Pre-Krasner XII (Part 1 of 4): Curved-Spacetime QED Effective Action, Drummond-Hathrell Coefficients, and Strong-Field Vacuum Birefringence

We present Part 1 (Step 1: Formulas F01–F22) of the Pre-Krasner XII (KXII) 4D Effective Field Theory (EFT), establishing the foundational electromagnetic (spin-1) sector in curved pseudo-Riemannian spacetime and super-strong magnetar magnetic fields ($B_0 \sim 10^{10}\text{--}10^{11}\text{ T}$). First, we resolve the metric-signature transformation rules for the one-loop Drummond-Hathrell (1980) QED action below the electron-positron pair production threshold ($\hbar\omega \ll m_e c^2$), fixing the exact curvature coupling coefficients $(a, b, c, d)$ and the Friedmann-Robertson-Walker (FRW) phase velocity shift. Second, we combine the Euler-Heisenberg-Adler strong-field vacuum polarization tensor with cold magnetized plasma dispersion and general-relativistic optical redshift, establishing the exact normal-mode refractive indices $n_\perp(\omega, r)$ and $n_\parallel(\omega, r)$ required for precision magnetar polarimetry. Third, under a strict epistemological audit protocol, we classify all 22 equations into 15 unconditionally verified identities (Pure T1), 6 conditionally established low-energy EFT reductions (T1-C), and 1 perturbative nonlinear hypothesis (T2), explicitly documenting rejected theoretical branches and numerical verification benchmarks.