Determination of the 5-Dimensional Kissing Number via Golden-Ratio Eulerian Paths on a Machin-Nexus Manifold: Dimensional Inheritance and the 4/3 Packing Density Proof

Determination of the 5-Dimensional Kissing Number via Golden-Ratio Eulerian Paths on a Machin-Nexus Manifold: Dimensional Inheritance and the 4/3 Packing Density Proof 【Publication date / 公開日】 2026-01-16 【Resource type / リソースタイプ】 Publication / Journal article (Preprint) 【Abstract / 要旨】 The kissing number problem in five dimensions (\tau_5), currently bounded between 40 \le \tau_5 \le 44, remains a fundamental challenge in discrete geometry. This paper presents a novel analytical framework, "Nexus Theory," to determine the exact value of \tau_5. By introducing the "Tsumoto Ratio" (\lambda_n = (n-1)/(n-2)), we demonstrate a harmonic inheritance of packing density from 4-dimensional baselines (16 domains) to 5-dimensional manifolds (32 domains). Through the integration of Machin-like phase constants and Golden-Ratio (\phi) based Eulerian paths, we provide a structural proof that the maximum number of non-overlapping unit spheres touching a central sphere in 5D is exactly 42. This transition mirrors the harmonic decay from a "Perfect 5th" (1.5) in 4D to a "Perfect 4th" (1.33) in 5D. This study confirms that higher-dimensional geometry is governed by predictable, discrete inheritance patterns of informational density. 【Keywords / キーワード】 * Nexus Theory * Kissing Number Problem * 5D Geometry * Tsumoto Ratio * Machin's Formula * Golden Ratio * Eulerian Path * Sphere Packing * Dimensional Inheritance 【Access right / アクセス権】 Open Access (Creative Commons Attribution 4.0 International)