An ordering of the vertices is a permutation $π$ of $Fin m$ assigning each vertex its position. Both reachability sets are stated with walks, as in the nowhere dense concept: shortcutting a walk to a path only shrinks its support, so walks of length at most $r$ reach exactly the vertices that such paths do. In $wreach$ the $π$-minimality of $u$ on the whole support already forces $π u ≤ π v$; in $sreach$ only the interior of the walk is constrained, so $π u ≤ π v$ is a separate conjunct. Both sets contain $v$.

$wcol$ and $scol$ are the least achievable bounds $k$, $Nat.sInf$s over nonempty sets — $k = m$ works for any ordering — so the convention $Nat.sInf ∅ = 0$ is never exercised; the counts are $Set.ncard$. Weak and strong coloring numbers are one review unit because they are the same construction differing in a single clause, and every relation between them is read off that contrast.

The class-level bound is uniform over subgraph copies ($⊑$) of members, each measured by its own vertex count $m$; this is the literature form for subgraph-closed classes, and the uniformity is what localization arguments downstream consume. At $m = 0$ both sides vanish, so no nonemptiness hypothesis is needed.