Theorem: Classical closure

The domain ${\mathcal{P}}_{\Sigma }$ is a unital commutative associative algebra under classical pointwise multiplication. It is stable under smooth directional differentiation and conjugation.

The subalgebra ${\mathcal{P}}_{\mathrm{gen}}$ generated by ${\mathcal{P}}_{\mathrm{mc}}$, all $B_f$, and all $J(a)$ is contained in ${\mathcal{P}}_{\Sigma }$ and has the same differential and conjugation stability. There is no bound on the finite number of factors in a pointwise word.

Proof. The coefficients of a product are finite sums of symmetrized exterior tensor products, so they remain compact distributions. For a direction list, the Leibniz rule expresses $D_{\mathbf{\eta }}(FG)$ as a finite sum $(D_{I}F)(D_{J}G)$. Its metric derivative at a fixed configuration is

The coefficients here are complex numbers, and the distributional factors belong to ${\mathcal{S}}_1$. Thus (eq:admission) holds for the product.

An additional direction adds an entry to the defining list. Conjugation preserves ${\mathcal{M}}_1$, ${\mathcal{Q}}_{\Sigma }$, and their trace. Finally, (eq:descendants) expresses derivatives of the sharp generators through coordinate generators, regular scalar linear functionals, and constants. The finite Leibniz rule therefore also preserves ${\mathcal{P}}_{\mathrm{gen}}$.