Compact smearing and locality

The support of $f$ is the support of a function of relational coordinate labels. The metric dependence of the observable includes the coordinate inverses and their initial-data prescription. Compact support of $f$ is therefore not, by itself, a localization theorem for an apparatus.

Donnelly and Giddings analyze gravitational dressing and its implications for locality under their asymptotic and charge assumptions DG. The present finite-time construction uses a smaller, initial-data-preserving gauge group. It neither proves nor bypasses their conclusions.

A localized measurement construction requires a system–probe coupling and a suitable scattering map. In the Fewster–Verch framework, causal factorization is a hypothesis supporting the composition theorem for instruments FV. No such coupling or instrument is constructed here.