Unit-Distance Graphs: 553-Vertex Proofs Tighten Plane Chromatic Bound to [5,7] — E8 Intelligence Research

FINDING: The chromatic number of the plane remains open; the best known lower bound is 5, achieved by explicit 553-vertex unit-distance graphs, with the Hadwiger–Nelson problem still unresolved (7 is the current upper bound). | MATH: Unit-distance graph: vertices in ℝ², edges iff |x−y| = 1. Chromatic number χ(ℝ²) ∈ [5,7]. The 553-vertex graphs are minimal under clausal proof minimization — no closed-form equation, but the key invariant is the independence ratio α(G)/|V| ≤ 1/χ. For χ=5, α ≤ 0.2|V|. The graphs are constructed via Moser spindle (7 vertices, χ=4) and Golomb graph (10 vertices, χ=4) as seeds, then iterated via graph substitution. | CONNECTION: The unit-distance constraint forces hexagonal lattice structure locally — the hexagonal tiling gives a 7-coloring (upper bound). The golden ratio φ=1.618 appears in the Moser spindle's coordinates (vertices at (0,0), (1,0), (1/2, √3/2), and rotated copies — the √3 factor is the hexagonal lattice constant). The ratio 0.618 = 1/φ appear Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com