After clearing substituted inputs, let B>=Delta bound the coefficient norm of Delta L_i; sparse evaluation and polynomial-size rows give polynomial log B. Put E=max(1,max_j ||e_j||_1). A cleared factor has norm at most B1=ell E(B+Delta+2)^(2ell-1), so log B1 is polynomial. With m factors, at most M_updates signed contributions have accumulator norm at most M_updates 2^n B1^m over Omega. Thus bit lengths grow with log(M_updates+1), not M_updates. Apply the earlier exact rational interpolation separately to the ell V-coordinates. Each H_s(0) is integral; summing and taking the residue recovers N. Sequential enumeration plus QJell stored coefficients gives polynomial space under the stated parameter bounds.