Theorem: Compressed attachment under the root cone input

Under (eq:Bcone), condition (eq:root-condition) holds for every distribution ${\bar{U}}_A[h]$ constructed above, regardless of its detailed wavefront set.

The product $\mathcal{G}[h]$ is compactly supported. Its pushforward $R[h]\in {\mathcal{E}}^{\prime }(M_w\times M_x)$ is defined. The same construction works on the off-diagonal domain before extension.

No normalized extension is thereby selected.

Proof. The lifted root wavefront covectors have the form

by (eq:Bcone). The lifted reduced distribution has covectors

with $(k,\ell )\ne 0$. For these two covectors to cancel, the $x$-component would force $p=0$, contrary to the nonzero root covector. Thus the product criterion is satisfied.

The root is compact in $(y,x)$. Every reduced term contains the interaction switch ${\chi }_{\mathrm{I}}(w)$, including the diagonal extension terms. Hence its $w$-support is contained in $\mathrm{supp}{\chi }_{\mathrm{I}}$. The product support is a closed subset of the compact set

The projection $\pi$ is therefore proper on that support, and the pushforward is defined.

Neither argument requires the root to be smooth or the reduced graph to be smooth at a null-related pair.