Turaev-Viro Invariants and SO(3) WRT: Root-of-Unity Decomposition — E8 Intelligence Research

FINDING: Turaev-Viro state sums at SU(2) root-of-unity are topological invariants whose 0-term decomposition yields the square of the SO(3) WRT invariant, with categorical defect structures tied to quantum group representations. | MATH: TV invariant \(Z_{TV}(M) = \sum_{\{j_e\}} \prod_{\text{tetrahedra}} \{6j\}_q \prod_{\text{edges}} d_{j_e}\), where \(q = e^{2\pi i/r}\) (root of unity), \(d_j = (-1)^{2j}[2j+1]_q = \frac{\sin((2j+1)\pi/r)}{\sin(\pi/r)}\). The 0-term summand equals \(|Z_{SO(3)}^{WRT}(M)|^2\). For \(r=5\) (or \(r=6\) in some conventions), \(\sin(\pi/5) = \frac{\sqrt{10-2\sqrt{5}}}{4} \approx 0.5878\), and the quantum dimension ratio \([2]_q/[3]_q\) yields \(\phi\)-related values. | CONNECTION: The root of unity \(q = e^{2\pi i/5}\) (order 5) directly links to the golden ratio: \(2\cos(\pi/5) = \phi = 1.618\), \(2\cos(2\pi/5) = \phi^{-1} = 0.618\). The quantum \(6j\)-symbols at \(r=5\) contain factors of \(\sqrt{\phi}\) and \(\sqrt{\phi^{-1}}\). The crystallographic connec Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com