Let delta=(R-1)!Q! clear the replacement-row denominators. An equation monomial has degree at most k, so Delta=delta^k clears L_i. Each term e_j(L_i)h(L_i)^j has denominator dividing Delta^(1+2j). Therefore Gamma=Delta^(2ell-1) clears every lifted factor and Omega=Gamma^m clears their product. Neutral constant-one factors do not change this. These deliberately loose denominators remain fixed as contributions are added; they are not multiplied again with each new summand. Their bit length is polynomial in m,k,ell,Q. This is a new lifting contract, not an assumption that the earlier explicit-rational-factor analysis already proved it.