Discrete Structure of a Four-Dimensional Ball: Unit-Cube Packing and the Asymptotic Volume Deficit (Paper 3)

English: We study integer-lattice packing of unit cubes inside a four-dimensional ball $B(R)$ of radius $R = 2k+1$. The count $N(k)$ is computed exactly for $k \le 60$. The volume deficit $\Delta(R) := V_4(R) - N(k)$ admits the asymptotic expansion $\Delta(R) = (16\pi/3) R^3 - 6\pi R^2 + O(R)$, derived from inclusion–exclusion, with leading constant $c = 8/(3\pi) \approx 0.84883$ confirmed numerically to within 0.024%. Connected to Lagrange–Jacobi four-square theory.

日本語: 半径 $R = 2k+1$ の4次元球 $B(R)$ への整数格子単位立方体充填を扱う。$N(k)$ を $k \le 60$ まで厳密計算。体積不足 $\Delta(R) = V_4(R) - N(k)$ は包除原理から $\Delta(R) = (16\pi/3) R^3 - 6\pi R^2 + O(R)$ という漸近展開を持ち、主要係数 $c = 8/(3\pi) \approx 0.84883$ は数値で 0.024% 以内で確認される。Lagrange–Jacobi 四平方理論と接続。