Finite-Field QED Corrections to Vacuum Birefringence and Magnetar Polarization Transport

{We study low-energy photon propagation in a constant magnetic field within the one-loop Heisenberg–Euler theory, retaining the refractive-index normalization ${\gamma }_s$ without expansion. Here ``finite-field’’ denotes exact dependence on $B\mathrm{/}{B}_{\mathrm{c}\mathrm{r}}$ within the one-loop, constant-field approximation. The resulting birefringence is propagated into magnetar polarization transport. In a centered-dipole model, the polarization-limiting radius is unchanged to better than ${10}^{-12}$ because mode decoupling occurs at $\sim {10}^2{R}_{\mathrm{N}\mathrm{S}}$, where $B\ll B_{\mathrm{c}\mathrm{r}}$. Near the surface, however, the weak-field Cotton–Mouton expression overestimates the accumulated birefringent phase by up to a factor $2.9$ at ${10}^{15}$~G. At the plasma–vacuum resonance, finite-field corrections reduce the resonance density by $32\mathrm{\%}$ and raise the adiabatic conversion energy by $14\mathrm{\%}$ for 1E~1547.0$-$5408; the corresponding changes are factors $2.6$ and $1.37$ for 1RXS~J1708$-$4009, and factors $9.7$ and $2.13$ for SGR~1806$-$20, the latter controlled by the strong-field asymptote. The resummed one-loop parallel-mode magnetic response remains positive and develops a broad maximum near $17B_{\mathrm{c}\mathrm{r}}$. The strictly truncated $\mathcal{O}(\alpha )$ response is monotonic; therefore the maximum is a structural prediction of the resummed one-loop constitutive model, while its detailed profile and precise location require higher-loop validation. These results identify vacuum-resonance observables as the most sensitive channel for testing finite-field QED in magnetars.