For a crossing row x and tile y0, write A_x=C(T)/(C_prime(0)(T-y0+x)) and R_x=2^(i+1)/(T-y0+x). The valuation bounds give v_2(A_x)>=a+1-i>=1 and v_2(R_x)>=1. With y=y0+2^(i+1)j, W_xy(T)=A_x/(1-R_x j). Modulo 2^p_t it suffices to sum A_x R_x^b j^b for b+1 below p_t. The finite geometric remainder has valuation at least p_t, because 1-R_x j is a unit. This is exact modular truncation, not floating-point approximation or an assumption about ordinary real convergence.