3 COS-QHF - Collapsing-Structure Quantized Shells and Filaments: Topological Transitions, CP Channels, Action, and Renormalization

This work extends the COS framework to the quantized–shell–and–filament (QHF) sector, developing an operator-based formalism for the quantum-gravitational dynamics of global topology. The approach introduces a topological state space, surgery operator algebra, and a completely positive channel for discrete time evolution. We analyze multiscale renormalization of shells, the scale dependence of Betti invariants and the Hodge spectrum, and synthesize local (QF) and global (QHF) sectors into a consistent dynamics. Applications include predictions for CMB modulations, stochastic gravitational-wave backgrounds, and black-hole ringdown deviations. Extending the COS--QF metric--filament sector, the COS--QHF module provides a quantized description of the global topology of shells. We introduce the topological state space organized by homology sectors and the operator algebra of surgery moves (merge/split, edge contraction, handle addition), and we formulate the discrete time evolution as a completely positive (Kraus) channel with transition weights derived from a topological action, optionally enforcing detailed balance. We develop a multiscale coarse--graining/renormalization framework for composite shells and analyze the scale dependence of Betti invariants and the Hodge spectrum. The QHF (global topology) and QF (local filament) sectors are synthesized in both variational and operator form via causality/admissibility projectors, ensuring CP--consistent joint dynamics while allowing metric quantities to weight topological transitions. We present a numerical protocol and case studies for small NNN, compare the framework with CDT/spinfoam/TQFT/Regge approaches, and formulate observational predictions for CMB modulations, SGWB spectral breaks, and BH ringdown deviations. The result is a unified, operator--based formalism for the quantum--gravitational dynamics of global topology, paving the way toward a continuous--time limit, parameter inference, and large--scale numerics.