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Fix $x\in S$ and let ${\sigma }_x$ be its principal vertex of a dual cube complex. The combinatorial distance in the dual complex is the wall metric:Consequently the wall-metric properness criterion says that the action on $C$ is metrically proper provided this number tends to infinity as $g$ leaves every finite subset of $G$. Equivalently, for every $R$, only finitely many $g$ separate $x$ from $gx$ by at most $R$ walls.

For a wallspace $(S,\mathcal{W})$, defineThe wallspace axiom makes this finite. It is always a pseudometric: it is symmetric, vanishes on the diagonal and satisfies the triangle inequality. For the last claim, every wall separating $x$ and $z$ must separate $x$ from $y$ or $y$ from $z$, so counting walls gives the inequality.^[ref]

It is a metric exactly when the walls separate every pair of distinct points. For example, on $S=\{a,b,c\}$ with only the wall $\{\{a,b\},\{c\}\}$, the distinct points $a,b$ have distance zero. Identifying pairs at distance zero gives a genuine metric space.

The principal vertex of a dual cube complex sends indistinguishable points to the same vertex. The wall-count distance equals the combinatorial distance between their principal vertices, making this pseudometric useful for the wall-metric properness criterion.