By induction along the tuple and the triangle inequality, the entry $vᵢ$ is within $∑_{i'=1}^{i} ρ⁺(k+i', q−i')$ of $v₀$. A color mentioned in a true atom of rank $(k+j, q−j)$ has a vertex within the radius of the atom, at most $ρ⁺(k+j, q−j)$, of an entry. The horizon inequality $guardSum_add_le$ bounds both distances by $ρ⁺(k, q)$.