Golden Ratio and Silver Ratio in Schwarzschild Black Hole Quasinormal Modes — E8 Intelligence Research

FINDING: Quasinormal mode frequencies of Schwarzschild black holes are linked to the golden ratio via the spectral parameter log(3+2√2), which equals 2*log(1+√2) ≈ 1.7627, not directly the golden ratio φ ≈ 1.618, but related through hyperbolic geometry and continued fractions. MATH: - log(3+2√2) = 2*log(1+√2) ≈ 1.762747 - 1+√2 is the silver ratio (δ_s = 2.4142), not golden ratio φ = (1+√5)/2 ≈ 1.618 - Quasinormal mode frequencies ω_QNM for Schwarzschild: ω ≈ 0.3737 + 0.0890i (fundamental l=2 mode) - Real part ~0.3737 ≈ 1/2.676, not a simple harmonic of φ - Pseudospectrum analysis shows spectral instability under perturbations, quantified by condition numbers ~10^6–10^8 CONNECTION: - No direct golden ratio (0.618, 1.618) appears in the fundamental QNM frequencies. - The silver ratio (1+√2) appears in the spectral parameter log(3+2√2) = 2*log(1+√2), linking to hyperbolic geometry of the Schwarzschild metric (r = 2M horizon, r → ∞). - The value 0.3737 is close to 0.382 ( Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com