The mixed loop and the comparison checklist

The insertion is bilinear, while the vertex is cubic. After one graviton cross-contraction, a scalar cross-contraction is still available. Two contractions consume four of the five legs and leave a scalar-linear coefficient. With the retarded prefactor, this is a potential ${\hslash }^1{\kappa }^2$ contribution.

A schematic term involves

with the appropriate stress derivatives and mass terms. Here $H_m$ denotes a not selected scalar Hadamard bisolution. A full retarded-product construction would also have to specify its covariance matrix and ordering conventions. XI-B supplied none.

We do not evaluate this expression. Its off-diagonal wavefront compatibility and its extension near observation-vertex coincidence require separate analysis. No divergence, subtraction, or unique diagonal extension is inferred from the drawing. Missing Hadamard data are a stop for this loop, not a no-go for Theorem thm:tree.

In particular, tree-exactness cannot be imported from X-C-v3, whose insertions were scalar-linear. The present one-line coefficient is designated a tree because it is the classical retarded coefficient. It is not a claim that the full response of $V_{1,g}$ and ${\mathcal{B}}_f$ has no positive-$\hslash$ terms.

There is also a distinct one-scalar-line classical response of this pair, with metric fields remaining. It is not the single graviton component defined by (eq:physicalsign). Equation (eq:A2sc) names that scalar sector for the background comparison; one graviton component alone cannot establish a complete second-order Ward theorem.