In recent literature, the source of these gauge ambiguities and corresponding resolutions (See Fig. ) has been thoroughly discussed from both an intuitive physical perspective and a rigorous mathematical perspective. In Refs. 42,73, Stefano et. al. and Garziano et. al. describe the source of gauge ambiguities in terms of the locality of the matter potential energy operator in the truncated Hilbert space, P V (x) P = V(x, x′ ). In other words, upon matter truncation to a finite basis, the potential energy operator is defined based on two positions in space, hence it is no longer local in space (only depending on x). Equivalently, by Fourier transforming in x′ , one can say that this operator is dependent on both the position and momentum operators. Fig. shows how for an n-level matter truncation, V(x, x′ ) gets increasingly non-local as n shrinks. Refs. 42,73 argue that this non-locality leads to gauge ambiguities since the expression in Eq. 66 contains the non-local potential, V(x, p), to which the gauge transformation has not been applied (due to the fact that V (x) commutes with Û ). This can be seen by rewriting Eq. 66 as,