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

We prove that every Voronoi cell of a unit-ball packing of ${\mathbb{R}}^4$ has volume at least $8$, with equality only at the $D_4$ configuration, so that the twenty-four-cell conjecture holds and the density of a sphere packing in four dimensions is at most ${\pi }^2\mathrm{/}16$. The proof runs through the number of contacts of a cell. Up to twenty-two a covering estimate suffices; at twenty-three the cell is bounded through an exact volume identity inside a ball and a semidefinite certificate for one inequality between pair angles; at twenty-four the configuration is the root system, which we prove from the second level of the semidefinite hierarchy with its equality case, the positive kernel verified in exact arithmetic.