Proposition: An infinite family can still have an empty query

The constant family $W_e=W_0$ belongs to ${\mathcal{P}}_{3q}$. For its datum $D_0$, every signalling strength vanishes, $E_{D_0}$ is an infinite antichain, and

Proof. Membership follows from Lemma lem:base. For a sender channel $S$, receiver element $B_z$, and other-party channel $M_c$,

This does not depend on the sender channel. Thus every strength is zero.

There are no raw edges, so each raw token is a singleton component and the quotient has no distinct comparable elements. For every map $\iota$, the generating set in (eq:generated) is empty. Hence $K_{\iota }=\{0\}$, which has empty interior in ${\mathbb{R}}^m$. It cannot satisfy (eq:proper-injective).