The old polynomial domain

Let ${\mathcal{P}}_{\mathrm{mc}}$ denote finite polynomial functionals with compact distributional kernels satisfying the usual microcausal exclusion: for a kernel of degree $n$,

with the total zero covector omitted. Let ${\mathcal{P}}_{\mathrm{reg}}\subset {\mathcal{P}}_{\mathrm{mc}}$ denote the finite polynomials with smooth compact kernels.

Only finite polynomials are used. Results about unrestricted smooth microcausal functionals are not presumed. In particular, the distinction between polynomial and general functional domains identified by Hawkins, Rejzner, and Visser is respected HRV. Their generalized-cone framework motivates a cone test, but supplies no premise asserting that $B_f$ passes it.