The existence of an admissible pair is the grouping iteration: start with all indices surviving at scale $r$, and while two surviving entries are within eight times the scale, drop one of them and multiply the scale by nine. At the end, entries of one group are within twice the scale $ρ$, hence within $4ρ − 2r$, and entries of different groups are farther apart than $6ρ ≥ 4ρ + 2r$.

For the detection property, if every vertex of $X$ is near the tuple, then the vertices of $S$ lie in distinct blocks, by the scattering of $S$. Conversely, if some vertex $x₀$ of $X$ is far from the tuple and no vertex of $X$ is at a distance between $r$ and $H$, then the blocks met by $X$ inject into $S$ minus the representative of $x₀$.