Proposition: Exact finite correction feasibility

A permitted correction makes every named residual zero if and only if

Equivalently,

When $c_{\ast }$ is one solution, all solutions are $c_{\ast }+\ker N$.

Proof. The evaluated residual is $w_0+Nc$, so exact feasibility is the equation $Nc=-w_0$. The orthogonal complement of $\mathrm{im}N$ is $\ker N^{\mathsf{T}}$. The final assertion is the usual affine solution-space description.