Abstract
Unlike classical elasticity, where stresses arise from deformations relative to a stress-free reference configuration, rigidity in amorphous systems is maintained by disordered force networks that generate internal prestress. Previously, we introduced a ''stress-only'' formulation, where mechanical equilibrium resembles Gauss's law in a rank-2 tensor electrostatics with vector charges, and demonstrated that the mechanical response of jammed solids is described by the dielectric response of this gauge-theoretic formulation. Here, we extend this framework by incorporating scale-dependent screening that captures both dielectric and Debye-type behaviour. This introduces a characteristic length scale in stress correlations as well as in the response to external forces. Through numerical simulations of soft-sphere packings, we show that this length scale is set by the particle size, thus providing a natural ultraviolet cutoff while preserving long-wavelength emergent elasticity. We show that this lengthscale remains finite for all pressures, with no evidence for an emergent Debye-like screening near the frictionless unjamming transition. We demonstrate that although individual realisations show strong fluctuations, disorder averaging at fixed macroscopic conditions yields a robust dielectric-like response that persists up to unjamming. Finally, we also provide a physical interpretation of the gauge field within the electrostatic mapping: relative grain displacements in response to localised external perturbations correspond to difference in the gauge field, linking the field-theoretic description to particle-level mechanics.