The Sphere Packing Problem in Dimension 7
Abstract
We prove that the optimal sphere packing density in \( \mathbb{R}^7 \) is \( \pi^3/105 \), achieved uniquely by the scaled root lattice \( \sqrt{2}\,E_7 \). The argument is local: it suffices to show that every Voronoi cell of any unit-ball packing in \( \mathbb{R}^7 \) has volume at least~\( 16 \), with equality characterised by the \( E_7 \) kissing configuration. The Weyl group \( W(E_7) \) of order~\( 2{,}903{,}040 \) partitions the configuration space into \( 115 \) chamber types; for each, positivity of the free-volume surplus \( F_\Omega(\theta) \)on \( (0,\pi/2) \) is certified by an exact rational \( 7\times7 \) \( LDL^\top \) decomposition (minimum pivot \( 823/3360 \)) together with a complete polynomial positivity argument covering the full angular range. The large-angle range follows from a geometric analysis of the freed-root body associated to each cap-cutting direction. The proof is unconditional and self-contained, carried out in exact rational arithmetic throughout.
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