Expand the product of the M affine row factors in the formal modes. For an exponent tuple beta, write Z^beta=product_Z Z^(beta_Z) and its weighted degree as sum_Z w_Z beta_Z. Every occurrence of a mode contributes its promised power of two, so the coefficient of Z^beta is divisible by two to this weighted degree. Substituting the integral source forms L_Z(X) cannot decrease that divisibility. Therefore only tuples with weighted degree below p_t can contribute modulo 2^p_t. Ordinary degree alone would miss the different precision costs of different modes.