Variational Backbone and Regime Closures XV (Atlas): One Part-I-Compatible Summary-to-Retained Representative, One Reduced Schur Operator, and the Stationary/Dissipative/Conservative Gate Atlas of a Low-Energy Geometric Realization

VBRC treats the low-energy regime atlas as a summary-interface atlas rather thana single master equation replacing classical, quantum, and relativistic laws. Under theInternal Invisibility Principle, those familiar laws appear as gate-dependent readouts ofone Part-I-compatible representative and its reduced Schur operator, not as independentprimitives.Following the Part I order—an admissible state/comparison arena C, a comparisonlanguage E, protocol-induced partial readout and IIP, the summary-entry discipline F =ΣH,P (unread content), with F = DI II only after a first-order representative has beendeclared, admitting only F on the retained side, the core density ecore read on the retainedeffective pair (IR, F), and only then gate-dependent regimes—this paper does not attemptto unify familiar classical, quantum, and relativistic laws by collapsing them into one finalequation. Instead, Part XV declares a low-energy geometric representative of ecore, studiesthe reduced Schur operator Seff obtained under the admissible summary-reduction gate, andapplies the three series readings R1, R2, R3 to the resulting retained law: the stationaryreading δEXV/δqA = 0, the dissipative reading ΓAq˙A + δEXV/δqA = 0, and the conservative(time-completed) reading MAq¨A + δEXV/δqA = 0. The conservative–dissipative equationthat contains both inertial and frictional terms is used in this Part only as a transitioncompletion containing the R2 and R3 limits in one notation; it is not a fourth primitivereading and does not replace the declared comparison representative. No claim of absoluteuniqueness of EXV is made beyond the declared first-order representative scope. Accordingly,Part XV should not be read as opening a fresh representative branch of the theory: itdeclares a low-energy geometric realization FXV = DI II of the Part I summary channel andstudies the gate atlas carried by the corresponding retained law, with classical, quantum,and relativistic equations treated as conditional comparison readouts rather than as newprimitives or as a new master equation. The recovery of familiar laws is conditional: eachnamed law appears only as a gate-dependent effective regime of that common structure,with explicit validity conditions and transition thresholds. Breakdown is therefore readnot as inconsistency but as passage into a neighboring regime. The characteristic matterterms of the effective laws are likewise not inserted by hand: effective mass/gap terms arisethrough summary reduction of the already licensed summary channel in Schur-type form onthe R3 (conservative) field branch, while effective stress-energy terms arise through metricvariation of the same geometric representative in its R1 metric branch. Its main contributionis therefore not a new local law but a single low-energy atlas of emergence, validity windows,and transition behavior for the declared representative, with the principal formal reductionsrecorded in the appendices. A common-source theorem internal to this Part shows, afteradmissible summary reduction, that the three readings on the retained branch are controlledby one and the same reduced Schur operator Seff that is derived (not inherited as a primitive)from the declared representative and gate: stationary softening, dissipative growth/decay, andconservative dispersion are then three reading-dependent interpretations of the sign-changesof one common reduced symbol. The threshold quantities fixed here are also the low-energystarting point of the later late-gate trilogy: the mixed-gate discriminant and the upper-gatecritical scale are different gate-specific realizations of the same transition grammar ratherthan disconnected additions.