where x = {x j }. To fix this problem, it was proposed to first truncate ĤM and then transform it by the projected PZW operator P Û P, which will gauge transform the non-local potential V(x, p). However, Ref. 75 points out that this does not formally solve the gauge ambiguities. Instead, it works specifically when the matter is truncated to a two level system, making the proposed solution just a rotation on a Bloch sphere. In Refs. 74,81, Taylor et. al. go further to propose a general resolution to gauge ambiguities for any matter system under the dipole approximation. The key insight discussed in these works is the concept of proper confinement of all operators in the truncated subspace. For a given projection operator, P, there is a complementary operator, Q, such that P + Q = 1M . Taylor et. al. describe a new gauge theory that is "properly contained" in the subspace defined by P. In other words, all the information of the truncated system lives entirely in the P subspace without any information in the Q subspace. For example, consider the case of P x2 P = P x( P + Q)x P ̸ = ( P x P) 2 . In this manner, P x2 P is not properly confined in P, since it contains x information from the Q subspace, P x Qx P.