Chern-Simons Invariants of Homology Spheres as Sums Over Roots of Unity — E8 Intelligence Research

FINDING: Integral homology spheres yield SU(2) Chern-Simons invariants as sums over roots of unity, linking topological invariants to number-theoretic phases. | MATH: Chern-Simons invariant \( CS(A) = 1/8π^2 ∫_M Tr(A dA + 2/3 A A A) \) mod \(Z\); for SU(2) on integral homology spheres, partition function \( Z_k(M) = ∑ⱼ₌₀ᵏ⁻² S₀ⱼ^2 e2π i (k+2) CS_j \) with \( S₀ⱼ = √2/k+2 sin(π(j+1)/k+2) \); roots of unity \( e2π i r/(k+2) \) appear. | CONNECTION: Summation over \( j \) involves \(sin(π(j+1)/(k+2))\) — ratios \(sin(π/5)/sin(2π/5) ≈ 0.618\) (golden ratio conjugate) appear for \(k=3\); base-60 emerges in modular \(S\)-matrix entries as rational angles; root system \(A_1\) (SU(2)) underlies lattice structure of weights. | DEPTH: 8 — directly ties 3-manifold topology to quantum invariants via roots of unity, echoing geometric harmony in modular data. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com