Variational Backbone and Regime Closures XIII: Gauge Forces as Operator Instances and RG Readouts Yang–Mills principal operator and asymptotic freedom as a scale-instance output

Part XIII derives the gauge-redundancy slot from the constitutional summary interface fixed in Part I. In the shorthand used here, S1–S5 supply a formal comparison arena C, a real-valued comparison language E : C → R, protocol-induced retained/unread coordinates, the prohibition of raw unread entry into retained laws, and the licensed summary rule Fsum = ΣH,P (unread content), with Fsum = DI II only in a declared first-order representative. Consequently admissible observables factor through the summary image, and representatives lying in the same summary fiber cannot be distinguished by the retained protocol. The summary-fiber quotient is the structural-interface source of the gauge-redundancy slot; a non-trivial gauge symmetry appears only when the declared representative realizes that slot nontrivially. Concrete gauge theory is a later realization of that quotient branch, not the source of the quotient itself. A compact group, principal bundle or associated representation, connection A, curvature FA, matter package, and gauge-fixing treatment are declared instance data on the already licensed retained branch. Thus gauge moves are summary-preserving re-descriptions, while the choice of G, the matter representation content, and the Standard-Model gauge dossier remain model-instance data rather than unique consequences of the constitutional layer alone. Once such a gauge instance is declared, the Yang–Mills principal-operator normal form [17] is not a tuning choice: inside the admitted local, parity-even, first-derivative quadratic, principal-symbol-preserving class, it is the IIP-admissible gauge-kinetic form forced by factorization through the retained curvature summary FA. Once a scale/coarse-graining instance is additionally declared, renormalization-group running is read as a downstream scale-readout protocol on the same retained gauge branch, not as a new microscopic law. Asymptotic-freedom-type statements therefore hold only inside the declared perturbative/sign gates. When the Standard Model gate package is further declared, the renormalizable local operator basis closes to the usual Standard-Model operator skeleton inside that package. This is a conditional compatibility and closure result, not a derivation of the Standard Model gauge group, matter package, or phenomenology from retained readability alone. However, within a declared chiral matter package, anomaly cancellation is not optional: it is the IIP-readability condition that prevents gauge-fiber choices from changing the retained current. The paper therefore separates: the gauge-redundancy slot derived from the Part I summary interface; Yang–Mills/RG as declared gauge/scale readouts with forced forms inside their gates; anomaly cancellation as a forced consistency wall inside the declared chiral gate; and Standard-Model data as compatibility information inside a declared instance.