4π-spinor-periodicity: Tr(R(2π)) = 2·cos(π) = −2; spin-1/2 states acquire (−1) phase

4π-spinor-periodicity theorem. From framework F8, the rotation R(n̂, θ) = cos(θ/2)·I − i·sin(θ/2)·(n̂·σ). At θ = 2π: cos(π) = −1, sin(π) = 0, so R(2π) = −I — a 2π rotation returns the spinor to its antipodal state (R(2π)|ψ⟩ = −|ψ⟩). Only…