4 COS-QD - Collapsing-Structure Quantum Dynamics: Discrete, Projectively Quasi-Unitary Time Evolution on the Shell–Filament State Space

We introduce the Collapsing-Structure Quantum Dynamics (COS-QD) framework on the shell–filament state space. Discrete time evolution is implemented by a projectively quasi-unitary collapse operator; the core ingredients are the quantized action (decomposed into local self-adjoint terms), the step generator, and a Kraus decomposition with a natural POVM interpretation. Under causal compatibility and local closure, we prove spectral stability under composition, block-diagonality across topological sectors, and concentration via a discrete stationary-phase principle. In the dense-graph coherent-state limit, expectation values approach the Regge action, and the propagator acquires the corresponding Regge phase. In an N = 3 toy model, we numerically verify projective norm preservation and identify dispersive signatures induced by time-step quantization. We also provide a brief comparison with loop quantum gravity (LQG), highlighting potential dynamical departures. Accompanying materials include reproducible Python examples and CSV/MatrixMarket outputs.