Variational Backbone and Regime Closures III (Conservative): Conservative Retained Laws from the Core Density R3 Time Completion, Summary Footprints, and Gate-Restricted Standard-Like Branches

This paper computes the R3, or time-completed, retained law generated by the VBRC variational core. Under the Internal Invisibility Principle, the unread sector cannot enter the retained dynamics as raw data. It contributes only through a licensed summary footprint F = Sigma (unread content). R3 time completion is imposed on the retained variable IR alone, while the unread variable is not assigned an independent kinetic term inside the core theory. The resulting retained Euler-Lagrange law has a standard conservative second-order backbone, but it also contains the licensed unread-summary footprint. Thus the generic VBRC R3 output is not simply a textbook Klein-Gordon, Schrodinger, or Hamilton-Jacobi equation. These equations arise only as downstream, gate-restricted readouts after the footprint is represented, localized, stabilized, or absorbed into effective lower-order data. The paper then separates admissible summary-channel treatments. One gate treats the summary as a protocol-given load; another imposes instantaneous summary stationarity and produces an effective projected lower-order retained law. An independent unread-side kinetic term is recorded only as a non-core extension gate. The main structural conclusion is that conservative retained dynamics in VBRC is a standard-like time-completed law together with a licensed summary entry, and that R2 selects or stabilizes the background branch on which R3 retained oscillations are studied.