.. is the residual momentum and μ ≡ P μ P is the truncated dipole operator. Note that Ĥ′ p•A (Eq. 66) as well as Ĥp•A (Eq. 45) only contain the vector potential Â up to the second order. This is no longer the case for Ĥp•A in Eq. 75. In fact, both the Pj term and the Û † V(x, p) Û term in principle contain infinite orders of Â. Hence, the consequence of level truncation on Ĥp•A is not just simply modifying the matrix elements of the momentum operator (as incorrectly indicated by Ĥ′ p•A in Eq. 66), but rather profoundly changing the structure of light-matter interactions through both the new potential Û † V(x, p) Û as well as the new momentum shift -∇ j μ Â + Pj , due to the mixing of the light and the matter DOFs through Û † and Û in the truncated subspace. It is clear that Ĥp•A (Eq. 76) will return to Ĥp•A (Eq. 45) under the complete electronic basis limit, such that μ ≡ P μ P → μ, thus ∇ j μ → ∇ j μ = z j , hence Pj → 0, as well as Û → Û , hence Û † P V (x) P Û → Û † V (x) Û = V (x). In Ref. 82, Gustin et. al. further generalizes the resolution of gauge ambiguities beyond the dipole approximation by defining Û in terms of the full matter polarization instead of the dipole operator. They then properly confine Û by truncating the polarization operator in terms of x. Unfortunately, Ĥp•A in Eq. 75 no longer remains in the minimum coupling form in Eq. 45 which only involves charges but not higher multipole moments. Of course, when approaching the complete electronic states limit, the minimum coupling form is restored. Nevertheless, Ĥp•A is invariant from Ĥd•E through the Û transformation, thus resolving the ambiguity between them.