Manuscript X-E@1 · X-E.tex · section · 293fae6a

Match table

E5: VII-C audit with a quantum graviton present › Match table · lines 1671–1720

{}1.14

Item	VII-C requires	X-E constructs
(E1)	The de Donder multiplet with $H_{hh} = P Δ_{0}^{+}$ as the fixed Wick matrix	A free gauge-fixed graviton with Hadamard form $H_{gr} = - 4 P \otimes H_{0}$ in the opposite signature, used only to define Wick ordering of graviton-quadratic bare terms. $P$ and $P$ agree under the dictionary; $H_{0}$ and $Δ_{0}^{+}$ are not identified, and no computed quantity uses either.
(E2)	Admission of the sharp zero-Cauchy insertion $Q_{Σ} (B_{f})$	The ramp insertion at fixed $ε$ . Its kernel is $Λ$ of (eq:D1), smooth-coefficient and admitted by X-B. The sharp limit is not taken; X-D's limit concerns classical backgrounds only.
(E3)	Normal-topology convergence of mollified sharp kernels	No approximating family. The regulator is fixed.
(E4)	Hypocontinuous product and extraction of the fully contracted $ℏ^{3}$ coefficient of $B_{f} ⋆ W_{g} ⋆ B_{f^{'}}$	One $⋆$ -contraction along one graviton line, at $ℏ^{0}$ , between $A_{1} [h]$ and $V_{1, g} [h, ϕ]$ . The vertex $V_{1, g} = - \frac{1}{2} \int g h_{μν} T^{μν}$ is linear in the graviton; it is not $W_{g} = \int g h_{00}^{2}$ . The graviton-quadratic terms $Q [h, h]$ are Wick ordered, not $⋆$ -multiplied, and no $ℏ^{3}$ coefficient is computed.

VII-C's theorem excludes the named extension with the stated admission and product requirements. The one-tree algebra of this manuscript is a smaller object: fixed regulator, $ℏ^{0}$ , one line, no $W_{g}$ . VII-C therefore does not apply and is not contradicted. Its reflected light-ray residue is not subtracted because no loop coefficient is formed; the mixed loop of Section sec:named is the first place such a coefficient would appear, and it is not constructed.