After filling each auxiliary copy, collect its measurements as a matrix C_hat on the original variables. Split every row into its contribution from each original group: C_hat x=sum_g C_g x_g. Fix each group's contribution separately, not merely the total. For bits x,y in different groups, fixing x+y=1 still couples them: permitted pairs are (0,1),(1,0), not a product of two local sets. Fixing x and y contributions separately restores the required product of fibers. Filling a row from D entries in {0,1,-1} adds coefficients for repeated variables and discards unused positions; by the triangle inequality its absolute coefficient sum is at most D. With h copies and c auxiliary rows, the split design has rhc rows, including zero rows.