The rate at which helium (${}^4\mathrm{He}$) and deuterium ($d$) fuse together to produce lithium-6 (${}^6\mathrm{Li}$) and a $\gamma$ ray, ${}^4\mathrm{He}(d,\gamma ){}^6\mathrm{Li}$, is a critical puzzle piece in resolving the discrepancy between big bang predictions and astronomical observations for the primordial abundance of ${}^6\mathrm{Li}$. The accurate determination of this radiative capture rate requires the quantitative and predictive description of the fusion probability across the big bang energy window ($30\mathrm{keV}\lesssim E\lesssim 400\mathrm{keV}$), where measurements are hindered by low counting rates. We present first-principle (or, ab initio) predictions of the ${}^4\mathrm{He}(d,\gamma ){}^6\mathrm{Li}$ astrophysical $S$ factor using validated nucleon-nucleon and three-nucleon interactions derived within the framework of chiral effective field theory. By employing the ab initio no-core shell model with continuum to describe ${}^4\mathrm{He}\text{-}d$ scattering dynamics and bound ${}^6\mathrm{Li}$ product on an equal footing, we accurately and consistently determine the contributions of the main electromagnetic transitions driving the radiative capture process. Our results reveal an enhancement of the capture probability below 100 keV owing to previously neglected magnetic dipole ($M1$) transitions and reduce by an average factor of 7 the uncertainty of the thermonuclear capture rate between 0.002 and 2 GK.