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

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 t