Remark: Pointwise jets, switch derivatives, and locality

Write $N_{\mathsf{k}}=|p|+|q|$. The derivative ${{\partial }_w}^{\gamma }$ in (eq:deltaCK) acts on the interaction switch and the residual coefficient ${L^{pq}}_{ij}(w;h)$, but not on $\Sigma (y)$. Its residual jet order is at most

Switch derivatives of order at most $N_{\mathsf{k}}$ may appear. This pointwise jet bound is not a bound on the order of the root distributional pairing.

The contact terms are propagated through $B_{f,g}$. They are not identified with variations of local bulk gravitational counterterms. Locality or nonlocality of the resulting functional is not established.

For $\mathsf{k}=\{0,0\}$, only $\gamma =0$ occurs and (eq:deltaCK) reduces exactly to the propagated $c\delta$ ambiguity of XVI-J. The derivative defects of Lemma lem:defect are fixed by the selected prescription; they are not additional freely chosen parameters.