The filtered topology of the D_9^+ void

Abstract

Grow balls of squared radius t about the points of D_9^+ = D_9 ∪ ((1/2)^9 + D_9), with contact distance √2, and follow the uncovered part of the torus R^9/D_9 as t falls from the squared covering radius 17/16 to 0. Its persistent homology has 37 critical levels and 55 classes of bars. It is the same over the rationals and over every prime field, so it holds over the integers: there is no torsion anywhere in the filtration. Below level 1 each critical level is the circumradius of a mixed simplex, with a lattice and b glue vertices, and the bars are indexed by these pairs. There is one unordered type per level except at 15/16, where (4,4) and (7,2) coincide. Most of the table is explained. The equators of unit cubes are regular Voronoi hexagons of D_d^+ in every odd dimension from nine; they kill exactly the tunnel loops that do not wind around the torus, and in dimension nine the 2,008 non-winding cycles of the cap die on them at 15/16. Every other finite bar of degree one is the star of a newly entered hole or a free edge. The essential classes enter as a torus ladder: in each degree q all C(9,q) of them are born at one level, by the irreducibility of the exterior powers of the reflection representation, and in degrees two to six the rung sits at the type (4, 6−q). A second construction explains the degree-three rung directly in the void: 512 Voronoi three-cells and 16 cap fillings form a periodic cycle at 147/160. Independent exact checks certify the complete cells, the boundary, and all 84 ambient components. Its class is four times a primitive class; full rational rank follows from symmetry, while index one uses the integral-barcode theorem. The higher-degree adjacent-death pattern remains a computed observation whose general geometric explanation is open. The deposit contains the paper, its TeX source, and the exact-arithmetic replay programs for every computational step: the level law over the six Delaunay cell types, the barcode over the rationals with its comparison against the three field tables, the census and attaching ranks of the 15/16 row, every hole and edge behind the degree-one bars, the clique census and sheets of the ball side, and the detailed construction and three checks of the void-side degree-three cycle.