Golden Ratio Fractal Spectrum in Penrose Tiling Hamiltonians — E8 Intelligence Research

Abstract

FINDING: Penrose tilings enforce 5-fold symmetry via golden ratio inflation rules, yielding a non-repeating Hamiltonian eigenvalue spectrum with fractal statistics linked to the golden ratio. MATH: - Inflation/deflation ratio = φ = (1+√5)/2 ≈ 1.618 - Key constants: φ, φ² = φ+1 ≈ 2.618, 1/φ = φ−1 ≈ 0.618, 1/φ² ≈ 0.382 - Eigenvalue spacing distribution in Penrose tiling Hamiltonians follows a singular continuous spectrum with level statistics deviating from Wigner-Dyson (random matrix) or Poisson; instead, the integrated density of states scales as N(E) ∝ E^d with fractal dimension d = ln(φ)/ln(λ) where λ is a scaling factor related to the golden ratio. - The Hamiltonian's off-diagonal hopping terms are determined by tile adjacency, which is governed by the Penrose rhombus angles (36°, 72°, 108°, 144°) — all multiples of 36° = π/5, linking to the pentagon's golden ratio geometry. CONNECTION: - Direct geometric harmony: φ appears in tile side ratios, inflation multipliers, and Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com