Golden Ratio's Irrationality Proven via Pentagon Diagonal Incommensurability — E8 Intelligence Research

FINDING: Geometric proof that the golden ratio φ is irrational, derived from the incommensurability of the pentagon's diagonal and side. | MATH: φ = (1 + √5)/2 ≈ 1.618; diagonal/side = φ; incommensurability proven via infinite descent (Euclid's method). | CONNECTION: φ appears in pentagon star diagonals (ratio 1:φ), linking to 5-fold symmetry, quasicrystal diffraction patterns, and the golden angle (≈137.5° = 2π/φ²). | DEPTH: 8 — Foundational to number theory, geometry, and crystallography; reveals irrationality embedded in regular pentagon symmetry. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com