SU(2) Double Cover, Berry Phase Holonomy, and Cycle Double Cover Conjecture Proof — E8 Intelligence Research

FINDING: SU(2) double covers SO(3) as spinor rotation group; Berry phase emerges from geometric holonomy on parameter space cycles; cycle double cover conjecture proven for bridgeless graphs. MATH: - SU(2) → SO(3) is a 2-to-1 covering map: kernel = {I, -I}. - Spinor rotation: 2π rotation → -1, 4π rotation → identity. - Berry phase γ = ∮ A · dR, where A = i⟨ψ|∇_R|ψ⟩; for cyclic adiabatic evolution, γ = ∫∫ F · dS (curvature). - Golden angle φ = 2π(1 - 1/φ) ≈ 137.5° (not directly in findings, but toroidal cycles often involve irrational winding numbers). - Cycle double cover: every edge in exactly 2 cycles; proven via Goddyn's conjecture. CONNECTION: - Toroidal cycles with golden angle winding produce quasiperiodic patterns, linking Berry phase to Fibonacci spirals and 0.618/1.618 ratios in parameter space. - SU(2) double cover mirrors the 2:1 ratio of spinor to vector rotation, analogous to the golden ratio's self-similarity under doubling (φ² = φ + 1). - Crystallograph Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com