The Sphere Packing Problem in Dimension 4

We prove that the optimal sphere packing density in \(\mathbb{R}^4\) is \(\Delta_4=\pi^2/16\), achieved uniquely by the \(D_4\) root lattice. The proof is local: every Voronoi cell of a unit-ball packing satisfies \(\mathrm{vol}(V_c)\ge 8\), with equality precisely when the active neighbours form the \(D_4\) kissing configuration at distance~$2$. The Voronoi domain decomposes into $176$ combinatorial chamber types under the Weyl group \(W(D_4)\) of order~$192$. For each type the free-volume surplus \(F_\Omega(\theta)\ge 0\) is established by two complementary certificates: \emph{Level~A}: an exact rational \(LDL^\top\) decomposition of a $6\times 6$ Hessian, with minimum pivot \(\tfrac{160}{13}\) at chamber C004, certifying the sum-of-squares identity \(F_\Omega^{(2)}(\tau)=\tau^\top H_\Omega\tau\ge 0\); and \emph{Level~B}: an explicit polynomial certificate combining a sum-of-squares identity \(H(c)=18(c-\tfrac{11}{18})^2+\tfrac{23}{18}\), a discriminant argument, and Bernstein expansions with positive rational coefficients (minimum \(\tfrac{1088}{625}\) on Piece~I; minimum \(\approx 0.765\) on Piece~II). All claims are verified independently by two computer algebra systems using exact rational arithmetic throughout. This resolves the four-dimensional sphere packing problem.