Primal-dual residual gap bound for a simplex candidate

For a real standard-form problem with $A\in {\mathbb{R}}^{m\times n}$ and candidate vectors $x\in {\mathbb{R}}^n$, $y\in {\mathbb{R}}^m$, define $r_p=Ax-b$, $r_d=c-A^{\mathsf{T}}y$, and $g=c^{\mathsf{T}}x-b^{\mathsf{T}}y$. When the data and candidates are finite stored binary floating-point values, Exact residual audit of finite-precision simplex candidates computes precisely these quantities, interpreting the stored values as exact dyadic rationals. For all the stated real data, $g=x^{\mathsf{T}}{r}_d+y^{\mathsf{T}}{r}_p,$ and consequently $\mathrm{\mid }g\mathrm{\mid }\le \mathrm{\parallel }x{\mathrm{\parallel }}_1\mathrm{\parallel }{r}_d{\mathrm{\parallel }}_{\mathrm{\infty }}+\mathrm{\parallel }y{\mathrm{\parallel }}_1\mathrm{\parallel }{r}_p{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{.}$ Here $\mathrm{\parallel }z{\mathrm{\parallel }}_1={\sum }_i\mathrm{\mid }{z}_i\mathrm{\mid }$ and $\mathrm{\parallel }z{\mathrm{\parallel }}_{\mathrm{\infty }}={\max }_i\mathrm{\mid }{z}_i\mathrm{\mid }$, with both norms equal to zero on the empty vector. This bounds the primal-dual objective difference; when either vector is infeasible it does not bound distance to an optimal value. If $Ax=b$, $x\ge 0$, $c-A^{\mathsf{T}}y\ge 0$, and $g=0$, then $x$ is optimal.