Hopf Fibration Links Qubit Geometry to Golden-Ratio Entanglement Symmetry — E8 Intelligence Research

Abstract

FINDING: The Hopf fibration (S³→S²) is the geometric backbone linking quantum qubit states to spatial rotations, with the binary icosahedral group as its discrete symmetry — directly encoding golden-ratio structure in entanglement geometry. | MATH: Hopf map: \(S^3 \xrightarrow{h} S^2\), fiber \(S^1\); qubit Bloch sphere = \(S^2\) (pure states), \(S^3\) = normalized spinor (SU(2) double cover of SO(3)); holonomy of rotating disc = \( \oint \omega = 2\pi(1-\cos\theta) \) (solid angle, from arXiv:2512.04481v2); binary icosahedral group \(2I\) ⊂ SU(2) has order 120, contains golden-ratio elements: rotation angles \( \frac{2\pi}{5}, \frac{4\pi}{5} \) with cosines \( \frac{\sqrt{5}\pm1}{4} \) → \( \cos(2\pi/5) = \frac{\sqrt{5}-1}{4} = \frac{\phi-1}{2} \approx 0.309 \), \( \cos(4\pi/5) = -\frac{\phi}{2} \approx -0.809 \); quaternion norm 1 → \(a^2+b^2+c^2+d^2=1\) (S³). | CONNECTION: **Direct golden-ratio link**: The binary icosahedral group is the preimage of the icosahedral rotation group un Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com