies the hierarchical architecture of the framework. The result is a minimal and internally consistent operator foundation for the SVFT spectral framework.