Abstract

This document closes four scope items left as statements in Landmark X-B, where the free quantum field is the scalar $\varphi$ and the metric coefficients $h,k$ are classical off-shell backgrounds. First, a scope proposition records that every object constructed in X-B is a field-degree-one element of the free algebra with polynomial classical background dependence and no $\hslash$; the non-goals are listed. Second, the one-contraction check is placed on its exact boundary: the auxiliary kernel ${\Delta }_{\mathrm{F}}={\Delta }_{+}-i{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}$ is proved symmetric, the check is re-derived, the pair passes the restricted causal-factorization and symmetry requirements, and the pointwise square of ${\Delta }_{\mathrm{F}}$ is proved to fail the wavefront product criterion at every diagonal point, so the tools of the sheet define no second pair and no time-ordered family. Third, the normalization sheet is recorded on the whole off-shell insertion domain and proved to determine the single map $R_1(V_{1,g}[h];\cdot )$: a degree-one class in the free algebra vanishes if and only if it is annihilated by every spatially compact smooth solution, hence any contact modification compatible with the classical normalization is zero in the free algebra, and no contact can repair or break the first-order identity. Fourth, the data that an independent Branch B would require are listed, with a non-relabeling rule; no existence, failure, or correction is asserted for that branch. Nothing is constructed beyond X-B.