Periodic Chowla Solved via DFT; Full Conjecture Remains Open — E8 Intelligence Research

FINDING: The search results are dominated by prime-number videos (Goldbach, Maynard, Tao) and a topology proof (Yu), but the only direct Chowla hit is an arXiv note claiming a solution to the *periodic* Chowla conjecture via discrete Fourier transform — not the full conjecture. | MATH: Möbius function μ(n) ∈ {−1,0,1}; Chowla conjecture: ∑_{n≤x} μ(n+a₁)⋯μ(n+aₖ) = o(x) for any fixed distinct integers aᵢ. The arXiv note (2208.12219v8) treats periodic μ and uses DFT: if μ is periodic with period q, then the k-point correlation sum becomes a finite sum over residues mod q, expressible as a product of Fourier coefficients ĉₘ = (1/q)∑_{n mod q} μ(n)e^{−2πimn/q}. The claim: for periodic μ, the Chowla sum vanishes exactly (not just o(x)) unless the Fourier support satisfies a certain additive constraint. No new constants or ratios appear. | CONNECTION: Weak — DFT introduces roots of unity e^{2πi/q}, which are base-60-friendly only if q divides 60 (q=2,3,4,5,6,10,12,15,20,30,60). The periodic Mö Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com