The Bootstrap Hydrogen Spectrum: Fine Structure Encoded in Gross Structure

We demonstrate that the gross hydrogen spectrum (Rydberg formula) contains the fine structure coefficient embedded in its numerator. The Lyman series ratio (F+V)/D³ = 32/27 decomposes as F + V = (E − χ²) + 3χ, where E = 30 edges, χ = 2 is the Euler characteristic, and the term (E − χ²)/D² = 26/9 is precisely the α² binding correction c₂ from icosahedral perturbation theory. All seven spectral series numerators are powers of icosahedral primes {χ, D, d, V−d} = {2, 3, 5, 7}: Lyman gives 32 = χ⁵, Balmer gives 27 = D³, Humphreys gives 343 = (V−d)³, Hansen-Strong gives 2048 = χ¹¹. The perturbation coefficients show electron-proton symmetry: c₃ = +χ/(F+χ) and c₇ = −χ/(V+χ). This decomposition shows that gross and fine structure are not independent—both emerge from the same icosahedral geometry.