Definition: Complete-source decorated coefficients

For $A\in \{C_2,M_2,J_3\}$ and constant symmetric tensors $X,Y$, write

for the $(a,b)$-bihomogeneous piece of ${{\mathfrak{s}}_{\mathfrak{A}}}^{\mu \nu }[X,Y](y)$ (Proposition prop:jet). For multi-indices $a,b$ in the support of $P_A$ and vertex multi-indices with $|p|+|q|\le 2$, put

(The jet orders $a,b$ are spent on the edge kernels by Lemma lem:Lsign; the dual-slot tensors are constant, as in XVI-K, Definition def:a.) Set

An unordered numerical label is $\lambda =\{\alpha ,\beta \}$ with $\alpha =a+p$, $\beta =b+q$, and ${q^{\circ }}_{\lambda }:={q^{\circ }}_{\alpha \beta }$. The corresponding coefficient ${a^{A\mu \nu }}_{\lambda }$ is obtained by accumulating every ordered contribution with $\{a+p,b+q\}=\lambda$ (no universal combinatorial factor $2$ is inserted for $\alpha \ne \beta$; when both orderings contribute equally the factor $2$ appears by summation, as in XVI-K, eq:ak). Write $\lambda =(A;a,b;p,q)$ for an ordered placement when the sector and both ends are kept labelled.