Theorem: Classical closure and arbitrary finite words

The space ${\mathcal{P}}_{\Sigma }$ is a unital commutative algebra under pointwise multiplication. It is closed under every smooth directional derivative and under complex conjugation.

Let ${\mathcal{P}}_{\mathrm{gen}}$ be the pointwise algebra generated by ${\mathcal{P}}_{\mathrm{mc}}$, all $B_f$, and all $\mathsf{J}(a)$. Then

and ${\mathcal{P}}_{\mathrm{gen}}$ is itself stable under all smooth directional derivatives and complex conjugation. There is no bound on the number of factors in a word.

Proof. For $F,G\in {\mathcal{P}}_{\Sigma }$, apply the iterated directional Leibniz rule, and then take a metric derivative. Every term is a complex scalar value of a directional derivative of one factor multiplied by a first metric derivative of a directional derivative of the other. The latter belongs to ${\mathcal{S}}_1$. Since ${\mathcal{S}}_1$ is a vector space, (eq:admission) holds for $FG$.

Directional stability follows because a further direction simply adds an entry to $\mathbf{\eta }$. Conjugation follows from reality of $L$ and the conjugation invariance of the wavefront condition defining ${\mathcal{M}}_1$.

For ${\mathcal{P}}_{\mathrm{gen}}$, derivatives of bulk generators remain bulk. Proposition prop:generators expresses derivatives of sharp generators in terms of bulk scalar linear functionals, coordinate generators, and constants. The ordinary Leibniz rule then proves closure for every finite word.