Lemma: Three distinct attachment steps

Under Assumption ass:root, for every $\mathsf{k}\in \mathcal{N}$:

1. The pullbacks exist and satisfy

   $$\begin{array}{rl}\mathrm{WF}({\pi }^{\ast }{B}_{f,g,\mu \nu })& \subset \{(y,x,w;\xi ,-\xi ,0):\xi \ne 0\},\\ \mathrm{WF}(r^{\ast }{\stackrel{ˉ}{q^{{\mathfrak{R}}_{\mathrm{K}}}}}_{\mathsf{k}})& \subset \{(y,x,w;\kappa ,0,-\kappa ):\kappa \ne 0\}.\end{array}$$

2. The product

   $$T_{\mathsf{k},\mu \nu }:={\pi }^{\ast }{B}_{f,g,\mu \nu }{r}^{\ast }{\stackrel{ˉ}{q^{{\mathfrak{R}}_{\mathrm{K}}}}}_{\mathsf{k}}$$

   exists on all $X$, including $y=w$. There is no interaction switch in $T_{\mathsf{k},\mu \nu }$.

3. For smooth $h$,

   $${U^{{\mathfrak{R}}_{\mathrm{K}}}}_{\mathrm{K}}(h)=\sum _{\mathsf{k}\in \mathcal{N}}\sum _{\mu ,\nu }{a^{\mu \nu }}_{\mathsf{k}}(y,w;h)T_{\mathsf{k},\mu \nu }$$

   has support contained in the fixed compact set

   $$K_U=\{(y,x,w):(y,x)\in \mathrm{supp}{B}_{f,g},w\in \mathrm{supp}{\chi }_{\mathrm{I}}\}.$$

   Consequently ${\pi }_x$ is proper on its support and $({\pi }_x{)}_{\ast }{U^{{\mathfrak{R}}_{\mathrm{K}}}}_{\mathrm{K}}(h)\in {\mathcal{E}}^{\prime }(M)$ is defined.

The pullback and product conclusions hold for any distribution in the relative variable, not just the selected extensions.

Proof. Pullbacks. The maps $\pi ,r$ are linear submersions. Hörmander's pullback theorem applies without a conormal obstruction Hormander (Theorem 8.2.4). The root bound (R3) lifts $(\xi ,-\xi )$ to $(\xi ,-\xi ,0)$. The relative pullback lifts every nonzero covector $\kappa$ to $(\kappa ,0,-\kappa )$. These shapes remain valid at coincidence.

Product. A sum of the two lifted shapes is

If it vanishes, its $x$- and $w$-components force $\xi =\kappa =0$. Thus no opposing nonzero wavefront covectors exist, and the product criterion applies Hormander (Theorem 8.2.10). This step uses the imported wavefront premise, not compactness.

Support and properness. The coefficients $a_{\mathsf{k}}(h)$ are smooth and contain one factor ${\chi }_{\mathrm{I}}(w)$. Their multiplication with the defined products and finite index contraction give (eq:UK). Root compactness (R2) bounds $(y,x)$, and the interaction switch bounds $w$, giving (eq:support). Compactness makes ${\pi }_x$ proper on the support and permits the distributional pushforward.