Spectral Rigidity and Capacity Constraints in Canonical Spectral Quantum Gravity: Compact Resolvent Stability, Weyl Invariance, and OS-Derived Trace Inequalities

This paper establishes the spectral rigidity layer of a canonical spectral framework for quantum gravity. Building on the construction of a background-covariant spectral bridge and Osterwalder–Schrader (OS) positive coupling developed in the preceding construction paper, this work derives the operator-theoretic consequences that follow once the canonical perturbation is fixed. The focus is not on constructing the theory, but on proving what becomes rigid once the canonical coupling is in place. The main results include: Principal symbol invariance: the canonical perturbation does not alter the Dirac operator’s principal symbol, ensuring exact Weyl asymptotic stability. Compact resolvent stability: under a precise geometric dichotomy (compact or noncompact confining), the coupled operator has purely discrete spectrum. Uniform eigenvalue shift bounds: deterministic comparison between perturbed and unperturbed counting functions. Capacity-driven band counting law: a monotone capacity inequality yields explicit bounds on scaled spectral concentration within admissible physical windows. Scaling-local spectral density constraints: eigenvalue clustering is controlled in effective bands without violating global Weyl growth. Operator trace inequalities derived directly from OS half-space factorization, converting reflection positivity into concrete spectral ordering relations. Explicit coupling between the reversible spectral channel (Dirac plus canonical perturbation) and an irreversible dissipative generator governing a monotone capacity budget. Together, these results show that once the canonical spectral bridge is fixed: Ultraviolet growth is rigidly geometric. Intermediate-scale spectral concentration is bounded. No arbitrary normalization freedom remains. Reflection positivity yields verifiable operator inequalities. This paper forms the rigidity layer in a five-part structural program: Construction layer – canonical spectral bridge and OS positivity. Rigidity layer – spectral compactness, Weyl invariance, and capacity constraints (this paper). Structural closure layer – algebraic collapse, modular rigidity, primitive completeness. Universal reduction theorem – derivation that minimal relativistic QFT conditions reduce to the canonical framework. Universality theorem – forcing of Einstein dynamics at leading order. The present work strengthens the analytic backbone of canonical spectral quantum gravity by proving that the coupled operator channel is spectrally stable, density-controlled, and positivity-constrained within the admissible regime. No empirical parameters or adjustable normalization constants are introduced.