[3] COS-QSF - Collapsing-Structure Quantized Shells and Filaments: Topological transitions, CP-compatible channels, and renormalization

This work develops the COS–QSF module, the global topological sector of the Collapsing Structures (COS) program. Building on the local metric-filament layer of COS–QF, QSF introduces an operator-based framework for shell topology, homological sectors, admissible Pachner/surgery-type transitions, topological projectors, and multiscale coarse-graining. The manuscript defines a topological state space organized by Betti and Euler invariants, formulates discrete topological transitions in Kraus form as completely positive, generally trace-nonincreasing maps, and describes how these maps may be extended to CPTP dynamics on enlarged spaces. It also develops effective topological transition weights, conditional detailed-balance structures, Hodge–Laplace spectral diagnostics, and renormalization/coarse-graining tools for composite shells. The QSF and QF sectors are connected through admissibility, compatibility, and causality-related projectors, allowing local metric data to weight global topological transitions while preserving CP-compatible joint dynamics. Numerical protocols and small-scale N = 3–5 case studies illustrate the framework. Candidate observable channels include CMB modulation patterns, stochastic gravitational-wave background features, and black-hole ringdown deviations; these are presented as open phenomenological targets rather than closed result claims.