Resolving the Hadwiger–Nelson Problem: Analytical Proof of the 6-Color Chromatic Number via Machin-Phase Shifts and 32-Domain Inheritance

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)