16 COS-PRI - Collapsing-Structure Time Evolution and Spacetime Limit: CPTP channels, conditional Regge--Einstein limit, and Lieb--Robinson causal envelope on discrete networks

COS-PRI develops the time-evolution and spacetime-limit interface of the Collapsing-Structure (COS) program. The module investigates how discrete operative time evolution, CPTP channel dynamics, GKSL continuous-time limits, conditional trajectory descriptions, and quasiclassical spacetime reconstruction can be organized within a common shell-filament framework. The manuscript formulates COS time evolution through quantum instruments and branch-ignored CPTP channels on the physical sector, carefully distinguishing unconditional physical dynamics from conditional Kraus branches, unravelings, and postselected trajectory descriptions. In the continuous-time scaling regime, the channel dynamics is related to GKSL generators, entropy contraction, mixing diagnostics, and residual-monitoring mechanisms. COS-PRI also records the conditional bridge from discrete shell-filament dynamics to Regge-type and Einstein-Hilbert continuum structures. The classical spacetime limit is treated as a controlled variational and reconstruction program, relying on admissibility, compactness, regularity, coercivity, and scaling assumptions. It is not presented as an unconditional classicization theorem or a global Einstein-attractor result. The module further discusses constraint-residual monitoring, quasiclassical trajectory selection, Γ-convergence interfaces, Regge-to-Einstein reconstruction, and large-deviation-inspired trajectory interpretations. These elements are separated by status: operative constructions, controlled conditional bridges, and open programmatic components. A dedicated causality layer formulates Lieb-Robinson-type influence estimates and finite-speed propagation bounds for locally implementable admissibility filtering and quasi-local GKSL decompositions. These claims apply only within the specified local channel class; global nonlocal projectors or unrestricted microdynamics are not automatically covered. COS-PRI therefore serves as the operative time-evolution, causality-envelope, and spacetime-limit status map of the COS program, linking COS-QD, COS-CL, COS-STAB, COS-CNS, and related modules while maintaining explicit distinction between proved constructions, conditional continuum/locality bridges, heuristic calibration interfaces, and open problems.