The Sphere Packing Problem in Dimension Four

The maximum sphere packing density in $\mathbb{R}^4$ is $\pi^2/16$, achieved uniquely by the $D_4$ root lattice. The proof establishes a local Voronoi cell bound: every packing cell satisfies $\mathrm{vol}(V_c)\ge 8$, via a four-layer argument (shell localisation, root-alignment, radial monotonicity, chamber positivity for all $176$ Weyl-orbit types) with positivity certificates exact over $\mathbb{Q}$.