Abstract

At fixed regulator width, we study the selected pair consisting of the switched metric–scalar interaction functional and the already admitted second-order composite. We prove admission of the vertex and construct the classical binary retarded coefficient. A balanced kernel calculation proves the joint one-line composition, properness, compact support, and microcausal admission. Its scalar branch is the inherited switched scalar response; its metric branch has the inherited inverse coefficient and agrees with the first-completion root normalization wherever that branch is used.

Two corrections are indispensable. Causal factorization is a normalization of the binary time-ordered datum, not the assertion that the retarded coefficient equals the ordinary quantum product. The latter assertion is incompatible with the stated retarded definition and field-degree normalization. Moreover, retarded support concerns the contracted attachment variables, not every uncontracted variable of a nonlocal composite.

The coefficient linear in $\hslash$ requires a joint two-line extension. We freeze a conservative total-diagonal counterterm class and identify the additional internal collision problem, but do not claim an extension that has not been constructed. The corrected quantum release therefore remains Unresolved at that gate. We give precise conditional admission and factorization statements, explicit classical zero-evaluation subclasses, and the free-differential display. No restriction quotient or gauge-consistency defect is formed.