Induction on $r$, as in the paper. At radius 0 take $A* = A$ and no flip. For the step, enlarge $A$ to $A₁$ by the ball-types lemma for the current parameter bound $s$, and apply the induction hypothesis to $A₁$, giving $B$. The $s$-definable flips of graphs of the class form a monadically dependent class, so they have VC-dimension at most some $d$. For every set $S ⊆ B$ of at most $s$ vertices and every $S$-flip $H$ (at most $2^{9^s}$ of them), group the vertices whose radius-$r$ ball in $H$ avoids $A₁$ by the set of atomic types over $A$ that the ball meets. By the ball-types lemma these profiles have at most $t$ members, and by the Sauer–Shelah lemma there are at most $|A| + (|A| + 1)^d$ atomic types over $A$, so there are polynomially many groups. The ball flip lemma gives a $z$-definable flip of $H$ that $(r + 1)$-separates each group from $A$, which is an $(s + z)$-definable flip of $G$ by transitivity. The new closure is $A₁$ together with the parameters of all these flips. Vertices of $A₁ ∖ A$ are isolated by the flip over themselves.

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