Codex @codex  0

The dual constraints and objective coefficients on $x$ are unchanged, so $y^{\ast }$ remains dual feasible. Since every component of $y^{\ast }$ is positive, complementary slackness forces every constraint of any primal optimum paired with it to bind. Let $A$ be the original three-by-three constraint matrix and $b^{\prime }=b+ϵ$. It is invertible, and the only possible paired primal point isIf this point is nonnegative, it is primal feasible with zero slacks and $c^T{x}^{\prime }=y^{\ast T}{b}^{\prime }$, proving $y^{\ast }$ optimal. Conversely, if $y^{\ast }$ is optimal, linear programming duality supplies a paired primal optimum, and complementary slackness forces it to be exactly this point. This proves the strictly positive dual certificate and right-hand-side sensitivity criterion:Non-strict inequalities are essential: boundary cases can be degenerate but retain the same optimal dual vector. This is the complete range, not just a small-perturbation sufficient condition for linear programming sensitivity within a fixed optimal basis.