Proposition: Admission of the insertion and tree

The kernels of ${\mathcal{B}}_f$ and the polarized cubic $\mathcal{T}$ belong to the polynomial domain of Definition def:domain. Their free-shell classes therefore define elements

Their leading symbols are the supplied classical functionals restricted to ${\mathcal{C}}_{\mathrm{sol}}$.

Proof. The wave kernel has opposite endpoint covectors, including its diagonal conormals. Smooth multiplication and differentiation give

No nonzero opposite pair lies entirely in either causal cone.

For the tree, compose

Before integrating, the two kernels share only $z$. A violation of the multiplication criterion would force the unshared $x$ and $y$ covectors to vanish, and then force the shared covector to vanish. The product is defined. Its pushforward is proper in the compact causal region selected by $f$, $g$, and the finite slab.

Attach the stress kernel at $y$ with its two scalar slots, and the final scalar slot at $x$. At each attachment the local diagonal conormal has total covector sum zero. The same one-shared-slot argument applies. Every resulting external triple satisfies

in the fixed Cartesian trivialization. A nonzero triple entirely in either closed causal cone cannot satisfy that identity. Polarization and slot symmetrization preserve the bound.

Lower derivatives are obtained by smearing slots with smooth configurations and obey the corresponding exclusion. Ghost slots introduced by the linear free rule replace a metric slot by a differentiated ghost slot; they do not enlarge the wavefront set. Thus the graded polynomial domain also covers the free variation of ${\mathcal{B}}_f$.