Choose a group schedule a_v for every original variable and an assigned-group schedule d_f for every factor. Occurrence (f,v) receives label (a_v,d_f). If several occurrences share a label, put them in different copies of the auxiliary array. Fill the designated coordinate with x_v and unused coordinates with zero. Repeated appearances of x_v always use the same original bit: copies are storage positions, not independent variables. Apply the auxiliary replacement row in the copy holding that occurrence. It restores x_v at zero and preserves the degree bound. All inputs of factor f use d_f(t), so the factor has one assigned group. The separate balancing lemma must bound group sizes and the number of copies.