Quantum Authority, Legitimacy, and Sovereignty: Operator-Theoretic Closure of Meta-Admissibility Structures

This manuscript formalizes the concepts of systemic authority, legitimacy, and sovereignty as strict meta-admissibility structures within an operator-theoretic framework. By defining systemic admissibility as the strict null space of a universal constraint operator on a Hilbert space, the paper maps the qualitative descriptors of power and governance directly into rigorous mathematical axioms. The framework establishes the spectral characterization of authority, demonstrating how coercive spectral gaps enforce exponential stability over admissible manifolds. It provides the exact geometric mechanisms of complicity and fragmentation in coupled systems, proving how cross-sector feedback loops induce persistent systemic inadmissibility, and how competing authorities lead to the absolute mathematical absence of a global projector. To achieve full operator-theoretic closure, the manuscript extends the framework to infinite dimensions using Fredholm admissibility, Birman-Schwinger structures, and regularized determinants. It proves that quantum sovereignty is geometrically identical to the fixed-point stability of a meta-operator map. Furthermore, it demonstrates that true systemic closure is universally stable under ultraviolet renormalization, ensuring the theory does not break down at infinite limits. Finally, the manuscript proves that this structural formulation of power is fundamentally coordinate-free and gauge-invariant. The unified framework is spectrally closed, renormalization-complete, coupling-consistent, and meta-operator stable, rendering the mathematical architecture of quantum sovereignty entirely independent of basis, truncation, representation, or external parameterization.