Fourier Analysis Yields Explicit Cyclotomic Values at Non-Primitive Roots — E8 Intelligence Research

FINDING: The search results are dominated by pedagogical material on cyclotomic polynomials (Φₙ(x)), with one substantive arXiv paper (1611.06783v2) deriving explicit values of Φₙ at non-primitive roots of unity via finite Fourier analysis. No direct hit on arXiv 1105.0669 (likely a different paper on FLT distribution) — the search returned only generic cyclotomic content. MATH: - Φₙ(x) = ∏_{1≤k≤n, gcd(k,n)=1} (x − ζₙᵏ), where ζₙ = e^{2πi/n}. - Degree of Φₙ = φ(n) (Euler totient). - Key identity: xⁿ − 1 = ∏_{d|n} Φ_d(x). - The arXiv paper (1611.06783) gives: for m | n, m ≠ n, Φₙ(ζₘ) = p^{φ(n)/φ(p)} if n/m is a power of prime p, else 1 (up to sign). This is a clean, explicit formula — a rare closed form for cyclotomic values at non-primitive roots. - No golden ratio, no 0.618/1.618, no base-60, no crystallographic constants appear in these results. CONNECTION: - Cyclotomic polynomials are the building blocks of root systems of type Aₙ (via the discriminant of xⁿ−1) and app Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com