To dig deeper into the meaning of the gauge choice in the exact electron factorization, we take a few steps back and derive the equation for the marginal before imposing any gauge: the phase factor in the marginal amplitude and in the conditional amplitude can be anything, not just zero. We call it S (r). To distiguish these different marginal and conditional amplitudes from the χ and Φ that we have considered so far, we will label them χ and Φ . Because the electronic vector potential depends on the conditional amplitude, we shall consider also a primed electronic vector potential A . In formulas, we are considering the following relations