The Sphere Packing Problem in Dimension 4
Abstract
We settle the sphere packing problem in $\RR^4$ by proving that the density of any unit-ball packing is at most $\pi^2/16$, with equality achieved precisely by the scaled lattice $\sqrt{2}\,D_4$ up to isometries of $\RR^4$. The proof rests on a Gram-spectral square decomposition of the Voronoi volume defect. For every saturated packing cell we express $\vol V - 8$ as a ratio $F_\Omega/D_\Omega$, where $D_\Omega > 0$ on the chamber interior and $F_\Omega$ is a polynomial that factors as a finite sum of squared norms of explicit matrix-vector products over $\QQ(\sqrt{2})$, each multiplied by a non-negative packing quantity, plus terms that vanish on the chamber radical ideal $\mathfrak{i}_\Omega$. The non-negativity of $F_\Omega$ then gives $\vol V \geq 8$ for every Voronoi cell; since cells tile $\RR^4$, this yields the density bound. The radial-shell localisation of the $D_4$ root system restricts active neighbours to the compact annular shell $2 \leq \norm{y} < 2\sqrt{2}$, reducing the problem to a finite parameter space in which exactly $176$ Weyl-orbit types of cells arise. Each type is handled by an exact polynomial identity verified by Gröbner-basis reduction over $\QQ(\sqrt{2})$. No numerical approximation appears anywhere. A spectral rigidity argument shows that equality forces every cell to be the regular $24$-cell and the packing to be a translate of $\sqrt{2}\,D_4$. All supplementary calculations --- vertex coordinate formulas, explicit matrix entries for $B_0$,…
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