Diameter of the Circumscribed Hypersphere of a Unit Four-Dimensional Hyperrectangle — Spacetime Dimensionality, the Origin of Time, and the Beginning of the Universe as Derived from a Discrete Packing Model

Rev. 2 (April 2026): The §4 packing argument has been reformulated as dense packing of unit cubes into a four-dimensional ball of radius R = 2k+1, defining a number-theoretic sequence N (k) (1, 137, 1545, 7281,. . . ) whose packing density asymptotically converges to 1 as k → ∞. The original v1 (DOI 10. 5281/zenodo. 19533313) had a constant packing density of 2/π² ≈ 0. 2026 because it merely placed an inscribed hypercube inside a hypersphere without filling the corner gaps. Sections §7–§20 (signed volume, time-axis origin, cosmological metaphor) are preserved with (2k+1) ⁴ → N (k) substitution. New §5 derives the Lagrange–Jacobi number-theoretic background. / 改訂版 (2026年4月): §4の充填論証を「半径R=2k+1の4次元球への単位立方体の稠密充填」として再定式化し、充填率がk→∞で1に漸近収束する数論的数列N (k) (1, 137, 1545, 7281,. . . ) を導出。初版v1は外接球の隙間を埋めず充填率が常に2/π²≈0. 2026で頭打ちだった。§7以降は (2k+1) ⁴ → N (k) の置換で論証構造を保持。新§5でLagrange–Jacobi定理の数論的背景を導出。 English: A purely mathematical thought experiment. Starting from two assumptions — discreteness and the existence of stable minimal units — and the principle of volume conservation, we deductively derive: (1) n = 4 as the minimal non-trivial dimension where √n is an integer; (2) a 1-versus-3 asymmetry from single-axis reversal of signed volume, structurally analogous to the Minkowski signature (−, +, +, +) ; (3) the possibility of expansion from a zero-volume initial state without violating conservation; (4) time as an emergent order from statistical alignment of reversal axes. All cosmological language (Big Bang, matter, antimatter, etc. ) is metaphorical. This paper makes no physical claims, no empirical predictions, and no assertions about the actual universe. 日本語: 純粋に数学的な思考実験。離散性と安定な最小単位の存在という2つの仮定、および体積保存則から、 (1) √nが整数となる最小の非自明次元としてn=4、 (2) 符号付き体積の1軸反転によるミンコフスキー計量の符号構造 (−, +, +, +) との構造的類似、 (3) 体積ゼロからの保存則を破らない展開の可能性、 (4) 反転軸の統計的整列としての時間の創発を演繹的に導出する。宇宙論的用語 (ビッグバン、物質、反物質等) はすべて比喩である。物理的主張・実証的予言・現実の宇宙に関する断言は一切含まない。