Apply the earlier replacement forms to the lifted factors, retaining V-coefficients, and sum over the Boolean cube to obtain H(T) in Q[V]/(V^ell)[T]. Replacement degree at most R gives degree_T H<=kR(m+2(ell-1)). At T=0 each replacement restores the original Boolean input, so every V-coefficient of H(0) is an integer and total-order truncation gives S_ell(H(0)) congruent to the count modulo 2^ell. Recover each [V^s]H(0) by rational interpolation, add them, then take the least nonnegative residue. Intermediate rational values need not obey parity congruences; never reduce a rational denominator modulo a power of two as if it were automatically invertible.