Spectral Rigidity and Capacity Constraints in Canonical Spectral Quantum Gravity

This paper establishes the spectral rigidity layer of a canonical spectral framework for quantum gravity. Building on a previously constructed canonical spectral bridge with Fejér–Hardy normalization and Osterwalder–Schrader (OS) positivity, we prove a suite of operator-theoretic consequences that strengthen the analytic backbone of the framework. Let Q = D + BQG (gE) be the canonically coupled Euclidean Dirac operator on a spin 4-manifold of bounded geometry. Under a precise geometric dichotomy (compact or noncompact confining), we prove: • Compact resolvent stability and elimination of essential spectrum• Principal symbol invariance under canonical perturbation• Exact Weyl asymptotic invariance with explicit coefficient 1/ (8π²) • A deterministic eigenvalue shift bound under bounded coupling• An eigenvalue band counting law derived from a scale-dependent capacity inequality• Scaling-local spectral density constraints compatible with Weyl growth• Trace positivity inequalities extracted directly from OS half-space factorization The results show that the canonical spectral coupling preserves ultraviolet spectral structure while constraining intermediate-scale eigenvalue concentration through a monotone capacity budget associated with a positive dissipative generator K. No additional normalization parameters or ad hoc regularization are introduced. The reversible spectral channel (Q) and irreversible capacity channel (K) form a coupled rigidity structure governed by operator positivity and spectral invariance. This work strengthens the analytic structure of canonical spectral quantum gravity by converting structural positivity and capacity assumptions into explicit spectral and trace inequalities.