For one pair position, let e_ab be the vector that is one at label (a,b) and zero elsewhere. A swap fixes e_aa and e_ab+e_ba, but negates e_ab-e_ba (a<b). These are eigenvectors: the swap multiplies each by its eigenvalue +1 or -1. They are mutually orthogonal: the sum of coordinatewise products of two different vectors is zero. There are r(r+1)/2 positive and r(r-1)/2 negative vectors, together a basis: every vector has a unique linear combination of these r^2 directions. For Q positions, choose one vector per position and define psi(u)=product_s psi_s(a_u(s),d_u(s)). Inner products of these product vectors factor across positions. Thus the D nonzero orthogonal vectors form a basis of the D-coordinate auxiliary space. Pi_s negates a product vector exactly when its choice at position s is negative.