Robust infeasibility under uniform constraint relaxation ($\epsilon <1/{\sum }_i{\lambda }_i$)

Suppose a linear program has a Farkas' lemma certificate $\lambda \ge 0$, $A^T\lambda \ge 0$, $b^T\lambda =-1$. A purported feasible point of $Ax\le b+\epsilon 1$, $x\ge 0$ would imply $0\le {\lambda }^{T}Ax\le -1+\epsilon {\sum }_i{\lambda }_i$. Hence every relaxation below the displayed positive threshold remains infeasible. For rational inputs, bounds on the encoding length of a certificate supply the precision gap used by the ellipsoid method.