For each nonzero M[u,y], form phi_{u,y}(T)=product over mismatched points s of (1-T/s). Define P_u(T)zeta=((I-M)zeta)u+sum_y M[u,y]phi{u,y}(T)zeta_y. The swap construction makes M fix the kernel of C, so the first term is determined by C zeta. Its support proof bounds each multiplier's degree by R-1. At zero all multipliers equal one, giving P_u(0)zeta=zeta_u. At an evaluation point t, terms using coordinates outside the assigned group vanish. Thus the replacement has degree at most R-1 and can be evaluated using its assigned group plus shared measurements. These are auxiliary coordinates; transfer to original variables is still needed.