Theorem: Pullback, product, and proper pushforward

Under the root hypotheses (eq:rootinput):

1. The pullbacks $p^{\ast }{B}_{f,g,\mu \nu }$ and $r^{\ast }{\bar{q}}^{\mathfrak{R}}$ exist.

2. Their products exist on all $X$, including $y=w$.

3. For smooth $h$, the distribution

   $$U^{\mathfrak{R}}(h)=\sum _{\mu ,\nu }{p}^{\ast }{B}_{f,g,\mu \nu }{r}^{\ast }{\bar{q}}^{\mathfrak{R}}{\chi }_{\mathrm{I}}(w){\mathcal{K}}^{\mu \nu }(y,w;h)$$

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

Proof. Both maps $p,r$ are submersions, so their pullbacks exist for every distribution on their respective target spaces.

The lifted root covectors and relative-kernel covectors have the respective forms

For the root, $\xi \ne 0$ follows from its $Z_2$ bound. For the relative kernel, $\kappa \ne 0$, with no restriction on its orientation needed. Their sum is

Vanishing of the $x$- and $w$-components forces $\xi =\kappa =0$, impossible for opposing nonzero wavefront covectors. Hörmander's product criterion therefore applies, including at coincidence. This argument uses the root wavefront premise, not its compactness.

The tensor coefficient is smooth. Multiplication by it and finite component contraction preserve existence. The support is contained in

which is compact. Compactness in $(y,x)$ comes from the root; compactness in $w$ comes from the interaction switch. The projection is therefore proper on the support, proving the pushforward claim.