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

v2 (April 2026): Reference list refined per paper. External references corrected and finalized; no self-citations. v1 (DOI 10.5281/zenodo.19837592) remains the original publication. 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 ≤ 60. The volume deficit Δ(R) := V₄(R) - N(k) admits the asymptotic expansion Δ(R) = (16π/3) R³ - 6π R² + O(R), derived from inclusion–exclusion, with leading constant c = 8/(3π) ≈ 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 ≤ 60 まで厳密計算。体積不足 Δ(R) = V₄(R) - N(k) は包除原理から Δ(R) = (16π/3) R³ - 6π R² + O(R) という漸近展開を持ち、主要係数 c = 8/(3π) ≈ 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。