Reconstruction and Globalization of Abelian Einstein–Dirac–Maxwell Multiplets

We study the inverse problem of recovering Abelian gauge couplings from the local response of labelled matter species in a classical Einstein–Dirac–Maxwell multiplet. With geometry, spin structure, masses, and the source protocol fixed, the Gram matrix G = QK⁻¹Qᵀ classifies kinetic and charge data up to real changes of field basis. It can be recovered from the full linear response with a common solution operator or from Coulomb energies; finite families of controlled sources yield explicit stability bounds. Anchor species and Schur complements reveal the remaining couplings and additional channels.We then characterize which local responses admit a compact gauge realization. The criterion is rationality of the image of G in the integer frame of the labelled species. The pair consisting of G and its charge lattice classifies minimal compact realizations. A bound on lattice height gives an explicit resolution threshold for recognizing admissible images from approximate data. Its B⁻² order is optimal in the worst case for separating a bounded-height catalogue. The result connects local response measurements with the arithmetic and global structure of Abelian gauge fields.