Represent factor i by B_i(V)=sum_{j<ell} e_j(L_i)h(L_i)^j V^j. The formal marker V records total h-power; it is not a Boolean input. Work in Q[V]/(V^ell): store the ell coefficients of degrees 0 through ell-1 and discard higher powers after multiplication. Use the SAME V for all factors. Only after multiplying all factors, add the retained coefficients with S_ell(sum b_s V^s)=sum b_s. S_ell is linear but not multiplicative. For ell=2, V times V becomes zero, although S_2(V) times S_2(V)=1. Summing each factor first would therefore lose the shared truncation and its degree saving.