Proposition: Evaluation descends to the action quotient

The response map

is well defined. Its image is precisely

Proof. The differentiated coefficient of a total derivative has an overall factor $i{\sum }_j{k}_{j\mu }$, which vanishes on $\mathcal{X}$. A genuine tensor identity vanishes before and after differentiation. Thus equivalent density representatives have identical responses.

Raw coverage gives every allowed response as a constant combination of the $n_{I,P}$, and every such combination comes from allowed raw contacts.