Proposition: Graph factor for the complete source

The combinatorial graph factor of the specified two-edge graph, with the complete source ${\mathcal{J}}_{\mathrm{D}}$ at $y$ and the full cubic vertex at $w$, is

It carries no power of $\kappa$ or $\hslash$. Coupling and loop weights are not determined in this note and are kept separate from (eq:L-factor).

Proof. Re-derive from the same first principles as XVI-I, without transferring XVI-K's factor as a black box.

1. Two identical graviton edges. The sector joins the two tensor slots of $F_t$ to one compactly switched copy of ${\ell }_3$ by two graviton edges, each carrying $D_F$ (XVI-I, Proposition prop:sector). Fredenhagen–Rejzner, arXiv:1208.1428v2, equation (82), place $l_{ij}!$ in the denominator for $l_{ij}$ lines between the same pair of vertices; here $l_{ij}=2$, so the coefficient is $1/2!$, declared once (XVI-I, Proposition prop:signs(6)).

2. {Second variation of $F_t$.} ${D_h}^2{F}_t$ contributes the factor $2$ (XVI-I, eq:Dh): the count of the two identical tensor slots, not a hand-inserted normalization.

3. Independence of the source completion. Replacing the selected source $2C_2\partial \partial {\sigma }_{\epsilon }$ by the complete ${\mathcal{J}}_{\mathrm{D}}={\mathfrak{s}}_{C_2}+{\mathfrak{s}}_{M_2}+{\mathfrak{s}}_{J_3}$ changes only the source-end contraction at $y$. It does not change the number of edges, the identity of the two edges, or the second tensor derivative of $F_t$. Hence neither factor in (eq:L-factor) is altered.

4. The product is $1$, as in XVI-I, Proposition prop:sector, and as used for the $C_2$ sector in XVI-K, Proposition prop:QK.

No coupling constant and no loop weight enters this count; if such weights are fixed later they multiply the whole amplitude and do not redefine (eq:L-factor).