At each evaluation point t, every variable has a group and every whole factor has an assigned group; these choices may change with t. An auxiliary coordinate label u records a pair (a_u(t),d_u(t)) at each of Q points: the coordinate's group and its replacement's assigned group. For integers r>=2 and Q>=1, with r group choices there are D=r^(2Q) possible labels. The construction initially supplies a placeholder zeta_u for every label, not a new Boolean variable. A replacement for zeta_u may use zeta_y directly at point t when a_y(t)=d_u(t). Otherwise that dependence must vanish or be accessible through the shared measurements.