Fibonacci Word Quasicrystal: Dense Bragg Peaks Indexed by Z[φ] — E8 Intelligence Research

FINDING: The Fibonacci word, when used as a cut-and-project rule, generates a 1D quasicrystal whose Fourier spectrum is a dense set of Bragg peaks indexed by Z[φ], with intensities governed by a product of sinc-like functions — the core of the "Fibonacci quasicrystal" diffraction pattern. | MATH: Let φ = (1+√5)/2 ≈ 1.618034. The Fibonacci word w_n = floor((n+1)φ) − floor(nφ) ∈ {0,1}. Its Fourier transform (discrete) has support on frequencies k = 2π(m + n/φ) for m,n ∈ Z, i.e., the Z-module Z[φ] = {a + bφ : a,b ∈ Z}. The structure factor S(k) = |Σ_n w_n e−2πikn|² scales as N² · sinc²(πN(k − k₀)) · |F(k₀)|² near each Bragg peak k₀, where |F(k₀)|² ∝ 1/(φ + 2 − 2cos(2πk₀)) — a Lorentzian envelope. The diffraction measure is a pure point measure: μ = Σk∈Z[φ] I(k) δ_k, with I(k) = (sin²(πkφ)/(φ² sin²(πk))) · (sin²(πk)/(φ² sin²(πk/φ))) — a product of two squared Dirichlet kernels. | CONNECTION: The golden ratio φ appears as the slope of the cut line; the acceptance window has width 1/φ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com