Fourier Formula Reveals Hidden Modular Structure in Cyclotomic Polynomials — E8 Intelligence Research

Abstract

FINDING: Cyclotomic polynomials define minimal polynomials of primitive roots of unity; a Fourier-analytic formula evaluates them at non-primitive roots, revealing hidden modular structure. | MATH: Φₙ(x) = ∏_{1≤k≤n, gcd(k,n)=1} (x − ζₙᵏ), ζₙ = e^{2πi/n}; deg Φₙ = φ(n); Φₙ(x) = ∏_{d|n} (x^d − 1)^{μ(n/d)} (Möbius inversion); at roots of unity ζₙᵐ with gcd(m,n)=d>1, Φₙ(ζₙᵐ) = Φ_{n/d}(1)^{φ(d)/φ(n/d)}·(sign/unit factor) — exact formula from finite Fourier transform (arXiv:1611.06783). | CONNECTION: Roots of unity form cyclic group Zₙ — a 1D lattice with rotational symmetry Cₙ. The Fourier transform on this group is the discrete Fourier transform (DFT), whose eigenvalues are Gauss sums — intimately tied to quadratic residues and crystallographic root systems (Aₙ lattice). The ratio φ(n)/n (Euler totient density) governs the sparsity of primitive roots; for n=60 (base-60), φ(60)/60 = 16/60 = 0.2667, and the primitive 60th roots include angles 6°, 18°, 30°, 42°, 54°, 66°, 78° — multiples of 6 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com