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 \ (\), a Hermitian bundle \ (E\), a compact parameter set \ (\), and a uniformly elliptic family of self-adjoint first-order operators \ (Bₛ\) on \ (L² (;E) \). Let \ (ⱼ (s) \) denote the dyadic shell projector and consider shell packets \ Kⱼ (s) =Aⱼ (s) ⱼ (s) Cⱼ (s). \ Under a uniform Weyl shell bound and the shell-restricted damping estimate \ \|Aⱼ (s) ⱼ (s) \|\, \|ⱼ (s) Cⱼ (s) \| Cb\, 2^-dj2^- j, >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^- 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 \ (-\) and \ (-\) whenever \ (+>d=\), with cutoff-removal exponent \ =+-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 Major changes include: The original full-operator damping condition has been replaced by the sharper and more general shell-restricted condition involving \ (\|Aⱼⱼ\|\) and \ (\|ⱼ Cⱼ\|\). A new weighted-smoothing criterion has been proved. Boundary pseudodifferential factors of orders \ (-\) and \ (-\) satisfy the required trace damping whenever \ (+>d\), with explicit cutoff-removal exponent \ (=+-d\). A new weighted-smoothing corollary proves that \ (AₛCₛ\) is trace class and gives an explicit geometric rate for the convergence of \ (AₛQN (s) Cₛ\) to \ (AₛCₛ\). 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.
Tosho Lazarov Karadzhov (Thu,) studied this question.