For t in S, write C(t+alpha z)=alpha z product_{q!=t}(t-q+alpha z). Each noncentral factor is coefficientwise divisible by 2^(v_2(t-q)). Their total valuation cancels the power of two in C_prime(0), leaving alpha z and an odd denominator. To obtain W_xy, omit the factor with q=y-x. If q=t, this removes alpha z and leaves a polynomial with odd-denominator rational coefficients. Otherwise alpha z remains and the omitted factor costs at most a powers of two, leaving every coefficient divisible by two. Thus W_xy(t+alpha z) has coefficients in Z_(2).