Self-Similar Golden Ratio Peaks in Fibonacci Quasicrystals via Supersymmetric Oscillators — E8 Intelligence Research

Abstract

FINDING: Fibonacci quasicrystal Fourier spectra exhibit self-similar Bragg peaks at powers of the golden ratio, with a rigorous quantum-calculus framework linking Fibonacci divisors to N=2 supersymmetric golden oscillators. MATH: - Golden ratio φ = (1+√5)/2 ≈ 1.6180339887; reciprocal φ⁻¹ = φ−1 ≈ 0.6180339887; φ² = φ+1 ≈ 2.6180339887. - Fibonacci quasicrystal (1D) Fourier transform: peaks at k = 2π(n + mφ) for integers n,m; intensity scales as |F(k)|² ∝ (sin(π(n+mφ)) / π(n+mφ))² — but for the *Fibonacci word* (Sturmian), the Fourier spectrum is *pure point* with peaks at frequencies n + mφ, and the peak amplitudes follow a power law in φ: |F(k)|² ~ φ^{-2|m|} for large |m| (self-similarity under scaling by φ). - Quantum calculus (arXiv:2410.04169v2): defines Fibonacci divisor derivative D_q f(x) = [f(φx) − f(φ⁻¹x)] / [(φ−φ⁻¹)x], with q = φ. Binet formula: F_n = (φⁿ − (−φ)⁻ⁿ)/√5. The Fibonacci divisor number operator N_F = (φ^{N} − (−φ)^{−N})/√5 acts on Fock space, yielding energy Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com