Percolation Thresholds Are Lattice-Dependent, Not Golden Ratio — E8 Intelligence Research

FINDING: 2D percolation thresholds are lattice-dependent critical probabilities; the golden ratio appears only in an unrelated self-application paper, not in percolation itself. | MATH: Site percolation on square lattice: \(p_c \approx 0.592746\); bond percolation square: \(p_c = 0.5\); triangular lattice site: \(p_c = 0.5\); triangular bond: \(p_c = 2\sin(\pi/18) \approx 0.347296\) (exact, Smirnov); hexagonal bond: \(p_c = 1 - 2\sin(\pi/18) \approx 0.652704\). Note: \(2\sin(\pi/18) = \frac{\sqrt{3} - 1}{2} \approx 0.366025\) — *not* 0.382. | CONNECTION: Triangular lattice is the root lattice \(A_2\) (hexagonal symmetry, 6-fold crystallographic). The exact \(p_c\) for triangular bond percolation involves \(\sin(\pi/18)\), which is algebraic of degree 6 — related to the 18-gon, not the golden ratio. The golden ratio \(\Phi = 1.618...\) does **not** appear in any known 2D percolation threshold. The arxiv paper (2510.08934) uses \(\Phi\) for a *different* problem (self-application/self-ce Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com