Emergent electromagnetic and gravitational couplings from Peierls-Nabarro tunnelling in an FCC Cosserat lattice

We compute the Peierls-Nabarro tunnelling amplitude of a screw dislocation in an FCC Cosserat lattice, working within the Kleinert-Volovik programme of emergent gauge fields. The shear-channel amplitude generates a convergent perturbation series, comprising gauge-mode trapping, translational and rotational self-energies, and inter-valley scattering, where the lattice geometry fixes every coefficient with no adjustable parameters. The lattice spacing cancels to leave the purely dimensionless result $\alpha^{-1} = 137.035\,999\,177$, matching the measured fine-structure constant to all 12 known digits ($