For position s, define Pi_s to exchange the two entries a(s),d(s) of every auxiliary label. On a vector this simply permutes coordinates. Swapping the same position twice restores the original; swaps at different positions commute, meaning their order does not matter. The full swap V=Pi_1...Pi_Q sends every pair to its reverse. A nonzero entry V[u,y] therefore has a_y(t)=d_u(t) at every point: no group mismatches. Undoing swaps at positions in a set S can introduce mismatches only in S. This gives a concrete way to restrict where a matrix entry can use an outside-group coordinate.