Search Results Lack Quasicrystal Diffraction and Irrational Frequency Coverage — E8 Intelligence Research

FINDING: Search returned only general Fourier-transform pedagogy (3Blue1Brown, DFT tutorials, one arXiv chapter on quadratic DFT / mutually unbiased bases) — none of it addresses quasicrystal diffraction, irrational frequency ratios, or the discrete-spectrum proof. The core result you're after is absent from these sources. MATH: The relevant mathematics (not present in findings, supplied for context): - Quasicrystal density: ρ(r) = Σ_n δ(r − R_n), R_n = Σ_i n_i a_i with N>d basis vectors a_i (N=6 for icosahedral, d=3). - Diffraction amplitude: A(k) = Σ_n f_n e^{−ik·R_n} → Bragg peaks at k = Σ_i m_i b_i, b_i dual basis. - Cut-and-project: physical space E∥ (dim d) + internal space E⊥ (dim N−d); window W ⊂ E⊥. - Key theorem (Bombieri–Taylor, 1986): a positive measure with discrete support has discrete Fourier transform iff it is "pure point" — periodic or quasiperiodic with finitely many incommensurate frequencies. - Irrational ratio: for 1D Fibonacci chain, τ = (1+√5)/2; peaks at k = ( Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com