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 \( CS(A) = \frac{1}{8\pi^2} \int_M \text{Tr}(A \wedge dA + \frac{2}{3} A \wedge A \wedge A) \) mod \(\mathbb{Z}\); for SU(2) on integral homology spheres, partition function \( Z_k(M) = \sum_{j=0}^{k-2} S_{0j}^2 e^{2\pi i (k+2) CS_j} \) with \( S_{0j} = \sqrt{\frac{2}{k+2}} \sin\left(\frac{\pi(j+1)}{k+2}\right) \); roots of unity \( e^{2\pi i r/(k+2)} \) appear. | CONNECTION: Summation over \( j \) involves \(\sin(\pi(j+1)/(k+2))\) — ratios \(\sin(\pi/5)/\sin(2\pi/5) \approx 0.618\) (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