Stereographic Projection: The Geometric Bridge from Bloch Sphere to Quantum States — E8 Intelligence Research

FINDING: Stereographic projection maps the Bloch sphere (quantum state space) to the complex plane, preserving angles but not areas, and is the geometric bridge between quantum two-level systems and projective space — while the golden ratio appears in the literature as a claimed harmonic constant, the lattice QCD result is a separate, concrete numerical computation. | MATH: Stereographic projection: \( z = \frac{x + iy}{1 - w} \) for sphere point \((x,y,w)\) with \(x^2+y^2+w^2=1\), mapping to complex plane \(z\); inverse: \(x = \frac{2\Re(z)}{1+|z|^2}, y = \frac{2\Im(z)}{1+|z|^2}, w = \frac{|z|^2-1}{|z|^2+1}\). Bloch sphere: pure state \(|\psi\rangle = \cos(\theta/2)|0\rangle + e^{i\phi}\sin(\theta/2)|1\rangle\) — the angles \((\theta,\phi)\) are exactly the spherical coordinates, and the cross-ratio of four points on the sphere is invariant under Möbius transformations (the automorphism group of the Riemann sphere). Golden ratio: no equation appears in the provided sources; the PBS vi Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com