Fix point t and an outside original group g. Let xi be a change supported only in g that leaves its measurements unchanged: C_g xi=0. Extend xi by zeros in other groups. Then C_hat xi=0, so the induced change in every filled auxiliary copy has zero auxiliary measurements. Every occurrence of the same original variable uses its same first schedule a_v; consequently each induced change is supported only in auxiliary group g. For a factor assigned elsewhere, its auxiliary replacement uses only its own group and auxiliary measurements. Both stay unchanged, so the replacement value stays unchanged. This holds for every input of the factor, proving the original-group separation condition despite correlated copies.