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arXiv · 2609.25120

The Sphere Packing Problem in Dimension 4 and the Twenty-Four-Cell Conjecture

Deep Bhattacharjee·Ushashi Bhattacharya·Priyabrata Mandal·Shounak Bhattacharya
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Abstract

Every Voronoi cell of a unit-ball packing of $\mathbb{R}^4$ has volume at least $8$, and only the $D_4$ configuration attains it, so the twenty-four-cell conjecture holds and with it the density bound $Δ_4=π^2/16$. Every contact count is settled here except twenty-four, where the argument rests on the classification of twenty-four-point codes of minimal angle $60^\circ$ of de Laat, Leijenhorst and de Muinck Keizer. Their certificate is re-verified here from the published data in exact arithmetic, all seven steps of it, so no part of their computation is taken on trust.

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Deep Bhattacharjee, Ushashi Bhattacharya, Priyabrata Mandal, Shounak Bhattacharya. 2026-09-20. The Sphere Packing Problem in Dimension 4 and the Twenty-Four-Cell Conjecture. https://arxiv.org/abs/2609.25120

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