To interact particles in nodes 18 and 19 with that at the origin, we move to " α 2 scale", i.e., halve the α value but preserve the same the same spherical t-design and Gauss-Hermite quadrature parameters as before. It is worth emphasizing that the identity on the first line of eqn. ( ) holds for all α, so the change in scale is valid as long as both short-and long-ranged terms are adjusted similarly. Even though the halving of α almost doubles the range of the short-ranged operator, it still spans less than two leaf widths because of the initial choice to allow the long-ranged operator span double the distance of the short-ranged term. By construction, the short-ranged operator therefore decays before the leftmost edge of node 18 in α 2 scale, nullifying any contribution it could make to interactions between the origin and particles in nodes 18 and 19 in this scale. Consequentially, all such interactions are entirely handled by the long-ranged term in α 2 scale, but with the same precision achieved for the α scale interactions. In a similar spirit, interactions between the origin and the particles of {20, 21, 22, 23} are treated using only α 4 scale long-ranged kernels, while the remaining particles residing in nodes 24-31 interact only using α 8 scale kernels. In summary, by offsetting the reach of the long-ranged kernel by a factor of two over the shortranged term, α may be rescaled at long distances in such a way that no short-range terms are required.