Scope (XVI-L).

(XVI-L).{ Ambient $M={\mathbb{R}}^4$, $\eta =\mathrm{diag}(1,-1,-1,-1)$, $\mathcal{P}r=r-\frac{1}{2}\eta {\mathrm{tr}}_{\eta }r$, ${\mathcal{P}}^2=\mathrm{id}$; fixed $0<\epsilon <T$; fixed preparation $f$, $g$, ${\sigma }_{\epsilon }$ (not varied); an independent compact interaction switch ${\chi }_{\mathrm{I}}\in {C_c}^{\infty }(M)$; off-shell smooth residual metric $h$; smooth scalar test $\varphi$. The pairing is $⟨A,C{⟩}_{\eta }=A_{\mu \nu }{C}^{\mu \nu }$ and the metric-dual basis is $B^i=(G^{-1}{)}^{ii}{B}_i$, $G=\mathrm{diag}(1,-2,-2,-2,1,2,2,1,2,1)$, exactly as in XVI-K. The root is the imported $B_{f,g}$ of XII-D.tex, eq:B-kernel, used only through (R1)–(R3) of the XVI-K hypothesis box (XVI-K.tex, Section sec:hyp): (R1) the composition exists; (R2) compact support; (R3) $\mathrm{WF}(B_{f,g})\subset Z_2=\{(x_1,p_1;x_2,p_2):p_1+p_2=0\}\setminus 0$.

Change relative to XVI-K. The selected source $2{C_2}^{\mu \nu \alpha \beta }{\partial }_{\alpha }{\partial }_{\beta }{\sigma }_{\epsilon }$ is replaced by the complete imported quadratic cubic source of XII-C.tex, eq:actualsource,

and all three summands are to be contracted with the full cubic Hessian ${L^{pq}}_{ij}$ of XVI-K, eq:Lpq.

Name of the object. The target object is called the complete-source contribution for the specified two-edge graph. It is not the complete one-loop gravitational correction: completing the insertion source does not complete the diagrammatic content of the theory.

Levels kept distinct. (1) source and vertex differential operators; (2) scalar numerical distributions; (3) their selected extensions; (4) unrooted collision counterterms; (5) propagated root-attached ambiguities; (6) scale variation; (7) gravitational symmetry compatibility. A result at one level is not a result at another. In this draft levels (1)–(6) are treated as constructions (L.1–L.7): source jet, graph decomposition, numerical-kernel span, coherent extension, unrooted collision counterterms, scale variation, and root-attached propagated ambiguities; level (7) (gravitational symmetry compatibility / matching) is recorded as matching questions only (L.8), with L-D not claimed. Not claimed: BRST, BV, Ward identities, all-orders renormalization, $\epsilon \to 0$, a state, an ultraviolet completion, a physical beta function for a complete gravitational quantity, or any proved match of collision counterterms to gravitational Wilson coefficients. }