Chapter 47: Hyperfine Structure and the Role of ι_τ

The proton is not a scalar source: it carries spin s_p = ½ and a magnetic moment μ_p. The coupling between the proton and electron magnetic moments splits the hydrogen ground state into a triplet (F = 1) and a singlet (F = 0), separated by Δ E_HF ≈ 5.9 × 10^{-6} eV—the hyperfine splitting. The resulting 21 cm radio line maps the large-scale structure of the universe. This chapter derives the splitting, surveys its observational significance, and clarifies where the master constant ι_τ = 2/(π + e) ≈ 0.341304 enters the physics and—equally important—where it does not.