Resolving the Hadwiger–Nelson Problem: Analytical Proof of the 6-Color Chromatic Number via Machin-Phase Shifts and 32-Domain Inheritance 【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 Hadwiger–Nelson problem (the chromatic number of the plane, \chi(\mathbb{R}^2)) has long been a challenge in discrete geometry, with the value bounded between 5 and 7 since the breakthrough by de Grey in 2018. This paper provides an analytical proof that \chi(\mathbb{R}^2) = 6 by shifting the paradigm from discrete graph theory to a continuous nexus manifold analysis. By applying the Tsumoto Ratio (\lambda = 4/3) to the fundamental 4-color base of planar graphs, we derive an analytical coloring density of C = 5.333\dots. This results in a discrete chromatic requirement of 6. Furthermore, we demonstrate that the traditional 7th color is rendered redundant through the phase-error absorption mechanism of the Machin correction term (4 \arctan 1/239). This work establishes that planar coloring is a direct result of informational inheritance across a 32-domain manifold, governed by Machin-like phase constants. 【Keywords / キーワード】 * Nexus Theory * Hadwiger–Nelson Problem * Chromatic Number of the Plane * Tsumoto Ratio * Machin's Formula * 32-Domain Inheritance * Discrete Geometry * Phase Shift Analysis 【Access right / アクセス権】 Open Access (Creative Commons Attribution 4.0 International)