The Fine-Structure Constant at Recombination: Microwave-Background Bounds and the Frame's Logarithmic Coupling

The cosmic microwave background records the physics of hydrogen recombination at redshift 1090, where the fine-structure constant enters the binding energies of hydrogen and the Thomson cross-section that couples matter to radiation. Analyses of the Planck data find alpha at recombination consistent with its laboratory value, with a 95 percent window of about 3.60e-03. Under logarithmic accumulation that window corresponds to a coupling of 5.15e-04 per e-fold of expansion - some 585 times weaker than the coupling the Oklo record sets at low redshift. A shift at the window's edge would move the Thomson cross-section by 7.20e-03. The microwave bound is thus the frame's early-universe anchor for alpha: loose in absolute terms, but earned at an epoch no other probe reaches directly. Computed from published analyses, not measured.