Theorem: Projected primitive-enlargement criterion

Let $H:Y_{\mathrm{old}}\to W$ have a complete left kernel $L$, let $E:Y_{\mathrm{new}}\to W$ be a specified additional primitive response, and let $b\in W$. Then

Let $Q_E$ be a complete left kernel of $LE$, and set

Then

If $D_{E}b\ne 0$, a row $\ell$ with $\ell b\ne 0$ satisfies

and is an exact separator for the enlarged class. For several targets $B$, a successful enlargement needs at least $\mathrm{rank}(LB)$ independent new image directions.

Proof. If $b=Hu+Ev$, then $Lb=LEv$. Conversely, $Lb=LEv$ implies $b-Ev\in \ker L=\mathrm{im}H$. Completeness of $Q_E$ gives

proving (eq:combinedseparator). The row identities follow from $LH=0$ and $Q_{E}LE=0$. The multiple-target lower bound follows by dimension.