Corrections to the proposed continuation

three readings of the X-E defect. X-F's exact identity

must replace the on-shell shortcut everywhere. A free gauge-fixed metric need not satisfy $G_{\mathrm{lin}}[u]=0$. The full background gauge correction packages both terms into (eq:commutator).

strength of “Moller response”. A retarded solution map for a prescribed equation and a variational Moller map are not synonyms. Theorem thm:obstruction separates them for the frozen source law. The direct tree identity can remain valid without the stronger variational identification.

graph-order correction. In the declared non-tadpole normal convention, background-dependent metric one-point loops first feed the relational scalar at ${\kappa }^3$, not ${\kappa }^2$. The order-${\kappa }^2\hslash$ scalar sector contains the two mixed contraction types of Proposition prop:inventory. This is a scheme-qualified statement.

scalar completion must be counted once. Applying the full scalar Bogoliubov map to already completed $d_1,d_2$ regenerates scalar corrections. A complete calculation must begin from a bare insertion or explicitly subtract the duplicated scalar response.

completed support bound. The $g{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}f$ local terms extend the scalar slot beyond $\mathrm{supp}f$. The corrected compact set is $O_f$ in (eq:correctsets).

Hadamard sign. No $+4$ replacement is made. The action-Hessian, equation, causal, and product-kernel conventions are distinguished in Section sec:free. A uniform joint Hadamard orientation must be checked, not inferred from a coefficient's sign.

wavefront logical direction. Failure of a sufficient product criterion neither constructs a residue nor proves nonexistence. No reflected light-ray residue is identified here.

formal constraint-ideal qualification. Suppose a constraint commutator gives $i\hslash a1$ with $a\ne 0$. A two-sided ideal containing that constraint then contains $\hslash 1$. Over $\mathbb{C}[[\hslash ]]$ this does not by itself imply it contains $1$. At fixed nonzero $\hslash$, or after adjoining ${\hslash }^{-1}$, the conclusion changes. No extra de Donder constraint quotient is used here.

matter support correction. The unrestricted off-shell breaking includes $(1-g)(P_m\varphi )\partial \varphi$. Its leading free-shell coefficient is supported where $g$ changes. This does not justify an unevaluated observable Ward-defect claim.

ordering data at quantum order. Unlike the classical tree coefficient, the loop contractions use the frozen Hadamard/Feynman product data. Such data are not a chosen state. Their transport and finite-ordering changes must be tracked, rather than declared irrelevant.