Theorem: Constrained simultaneous repair

For constructed $B,D_{\mathrm{sub}},L,T$, a permitted simultaneous correction exists exactly when

where the notation on the left is understood as the ordered pair $(-B,D_{\mathrm{sub}})$.

More explicitly, if $c_{\ast }\in \mathcal{C}$ solves the unconstrained equation $\mathcal{M}{c}_{\ast }=(-B,D_{\mathrm{sub}})$, then the complete permitted solution set is

If the allowed set is the whole vector space, this reduces to the affine actual-image criterion.

Proof. A simultaneous permitted repair means $B+Lc=0$, $D_{\mathrm{sub}}-Tc=0$, and $c\in {\mathcal{C}}_{\mathrm{allow}}$. This is exactly membership in the displayed image. The difference of two unconstrained solutions belongs to $\ker L\cap \ker T$, and every element of that common-kernel translate is an unconstrained solution. Intersect with the allowed set to obtain (eq:solutions).