The proof in the paper. Group $A$ by neighborhoods in the ball $B$ and $B$ by neighborhoods in $A$. The parameter set consists of representatives of all groups and of a bounded-VC duality sample for every pair of groups of $A$ and for every group of $B$ against the inner ball. The $S$-flip toggles exactly the pairs of atomic types joined by an edge of $G$ between $A$ and $B$. Then $A$ and $B$ are non-adjacent in the flip, the inner ball is unchanged, and it has no edges to the outside of $B$, so the ball of radius $r + 1$ still avoids $A$. The duality is used in a form without exception for the sample, obtained from the concept at the cost of doubling the sample.