Proposition: Covering and a conditional tight budget

Detection requires $E_{\pi }\subseteq S$ for some split pair. If ${\pi }^{\ast }$ minimizes edge size, has $v_{{\pi }^{\ast }}(E_{{\pi }^{\ast }})>0$, and is the unique split pair whose edge is contained in $E_{{\pi }^{\ast }}$, then

Under these hypotheses the minimal edge-size lower bound is attained.

Proof. The necessity is Corollary cor:budget. At $S=E_{{\pi }^{\ast }}$, all other split pairs are screened. The surviving pair has orthogonal record supports, so Proposition prop:screening gives the displayed equality.