Variational Backbone and Regime Closures XV: Regime Conditions, Breakdown Thresholds, and Transition Maps in the Curvature-Coupled Dual-Flow Backbon

Under the Internal Invisibility Principle, this paper does not attempt to unify familiarclassical, quantum, and relativistic laws by collapsing them into one final equation. Instead,Part XV organizes them as gate-dependent effective readouts of one fixed curvature-coupledhidden–open backbone, inherited from Part I through the declared hidden-to-retainedsummary discipline F = ΣH,P(II) = DIII. Within that same declared summary instance,this Part adopts one fixed curvature-coupled hidden–open backbone E, together with oneuniversal conservative–dissipative equation, as the natural minimal realization used in theseries within the declared first-derivative representative scope; no claim of absolute uniquenessis made beyond that scope. Accordingly, Part XV should not be read as opening a freshrepresentative branch of the theory: it studies the low-energy atlas and gate map carriedby one already licensed geometric representative, with classical, quantum, and relativisticequations treated as downstream effective readouts rather than as new primitives. Therecovery of familiar laws is conditional: each named law appears only as a gate-dependenteffective regime of that common structure, with explicit validity conditions and transitionthresholds. Breakdown is therefore read not as inconsistency but as passage into a neighboringregime. The characteristic matter terms of the effective laws are likewise not inserted by hand:effective mass/gap terms arise through elimination of the already licensed summary channelin Schur-type form, while effective stress-energy terms arise through metric variation of thesame backbone. Its main contribution is therefore not a new local law but a single low-energyatlas of emergence, validity windows, and transition behavior for the series backbone, withthe principal formal reductions recorded in the appendices. The threshold quantities fixedhere are also the low-energy starting point of the later late-gate trilogy: the mixed-gatediscriminant and the upper-gate critical scale are different gate-specific realizations of thesame transition grammar rather than disconnected additions.