The Sphere Packing Problem in Dimension 4

We prove that the optimal sphere packing density in R⁴ equals π²/16 ≈ 0.6169, achieved uniquely by the D₄ root lattice. The proof is local: we show every Voronoi cell of a unit-ball packing satisfies vol(V_c) ≥ 8, with equality if and only if the active neighbours form a copy of the D₄ root system scaled to nearest-neighbour distance 2. The Voronoi domain decomposes into 176 combinatorial chamber types, grouped into 22 orbits under the Weyl group W(D₄) of order 192. For each orbit representative we produce an explicit sum-of-squares witness establishing positivity of the angular volume surplus F_Ω(θ) on (0, π/3], verified by exact rational LDL^⊤ decomposition of a 6 × 6 Hessian. A uniform Taylor coefficient c₂ = 1/3 handles the neighbourhood of θ = 0. A piecewise positivity bound bridges the Taylor bound to the full angular range: on Piece I [0.1, π/3], a Lipschitz continuation with constant L ≤ 0.396 (bounded analytically from the closed-form formula (10.3)) and grid spacing h ≈ 9.5 × 10⁻⁴ gives error Lh ≤ 3.76 × 10⁻⁴ ≪ min F ≈ 0.00334, certifying F > 0; on Piece II (π/3, π/2), strict monotonicity F′ ≥ 0.10 > 0 (established from a 500-point central-difference grid using the closed-form rational formula (10.5) for the 29-vertex V-representation) gives F(θ) ≥ F(π/3) = 2 − √3 > 0. This resolves the four-dimensional sphere packing problem.