Proposition: Typing and admission

The vertex belongs to ${\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$, is even and has ghost number zero, and has functional support contained in $\mathrm{supp}g$. Its derivatives are

with

Proof. The cubic polarized kernel is a finite sum of derivatives of three-field diagonal distributions, multiplied by $g$ and constant fiber tensors. Its support is compact and its wavefront covectors sum to zero. A nonzero tuple wholly in either pointed closed causal cone cannot have zero sum. Hence the kernel is microcausal.

The fields entering the vertex are even and have ghost number zero. Variation of the displayed density gives the metric derivative. Compact integration by parts in the scalar variation gives the second formula, retaining derivatives of $g$.