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

We prove a uniform trace-norm cutoff-removal theorem for dyadic shell perturbations on a compact renormalized inner-horizon collar. The setting is a compact boundary spectral model consisting of a compact boundary cross-section \(\Sigma\), a Hermitian bundle \(E\to\Sigma\), a compact parameter set \(\Gamma\), and a uniformly elliptic family of self-adjoint first-order operators \(B_s\) on \(L^2(\Sigma;E)\). Let \(\Pi_j(s)\) denote the dyadic shell projector and consider shell packets \[ K_j(s)=A_j(s)\Pi_j(s)C_j(s). \] Under a uniform Weyl shell bound and the shell-restricted damping estimate \[ \|A_j(s)\Pi_j(s)\|\,\|\Pi_j(s)C_j(s)\| \le C_b\,2^{-dj}2^{-\kappa j}, \qquad \kappa>0, \] we prove that the shell series converges absolutely in trace norm, uniformly over the parameter set, and that its high-frequency tail is \(O(2^{-\kappa N})\). Consequently, finite shell cutoffs converge to a cutoff-free trace-class limit at an explicit geometric rate, and traces against uniformly bounded test families converge uniformly. A weighted-smoothing criterion verifies the hypothesis for uniformly bounded boundary pseudodifferential families of orders \(-\alpha\) and \(-\beta\) whenever \(\alpha+\beta>d=\dim\Sigma\), with cutoff-removal exponent \[ \kappa=\alpha+\beta-d. \] The result is a compact-collar Schatten estimate associated with the boundary spectral calculus of APS collar models. It does not address nonlinear mass inflation, global Kerr--Newman interior evolution, or observational questions. Version 2 is a substantial mathematical and expository revision. Major changes include:The original full-operator damping condition has been replaced by the sharper and more general shell-restricted condition involving \(\|A_j\Pi_j\|\) and \(\|\Pi_j C_j\|\). A new weighted-smoothing criterion has been proved. Boundary pseudodifferential factors of orders \(-\alpha\) and \(-\beta\) satisfy the required trace damping whenever \(\alpha+\beta>d\), with explicit cutoff-removal exponent \(\kappa=\alpha+\beta-d\). A new weighted-smoothing corollary proves that \(A_sC_s\) is trace class and gives an explicit geometric rate for the convergence of \(A_sQ_N(s)C_s\) to \(A_sC_s\). The main argument has been reorganized into a Hilbert--Schmidt shell estimate, a one-shell trace bound, and a unified uniform trace-norm cutoff-removal theorem. Uniform convergence of traces against bounded test families has been stated explicitly. The Weyl shell-rank argument and its endpoint treatment have been refined. The relation with the companion Paper~A has been expanded substantially. The APS sign projection defining the domain in Paper~A is now clearly distinguished from the high-frequency dyadic projectors used for Schatten estimates in Paper~B. The abstract, terminology, theorem references, bibliography, keywords, and scope statement have been revised and polished. The scope remains deliberately local and functional-analytic: no nonlinear or global black-hole evolution theorem is asserted.