Variational Backbone and Regime Closures XIII: Gauge Redundancy from the Internal Invisibility Principle Yang–Mills normal form and Standard-Model compatibility as declared instances

Part XIII derives the gauge-redundancy slot from the constitutional summary interfacefixed in Part I. In the shorthand used here, S1–S5 supply a formal comparison arena C, areal-valued comparison language E : C → R, protocol-induced retained/unread coordinates,the prohibition of raw unread entry into retained laws, and the licensed summary ruleFsum = ΣH,P (II ),with Fsum = DI II only in a declared first-order representative. Consequently admissibleobservables factor through the summary image, and representatives lying in the samesummary fiber cannot be distinguished by the retained protocol. The summary-fiber quotientis the structural-interface source of the gauge-redundancy slot; a non-trivial gauge symmetryappears only when the declared representative realizes that slot nontrivially.Concrete gauge theory is a later realization of that quotient branch, not the source of thequotient itself. A compact group, principal bundle or associated representation, connection A,curvature FA, matter package, and gauge-fixing treatment are declared instance data on thealready licensed retained branch. Thus gauge moves are summary-preserving re-descriptions,while the choice of G, the matter representation content, and the Standard-Model gaugedossier remain model-instance data rather than unique consequences of the constitutionallayer alone.Once such a gauge instance is declared, the Yang–Mills principal-operator normal form 8is recorded as the canonical gauge-kinetic representative inside the admitted principal class.Once a scale/coarse-graining instance is additionally declared, renormalization-group runningis read as a downstream scale-readout protocol on the same retained gauge branch, not as anew microscopic law. Asymptotic-freedom-type statements therefore hold only inside thedeclared perturbative/sign gates.When the Standard Model gate package is further declared, the renormalizable localoperator 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 Modelgauge group, anomaly package, matter content, or phenomenology from retained readabilityalone. The paper therefore separates: the gauge-redundancy slot derived from the Part Isummary interface; Yang–Mills/RG as declared gauge/scale readouts; and Standard-Modeldata as compatibility information inside a declared instance.