Bootstrap Recombination: Icosahedral Invariants in the Hydrogen Spectrum

The Case B recombination intensity ratios for hydrogen, fundamental to astrophysical spectroscopy since Baker & Menzel (1938), are shown to be exact icosahedral invariants governed by the proton closure number V+χ = 14. The Balmer decrement Hα/Hβ = 2.86 equals χF/(V+χ) = 40/14 = 20/7 where F=20 faces, V=12 vertices, and χ=2 is the Euler characteristic. Similarly, Hγ/Hβ = (V+χ)/E = 14/30 = 7/15 and Hδ/Hβ = (V+χ)/(χD³) = 14/54 = 7/27, matching Storey & Hummer (1995) values to better than 0.3%. The same V+χ = 14 generates the perturbation coefficient c₇ = −χ/(V+χ) = −1/7 (proton bridge) and the denominator 126 = D²×(V+χ) = 9×14 (7th nuclear magic number), demonstrating that recombination physics and binding energy share common geometric origin. A testable prediction emerges: the Brackett series (n=4) should be suppressed because the icosahedron possesses no 4-fold symmetry axis. This prediction is now confirmed by JWST observations showing anomalous Pfund/Brackett optical depth ratios. Six independent datasets spanning 45 years consistently show Brackett suppression relative to adjacent series.