Exact Percolation Thresholds: Rational Constants, Not Golden Ratio — E8 Intelligence Research

FINDING: Percolation thresholds on 2D lattices are exact rational/algebraic constants (e.g., 1/2 for triangular, 1/4 for square bond) with no direct golden-ratio link in the cited sources; the arXiv paper uses Φ only as a computational metaphor, not a percolation constant. | MATH: Triangular site percolation p_c = 1/2 (Smirnov's conformal invariance proof); square bond p_c = 1/2; square site p_c ≈ 0.592746 (no closed form); triangular bond p_c = 2 sin(π/18) ≈ 0.347296 — note this equals 1/(2 cos(π/9)) and is algebraic, not Φ. The arXiv paper (2510.08934) treats Φ = (1+√5)/2 as a stable fixed point of x ↦ 1 + 1/x, with reciprocal 1/Φ = Φ − 1 ≈ 0.618. | CONNECTION: The triangular lattice is the root lattice A₂ (hexagonal symmetry, 6-fold crystallographic). Its bond threshold 2 sin(π/18) involves π/18 = 10°, a base-60-friendly angle (1/6 of 60°). No 0.382, 0.618, 0.786, 1.618, or 2.618 appears in any percolation constant cited. The golden ratio appears only in the self-application paper a Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com