The old bulk domain

Let ${\mathcal{P}}_{\mathrm{mc}}\subset {\mathcal{P}}_{\mathrm{dist}}$ be the finite polynomial domain whose kernels satisfy

where the closed nonspacelike cotangent cones include zero in each slot and the total zero covector is omitted.

Let ${\mathcal{P}}_{\mathrm{reg}}$ be the subspace with smooth compact kernels.

Only finite polynomials are used here. No closure theorem for arbitrary smooth microcausal functionals is assumed. The distinction between fixed-kernel polynomial classes and general functional classes is relevant to the broader pAQFT domain problem HRV.