Lemma: Bulk descendants and classical products

Every fixed smooth directional derivative of a bulk polynomial is bulk. At each smooth configuration, its first metric derivative has no nonzero nonspacelike wavefront covector.

The classical pointwise product of two bulk polynomials is bulk.

Proof. Directional differentiation and evaluation contract slots with smooth fields. A forbidden nonspacelike tuple in the remaining slots would give a forbidden tuple of the original kernel by adjoining zero covectors in contracted slots. The convention in (eq:microcausal) therefore gives the descendant statement.

Pointwise products use symmetrized exterior tensor products. An all-future or all-past nonzero wavefront tuple of such a tensor product would already violate the condition in one nonzero factor.