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 (R): = V₄ (R) - N (k) admits the asymptotic expansion (R) = (16/3) R³ - 6 R² + O (R), derived from inclusion–exclusion, with leading constant c = 8/ (3) 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 (R) = V₄ (R) - N (k) は包除原理から (R) = (16/3) R³ - 6 R² + O (R) という漸近展開を持ち、主要係数 c = 8/ (3) 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。