Abstract
<jats:p>We examine the identity \(\Omega_{\gamma} = \alpha^{2}\) relating two independently measured dimensionless quantities of distinct physical origin: the CMB photon density parameter and the squared fine structure constant. Because the photon density parameter depends on the Hubble constant while \(\alpha^{2}\) does not, the identity is not an approximate numerical coincidence but an exact constraint: it maps the measured photon temperature onto a single value of the Hubble constant, \(H_{0} = {68.15{km}s^{- 1}{Mpc}^{- 1}}\), with a \(\pm {0.03{km}s^{- 1}{Mpc}^{- 1}}\) uncertainty propagated from the FIRAS temperature. The significance of the relation rests not on how closely it holds—at its predicted \(H_{0}\) it holds exactly, by construction—but on whether that independently selected value agrees with Hubble-constant determinations that use no distance ladder. The predicted value is consistent with the Planck 2018 determination (TT,TE,EE+lowE+lensing, \(H_{0} = {67.36 \pm {0.54{km}s^{- 1}{Mpc}^{- 1}}}\)) at \(1.5\sigma\), lies between early-universe and intermediate-distance determinations, and is in \(4.7\sigma\) tension with the SH0ES distance-ladder value. We make no theoretical claim regarding the origin of the identity, treating it as an empirical hypothesis that predicts a definite \(H_{0}\). We quantify the prediction’s sensitivity to the inputs and discuss the look-elsewhere effect. Gravitational-wave standard sirens and strong-lensing time delays will confirm or exclude the prediction within the decade.</jats:p>