Apply the closure theorem at radius $6r$ to obtain $A*$ and the sets $S(v)$. For every $T ⊆ A*$ of size at most $k·s₀$, the partition-metric conversion applied to $𝒫_T$, which has at most $3^{k s₀}$ parts, gives one flip $H_T$ whose radius-$r$ balls lie in the radius-$6r$ balls of the flip metric of $𝒫_T$. For a tuple $v̄$ outside $A$ take $T = S(v₁) ∪ … ∪ S(v_k)$. As $𝒫_T$ refines $𝒫_{S(vᵢ)}$, the flip metric of $𝒫_T$ dominates that of $𝒫_{S(vᵢ)}$, and the latter is at least the distance in the $S(vᵢ)$-flip from the closure theorem, which exceeds $6r$. So $H_T$ separates $v̄$ from $A$ at distance more than $r$. There are at most $(|A*| + 1)^{k s₀}$ sets $T$.

The argument is carried out for the graph classes of the archive and their monadic dependence, and transferred to the graph classes used here.