Penrose Tilings: 5D Cubic Projections Yield Critical Eigenstates via Golden Ratio — E8 Intelligence Research

Abstract

FINDING: Penrose tilings as 5D cubic lattice projections yield singular continuous spectra in tight-binding models, with critical (non-Bloch) eigenstates. | MATH: Projection from Z⁵ to 2D via irrational slope (golden ratio φ = (1+√5)/2 ≈ 1.618); tight-binding Hamiltonian H = -t Σ_{⟨i,j⟩} c_i† c_j on Penrose vertex graph; eigenstates exhibit power-law decay (criticality) rather than exponential localization or Bloch periodicity. | CONNECTION: Golden ratio φ (1.618) and its reciprocal φ⁻¹ (0.618) define the projection matrix; 5-fold symmetry links to icosahedral group (H₃ Coxeter); singular continuous spectrum implies fractal energy bands with scaling ratios related to φ. | DEPTH: 8 — Bridges aperiodic order, number theory (φ), and quantum criticality; shows that geometric harmony (φ) directly forces spectral singular continuity, a rare and profound mathematical-physics nexus. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com