 is gauge-invariant, and it cannot be further decomposed into quantities with support on patches that do not include the entirety of \(I. \)For instance, if there are two patches, each containing either \(\nu_x\) or \(\nu_y\), then that patch has insufficient charges to complete \(S(I)\) into a gauge-invariant function. This example is in line with the physics literature, which largely takes gauge-invariance (and in particular the existence of a generalized Gauss constraint), to signal some general type of nonlocality (cf. Strocchi (2013, Ch. 7)).