Local ambiguities can have nonlocal responses

A useful classical identity makes one potential shortcut unavailable. Let ${\mathsf{D}}_s=\mathsf{D}+s\delta \mathsf{D}$ have the same boundary prescription and let ${\mathsf{G}}_s$ be its zero-data inverse. Differentiating ${\mathsf{D}}_s{\mathsf{G}}_s=\mathrm{id}$ gives

The solution of the differentiated equation has zero data, which justifies the inverse on the left.

Thus a local change of an equation can produce a propagated, nonlocal response. Disjoint observation supports do not by themselves remove all action-counterterm responses.

The effective-field-theory distinction between analytic local terms and nonanalytic long-distance loop contributions is a valuable guide Donoghue, but it does not establish that every nonlocal smeared number is normalization independent. A separator must annihilate the complete propagated action-and-composite ambiguity space of the chosen response.