Dyadic Shell Estimates and Trace-Norm Cutoff Removal on a Renormalized Inner-Horizon Collar

This preprint proves a uniform trace-norm cutoff-removal theorem for dyadic shell perturbations on a compact renormalized inner-horizon collar. The paper works with a compact boundary spectral model consisting of a compact smooth boundary cross-section \(\Sigma\), a Hermitian vector bundle \(E\to\Sigma\), a compact parameter set \(\Gamma\), and a compact family of self-adjoint first-order elliptic operators \[ B_s: H^1(\Sigma;E)\subset L^2(\Sigma;E)\to L^2(\Sigma;E). \] Writing \[ \Lambda_s=(1+B_s^2)^{1/2}, \] the dyadic shell projectors are defined by \[ \Pi_j(s)=\mathbf{1}_{[2^j,2^{j+1})}(\Lambda_s). \] The paper studies shell packets of the form \[ K_j(s)=A_j(s)\Pi_j(s)C_j(s), \] where \(A_j(s)\) and \(C_j(s)\) are bounded operators satisfying the explicit dyadic trace-damping condition \[ \|A_j(s)\|,\|C_j(s)\| \le C_b\,2^{-dj}2^{-\kappa j}, \qquad \kappa>0. \] Under a uniform Weyl shell-rank bound \[ \operatorname{rank}\Pi_j(s)\le C_W2^{dj}, \] the main theorem proves that the shell series \[ K(s)=\sum_{j\ge0}K_j(s) \] converges absolutely in trace norm, uniformly for \(s\in\Gamma\), and that the high-frequency tail satisfies the geometric estimate \[ \left\|\sum_{j\ge N}K_j(s)\right\|_{\mathfrak S_1} \le C2^{-\kappa N}. \] Consequently, finite dyadic shell cutoffs converge to a cutoff-free trace-class limit, and traces against uniformly bounded test families converge uniformly. This paper is the second short theorem paper in the compact-collar series. Paper A established the APS/Fredholm collar theorem and index stability. The present Paper B supplies the trace-class shell-estimate layer: dyadic shell estimates, trace-norm cutoff removal, and uniform tested trace convergence. The result is a compact-collar Schatten theorem for shell-localized perturbations on the boundary spectral side. It does not claim a theorem on nonlinear mass inflation, global Kerr--Newman interior evolution, observational comparison, black-hole evaporation, Page curves, or the full black-hole information paradox.