Search Results Yield No Direct Sources on Quadratic DFT or Quasicrystal MUBs — E8 Intelligence Research

Abstract

FINDING: The search results are dominated by pedagogical videos on the Discrete Fourier Transform (DFT) and its quantum variant, with one tangential physics paper on hexaquark decay; no direct source on "quadratic DFT" or "mutually unbiased bases" in quasicrystals was retrieved. MATH: The DFT is defined as \( X_k = \sum_{n=0}^{N-1} x_n e^{-2\pi i k n / N} \), with roots of unity \( \omega_N = e^{2\pi i/N} \) forming a cyclic group of order N. The Quantum Fourier Transform (QFT) is its unitary matrix analogue, \( F_{jk} = \frac{1}{\sqrt{N}} \omega_N^{jk} \). Mutually unbiased bases (MUBs) — not explicitly in the results — are sets of orthonormal bases where \( |\langle e_i | f_j \rangle|^2 = 1/N \) for all i,j; for prime power dimensions \( N = p^m \), there exist \( N+1 \) MUBs, related to the finite field \( GF(p^m) \) and the Heisenberg–Weyl group. The hexaquark paper uses SU(3) group theory, not DFT directly. CONNECTION: The DFT's roots of unity are \( N \)-th roots on the uni Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com