Normalized integer Farkas infeasibility gap ($b^T\lambda \ge [(n+1)U{]}^{-(n+1)}$)

Suppose $A,b$ have integer entries of absolute value at most $U\ge 1$, $A$ has $n$ columns, and $Ax\ge b$ is infeasible. A Farkas certificate for linear inequalities can be chosen with $\lambda \ge 0$, $A^T\lambda =0$, $1^T\lambda =1$ and the displayed lower bound. Normalize an infeasibility certificate to obtain the last equality, then maximize $b^T\lambda$ over the resulting compact linear polyhedron. A maximizing extreme point has at most $n+1$ positive entries: otherwise the augmented columns $(A_i^T,1)$ on its support would be dependent, allowing feasible perturbations in both directions. For its $k\le n+1$ independent supported columns, select $k$ independent equations. Cramer's rule expresses the supported entries with a common nonzero integer determinant denominator. That determinant has absolute value at most $k!U^k\le [(n+1)U{]}^{n+1}$. The positive value $b^T\lambda$ therefore has a positive integer numerator over a denominator of at most this size, proving the bound.