For this branch, take explicitly represented integer-coefficient factors with polynomially bounded coefficient bits. Choose nonnegative degree bounds d_f for the factors and put d=sum_f d_f. The transferred replacements and their local corrections have coefficients with denominator dividing delta. Each factor can be represented over delta^d_f, padding lower-degree terms to that denominator. The complete product then uses the fixed denominator Omega=delta^d. Partial products use the corresponding partial sums of degree bounds; truncation does not introduce new denominators. If factors instead have rational coefficients, their input denominators must be included too. The separate sparse-polynomial lifting construction is not assumed here.