Geometric picture of the PCA physicality priors in two-dimensions, demonstrating why they necessarily allow unphysical regions of parameter space. The solid blue square is the physicality region in terms of the ionization fraction, x₁ and x₂, at individual redshifts. The solid orange square is the physicality region in terms of the mode amplitudes, m₁ and m₂. The two types of physicality priors corresponding to Eq. (4) and Eq. (5) are shown as the dashed square and dashed circle, respectively. Their intersection, shaded orange, is the full physicality region. The left panel takes a fiducial ionization history, x^fid  =  0.15, consistent with Mortonson & Hu (2008a) and subsequent works. The right panel shows that the undesired but allowed unphysical parameter space can be reduced (but not fully) with a better choice of fiducial model.