Proposition: Complete-source two-edge kernel off the diagonal

On $y\ne w$,

Equivalently, organizing by unordered numerical labels $\lambda$,

with $a_{\lambda }={\sum }_A{a^A}_{\lambda }$. The overall combinatorial factor is $1$ by Proposition prop:Lnorm. Restricting to $A=C_2$ and $a=b=0$ recovers XVI-K, Proposition prop:QK: ${Q^{\circ \mu \nu }}_{\mathrm{K}}={\sum }_{|p|+|q|\le 2}{a^{\mu \nu }}_{pq}{q^{\circ }}_{pq}$.

Proof. Insert the two edge slot fields, carrying source-end jets of orders $a,b$ and vertex-end jets of orders $p,q$, into ${\chi }_{\mathrm{I}}(w)D^2{\ell }_3(h)[\cdot ,\cdot ](w)$ via XVI-K, eq:Lpq, and contract the source ends by ${\mathfrak{s}}_A$ via (eq:jet). Lemma lem:Lsign converts the derivatives into $(-1{)}^{|p|+|q|}{q^{\circ }}_{a+p,b+q}(y-w)$. The dual-slot normalization is $-\mathcal{P}{B}^i$, $-\mathcal{P}{B}^j$ as in XVI-K, Definition def:a / XVI-J, eq:Kdef. Summing over the three source summands and over all placements allowed by Theorem thm:budget and XVI-K, Proposition prop:hess, gives (eq:QL). The $C_2$, $a=b=0$ truncation is Definition def:La with $P_{C_2}$ equal to XVI-K's polarized source (Proposition prop:jet(6)), hence XVI-K, eq:QK.