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 I_R 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.