A settling floor for the voids of periodic packings

Grow balls about the sites of a periodic packing and watch the uncovered part of the torus. For large balls its topology depends on every detail of the arrangement. For small balls it does not: the inclusion of the void into the torus is an integral isomorphism on H_q whenever the squared ball radius is at most (d−1−q)/(d−q), with contact distance √2, for every periodic packing in every dimension. The proof is Lefschetz duality together with a centroid bound on how many balls can share a point. In degree zero this is a floor (d−1)/d below which the void of every periodic packing is connected in R^d, and so percolates. FCC and E_8 percolate exactly at the floor. D_9^+ shows that percolation and complete connectivity are different events: its void percolates from 33/32 but becomes connected only at the floor 8/9, and there the bound is attained in every degree. The deposit contains the note, its TeX source, the exact barcode program for the D_9^+ face complex with its result tables over F_2, F_3 and F_(2^31−1), and a check of the sharpness table against them.