Localization as a Stability Phase of a Relational Functional

We study a class of discrete variational functionals defined on configurations consisting of a positive scalar field and an internal variable taking values in a compact Lie group. The interaction is governed by a quadratic pair term, a local saturation term, and a loop contribution defined through holonomies along cycles of the interaction graph. We establish existence of global minimizers using the direct method in the calculus of variations. We identify a critical density at which uniform configurations lose stability. Beyond this threshold, we construct non-uniform minimizers that are exponentially localized, proved via spectral properties of the linearized operator and a nonlinear fixed-point argument. Localized configurations are shown to exhibit rigidity: all nontrivial orientation variations increase the energy. A continuum limit is derived via Γ-convergence, yielding an effective scalar-internal field theory with emergent gradient and curvature structure. The results provide a complete variational framework for the emergence of localized and rigid configurations from purely relational interaction data.