A logarithmic redshift drift template for the fine-structure constant and its multi-messenger signatures

We propose a phenomenological template for testing an effective vacuum-relaxation framework through a logarithmic redshift drift of the fine-structure constant. While the bulk electromagnetic vacuum is assumed to remain impedance-protected, bound atomic states may retain a screened elastic mismatch with the relaxing background. This yields $$\frac{\Delta\alpha}{\alpha}(z)\simeq-\kappa_{\mathrm scr}\,\frac{\Delta d_L}{d_L}(z)$$ where $\Delta d_L/d_L\equiv(d_L^{\mathrm EM}-d_L^{\mathrm GW})/d_L^{\mathrm GW}$ is the fractional electromagnetic-versus-gravitational-wave luminosity-distance offset. Adopting the benchmark closures $\epsilon_{\rm rel}=\pi\alpha_0/2$ and $\kappa_{\rm scr}=2\alpha_0^2$, the model predicts$$\Delta\alpha/\alpha\simeq-\pi\alpha_0^3\ln(1+z)\simeq-1.22\ln(1+z)\ {\rm ppm}.$$The signal reaches $-1.34$ ppm at $z=2$, within the target range of ESPRESSO and ELT-ANDES, and is correlated with a sub-percent to percent-level standard-siren offset. This provides a fixed falsifiable consistency benchmark between quasar spectroscopy and gravitational-wave cosmology.