Definition: Polynomial microcausal domain

${\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$ consists of finite graded polynomials $F(\Phi )={\sum }_n⟨u_n,{\Phi }^{\otimes n}⟩$ on ambient configurations on $M$, with compact, graded symmetric distributional kernels $u_n$ on $M^n$ satisfying

where a total zero covector is never a wavefront point and finite field indices do not change the criterion. The free shell ${\mathcal{C}}_{\mathrm{sol}}$ is

on $M$; smooth slab solutions extend to $M$ by their Cauchy data. Admission is $Q_H(F)=[F{|}_{{\mathcal{C}}_{\mathrm{sol}}}]\in {\mathcal{E}}_H$. XI-E admitted ${\mathcal{B}}_f$ and $\mathcal{T}$; it did not admit ${\mathcal{C}}_3$.