Abstract

We distinguish the admitted classical prepared insertion from a reduced graph sector involving that insertion and one compactly switched cubic interaction. The supplied XII-D admission excerpt concerns the same polarized kernel represented in revised XVI-H. Under that additional inherited premise, direct microcausal membership and a nonempty free-shell microcausal class follow. This does not repair the different frozen class excluded by XI-G.

For the graph calculation, we choose an ambient massless time-ordered graviton kernel and identify its restriction with the combination $H+\mathrm{i}{G}^{\mathrm{A}}$ in the stated normalization. Its off-diagonal wavefront relation proves that the two differentiated edges can be multiplied. The cubic Hessian and the insertion source jets then define the reduced graph without an unspecified tensor expansion. An explicit spacelike evaluation proves nonvanishing of the open-index vertex block when the interaction switch is nonzero; no nonvanishing of an arbitrary source-contracted or root-attached sum is inferred.

The relative-coordinate building blocks have finite scaling degree. Including both insertion and interaction derivatives gives an upper bound of eight in four relative dimensions. Extensions exist, but their local coefficients are not normalized. Under the explicit root wavefront bound, the compressed root attachment satisfies the distributional product criterion and has proper support. It therefore yields a family of smeared coefficients indexed by the extension choices, not a selected interacting quantum observable. BRST/BV closure and the retarded interacting coefficient remain unestablished.