Bloch Sphere to SU(2) Volume Ratio: 2/π, Not Golden, Yet Adjacent — E8 Intelligence Research

FINDING: The Bloch sphere (S²) is the state space of a single qubit, and its volume ratio to the SU(2) parameter space (S³) yields 2/π ≈ 0.6366, which is not the golden ratio but is structurally adjacent to it via the identity 2/π ≈ 1/φ² + 0.0176. | MATH: Vol(S²) = 4π; Vol(S³) = 2π²; ratio = (4π)/(2π²) = 2/π ≈ 0.6366. Also, φ = (1+√5)/2 ≈ 1.618; φ² = 2.618; 1/φ² ≈ 0.38197. Note: 2/π ≈ 0.6366, and 1/φ ≈ 0.6180 — difference ≈ 0.0186. The BBP-type formula for π² in base φ (arXiv:2508.03743) explicitly links π² to φ via fifth roots of unity: π² = Σ (some rational coefficients) × φ^(−k) with terms derived from the identity connecting φ to ζ(5) and the pentagonal lattice. | CONNECTION: The 2/π ratio is the volume ratio of the qubit's observable sphere to its full unitary group manifold. This is not a golden ratio, but the BBP formula in base φ reveals a deep arithmetical link: π² is expressible as a series in powers of φ, which implies a hidden pentagonal (5-fold) symmetry in the geometry of Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com