Lemma: Tree composition

Under (S), the product $\Pi ={\prod }_{a\in V}{w}_a(x_a){\prod }_{\ell \in E(T)}{k}_{\ell }(x_a,x_b)$ exists in ${\mathcal{D}}^{\prime }(M^V)$ and is compactly supported. Every point of $\mathrm{WF}(\Pi )$ has covectors ${\zeta }_a={\sum }_{\ell \ni a}{ϵ}_{\ell ,a}{p}_{\ell }$ for some $p_{\ell }\in {\mathbb{R}}^4$, where ${ϵ}_{\ell ,a}=\pm 1$ according to the orientation of $\ell$ at $a$. The pushforward $t={\pi }_{X\ast }\Pi \in {\mathcal{E}}^{\prime }(M^X)$ satisfies

Proof. Pull each $k_{\ell }$ back to $M^V$ along the projection to $(x_a,x_b)$; its wavefront set consists of tuples with $(p,-p)$ at the endpoints of $\ell$ and zero elsewhere Hormander (Theorem 8.2.4). Order the lines ${\ell }_1,\dots ,{\ell }_N$ and let ${\Pi }_j$ be the product of the weights and the first $j$ lines. Claim: $\mathrm{WF}({\Pi }_j)\subset {\Sigma }_j$, the set of nonzero tuples of the form ${\zeta }_a={\sum }_{i\le j,{\ell }_i\ni a}{ϵ}_{{\ell }_i,a}{p}_{{\ell }_i}$. For $j=0$ this is empty. Suppose it holds for $j$. The product ${\Pi }_j\cdot k_{{\ell }_{j+1}}$ is defined if there is no $(x,\zeta )\in \mathrm{WF}({\Pi }_j)$ with $(x,-\zeta )\in \mathrm{WF}(k_{{\ell }_{j+1}})$ Hormander (Theorem 8.2.10). Such a pair would give an assignment $(p_{{\ell }_i}{)}_{i\le j+1}$, with $p_{{\ell }_{j+1}}=q\ne 0$ the line covector, satisfying ${\sum }_{i\le j+1,{\ell }_i\ni a}{ϵ}_{{\ell }_i,a}{p}_{{\ell }_i}=0$ at every vertex $a$. The subgraph formed by ${\ell }_1,\dots ,{\ell }_{j+1}$ is a forest. In a forest with at least one line there is a vertex of degree one; its unique line has zero covector; remove it and repeat. Hence all $p_{{\ell }_i}=0$, contradicting $q\ne 0$. So the product exists and, by the same theorem, its wavefront set lies in the sums of the two wavefront sets with zero allowed, which is ${\Sigma }_{j+1}$. Smooth weights do not enlarge wavefront sets. This proves the claim for $\Pi ={\Pi }_N$ and the covector description.

The support of $\Pi$ lies in $S$, compact by (S), so the projection to $M^X$ is proper on it and $\mathrm{WF}(t)\subset \{(x_X,{\zeta }_X)\ne 0:\exists x_I,(x_X,{\zeta }_X;x_I,0)\in \mathrm{WF}(\Pi )\}$ Hormander (Theorem 8.2.12). A point of $\mathrm{WF}(\Pi )$ with zero covectors at all internal vertices is a line assignment balanced at every internal vertex. Then ${\sum }_{a\in X}{\zeta }_a={\sum }_{a\in V}{\zeta }_a-{\sum }_{a\in I}{\zeta }_a={\sum }_{\ell }(p_{\ell }-p_{\ell })-0=0$. Hence $\mathrm{WF}(t)\subset Z_n$.