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: \( z = x + iy/1 - w \) for sphere point \((x,y,w)\) with \(x^2+y^2+w^2=1\), mapping to complex plane \(z\); inverse: \(x = 2(z)/1+|z|^2, y = 2(z)/1+|z|^2, w = |z|^2-1/|z|^2+1\). Bloch sphere: pure state \(|ψ = cos(θ/2)|0 + eiφsin(θ/2)|1\) — the angles \((θ,φ)\) 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