  We present an $\mathcal{O}(\alpha)$ Standard Model calculation of the inner
radiative corrections to Gamow-Teller $\beta$ decays. We find that \textit{a
priori} contributions arise from the photonic vertex correction and $\gamma W$
box diagram. Upon evaluation most elastic contributions vanish due to crossing
symmetry or cancellation between isoscalar and isovector photonic
contributions, leaving only the polarized parity-odd contribution, i.e., the
Gamow-Teller equivalent of the well-known axial $\gamma W$ box contribution for
Fermi decays. We show that weak magnetism contributes significantly to the Born
amplitude, and consider additional hadronic contributions at low energy using a
holomorphic continuation of the polarized Bjorken sum rule constrained by
experimental data. We perform the same procedure for the Fermi inner radiative
correction through a combination of the running of Bjorken and Gross-Llewellyn
Smith sum rules. We discuss heavy flavor, higher-twist, and target mass
corrections and find a significant increase at low momentum from the latter. We
find $\Delta_R^A = 0.02532(22)$ and $\Delta_R^V = 0.02473(27)$ for axial and
vector inner radiative corrections, respectively, resulting in
$\Delta_R^A-\Delta_R^V=0.60(5) \times 10^{-3}$, which allows us to extract
$g_A^0$ for the first time to our knowledge. We discuss consequences for
comparing experimental data to lattice calculations in beyond Standard Model
fits. Further, we show how some traditional $\beta$ decay calculations contain
part of this effect but fail to account for cancellations in the full
$\mathcal{O}(\alpha)$result. Finally, we correct for a double-counting instance
in the isospin $T=1/2$ mirror decay extraction of $|V_{ud}|$, the up-down
matrix element of the Cabibbo-Kobayashi-Maskawa matrix, resolving a
long-standing tension and leading to increased precision.
