Proposition: Zero-charge comparison classes

At every smooth configuration and after every finite smooth directional insertion,

for

Proof. For the partial contractions, the inherited metric derivative is compact smooth. Inserting a scalar direction replaces the smooth scalar configuration by that direction. Inserting a metric direction removes the only metric slot, so subsequent metric differentiation is zero. Higher derivatives vanish by polynomial degree. All nonzero metric kernels in this calculation are therefore compact smooth elements of ${\mathcal{M}}_1$, with zero charge.

For $F\in {\mathcal{P}}_{\mathrm{mc}}$, contraction of its higher functional derivatives against smooth directions is a proper pushforward after inserting compact cutoffs equal to one on the coefficient supports. A wavefront covector of the remaining one-slot kernel can arise only from a joint covector with all integrated covectors zero.

A nonzero causal covector in the remaining slot, together with those zeros, would belong entirely to one of the excluded closed causal cones. Thus the remaining metric kernel has only spacelike wavefront covectors. It belongs to ${\mathcal{S}}_1$ by carrier admission, and Lemma lem:MCcharge proves its charge is zero.