An Empirical Constraint on the Hubble Constant from the Relation Ω_γ = α²

This note examines the empirical identity Ω_γ = α², which equates the present-day CMB photon density parameter to the square of the fine-structure constant — two dimensionless quantities of unrelated physical origin. Because Ω_γ depends on the Hubble constant while α² does not, imposing the identity as an exact constraint converts the precisely measured CMB temperature (FIRAS) into a single predicted value, H₀ = 68.15 km/s/Mpc, with a ±0.03 band propagated from the temperature error. The prediction sits between the Planck (1.5σ) and SH0ES (4.7σ) determinations of H₀. No derivation from first principles is claimed: the relation is presented as a falsifiable empirical hypothesis, with its possible physical significance and the look-elsewhere caveat discussed explicitly. Gravitational-wave standard sirens and strong-lensing time delays are expected to confirm or exclude the prediction within the decade.