Solid Angle and the Fine-Structure Constant: Uncovering the Exact Geometric Relationship Between Hydrogen's Physical Scales

We show that the ratio of the hydrogen Lyman-limit wavelength λ_Ly to the Bohr radius a₀ satisfies the exact identity λ_Ly/a₀ = 4π/α = Ω_sph/α, where α ≈ 1/137 is the fine-structure constant and Ω_sph = 4π sr is the solid angle of a complete sphere. Using only CODATA 2022 values of the fundamental constants and four lines of algebra, we demonstrate that every factor in this identity carries a transparent physical meaning: α measures the ratio of the electron's ground-state speed to c; the factor 2π encodes one complete oscillation cycle (h = 2πℏ); and the remaining factor of 2 follows from the virial theorem applied to the Coulomb potential, ⟨T⟩ = −E_total, which sets E_R = ½ m_e c² α². These two factors—2π and 2—are structurally identical to the azimuthal and polar integrals whose product yields Ω_sph = 4π sr. We stress that this decomposition is tied to the convention of expressing photon energy through the linear frequency ν via the Planck constant h; in a natural-unit convention based on ℏ the factor 2π does not appear, and the decomposition changes accordingly. The result is verified numerically to better than one part in 10⁸ using CODATA 2022 data. We further present the hierarchy of all three electromagnetic length scales of hydrogen (r_e, a₀, λ_Ly), each separated by specific powers of α and the factor 4π, and we discuss implications for dimensional analysis, atomic units, and extensions to hydrogenic ions and two-photon spectroscopy. The finite proton-mass correction shifts the ratio by +0.054%, in agreement with the measured Lyman limit. No new physics is proposed; the paper illuminates a geometric structure that is exact, experimentally grounded, and largely absent from standard presentations of atomic physics.