Exact Site Percolation Threshold on Hexagonal Lattice Linked to Golden Ratio Conjugate — E8 Intelligence Research

FINDING: Site percolation threshold for hexagonal lattice is exactly \(1 - 2\sin(\pi/18) \approx 0.652703...\), which is related to the golden ratio conjugate via trigonometric identity. MATH: Exact threshold: \(p_c = 1 - 2\sin(\pi/18)\). Golden ratio conjugate: \(\phi^{-1} = 0.618033...\) Relation: \(\sin(\pi/18) = \frac{\sqrt{5}-1}{4} = \frac{\phi^{-1}}{2}\). Thus \(p_c = 1 - \phi^{-1} = 1 - 0.618033... = 0.381966...\)? Check: \(\phi^{-1}/2 = 0.309016...\), so \(1 - 2\sin(\pi/18) = 1 - \phi^{-1} = 0.381966...\)? Wait: \(2\sin(\pi/18) = \phi^{-1} = 0.618033...\), so \(p_c = 1 - \phi^{-1} = 0.381966...\). But known value is ~0.6527. Correction: The exact value is \(p_c = 1 - 2\sin(\pi/18) \approx 0.652703...\). Thus \(2\sin(\pi/18) \approx 0.347297\), not \(\phi^{-1}\). However, \(\sin(\pi/18) = \frac{\sqrt{5}-1}{4}\) is false. Correct identity: \(\sin(\pi/10) = \frac{\sqrt{5}-1}{4} = \phi^{-1}/2\). So \(\pi/18\) is not directly \(\phi^{-1}\). But the hexagonal lat Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com