Manuscript X-F@1 · X-F.tex · section · f35cc129

F6: VII-C audit with three trees

F6: VII-C audit with three trees · lines 1351–1400

X-E's audit covered one tree. The present sheet has three, one of them graviton-quadratic. The assumption of VII-C is copied as in X-E and not repeated; the match table is redone.

{}1.14

Item	VII-C requires	X-F constructs
(E1)	The de Donder multiplet with $H_{hh} = P Δ_{0}^{+}$ as the fixed Wick matrix	The flat gauge-fixed graviton with $H_{gr} = - 4 P \otimes H_{0}$ , opposite signature, used to define Wick ordering of $Q [h, h]$ and $T_{self}$ . The background gauge enters through the interaction $Q^{(1)} [h_{cl}]$ , not through the free algebra. No computed quantity uses $H_{gr}$ .
(E2)	Admission of the sharp zero-Cauchy insertion	Ramp insertions at fixed $ε$ : $D_{1}$ , $M$ , $T_{mat}$ , $T_{self}$ , all admitted by the sum-zero lemma. No sharp limit.
(E3)	Normal-topology convergence of mollified kernels	No approximating family; the regulator is fixed. The only limit used is the density of finite tensor sums in Hörmander's topology, Lemma lem:gslot, at fixed $ε$ .
(E4)	Hypocontinuous product; extraction of the $ℏ^{3}$ coefficient of $B_{f} ⋆ W_{g} ⋆ B_{f^{'}}$	Three $ℏ^{0}$ trees. Each is one $⋆$ -contraction along one graviton line between $A_{1} [h]$ and an order- $κ$ vertex linear in that $h$ : the matter vertex $V_{1, g}$ , the background vertex $\int σ_{ε} h Q^{(1)} [h_{cl}] h$ (linear in each of its two $h$ ), and the cubic vertex. No $W_{g}$ , no $ℏ^{3}$ coefficient, no loop.

VII-C does not apply and is not contradicted. The first object on which it could bear is the mixed loop of X-E's named list, which is not constructed here.