Gomory infeasibility certificate from a fractional slack row

For a nonnegative all-integer row $x_B+{\sum }_j{a}_j{x}_j=b$ with $a_j>0$, the Gomory fractional cut requires ${\sum }_j\{a_j\}{x}_j\ge \{b\}$. The original row gives ${\sum }_j{a}_j{x}_j\le b$. If a constant $\kappa$ satisfies $\{a_j\}\le \kappa a_j$ for every $j$ and $\kappa b<\{b\}$, these inequalities contradict the cut. A single row therefore certifies integer infeasibility. Slacks may be used as integer variables when the original constraint coefficients and right sides are integral and the decision variables are integral.