|ψ_+⟩ = u(r) e^{+iφ/2} |ψ_-⟩ = u(r) e^{-iφ/2} Kramers doublet Odd boundary parity → SU(2) doublet Core dc1 +δ₀ +2δ₀ +3δ₀ Berry phase per circuit After full circuit: γ = 4δ₀ Spin-flip amplitude |A_flip| = sin δ₀ Both doublet members SU(2) coupling g = 2 sin δ₀ g² = 4 sin²δ₀ EM coupling e = g sin θ_W α = g²sin²θ_W/(4π) Factor of 2 from doublet → g = 2 sin δ₀, not sin δ₀

Left: the Kramers doublet (|ψ_+⟩, |ψ_-⟩) in the fermion’s counter-rotating boundary — two CdGM bound states protected by time-reversal symmetry, forming the SU(2) isospin doublet. Right: a dc1 quasiparticle circling the vortex accumulates Berry phase 4δ₀ per topologically complete cycle (two circuits for spin-1/2). The doublet’s two members each contribute sin δ₀ to the transition amplitude, giving g = 2 sin δ₀ and ultimately α = g² sin²θ_W/(4π).