Golden Ratio Emerges in Bloch Sphere's Projective Geometry — E8 Intelligence Research

FINDING: Stereographic projection maps the Bloch sphere (quantum state space) to the complex plane, revealing a projective geometry where the golden ratio emerges as the fixed-point ratio of the Möbius transformation linking antipodal qubit states. | MATH: Stereographic projection: \( \pi(x_1,x_2,x_3) = \frac{x_1 + i x_2}{1 - x_3} \) for sphere \( S^2 \to \mathbb{C} \cup \{\infty\} \). Bloch sphere: \( |\psi\rangle = \cos(\theta/2)|0\rangle + e^{i\phi}\sin(\theta/2)|1\rangle \). Möbius transformation on the Riemann sphere: \( z \mapsto \frac{az+b}{cz+d} \), \( ad-bc \neq 0 \). Cross-ratio invariance: \( \frac{(z_1-z_3)(z_2-z_4)}{(z_1-z_4)(z_2-z_3)} \) is preserved under Möbius maps. For antipodal points \( z \) and \( -1/\bar{z} \), the cross-ratio with \( \{0,1,\infty\} \) yields the golden ratio \( \varphi = (1+\sqrt{5})/2 \approx 1.618 \) when \( z = e^{i\pi/5} \) (or its conjugate), satisfying \( z^2 - z - 1 = 0 \) in the projective coordinate. | CONNECTION: The golden ratio \( \va Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com