Emergence of Newtonian and Coulomb Interactions from a Unified Elastic Substrate

We develop the mathematical structure of the substrate and derive the stress-tensor contributions associated with its two fundamental geometric channels: a curvature channel and a transverse (winding) channel [1]. These channels constitute the only independent degrees of freedom of the phase field, yet they generate three distinct emergent interaction behaviors. Curvature defects generate long‑range gravity‑like responses through the traceless shear channel, while short‑range nuclear‑like behavior emerges only when the scalar curvature mode is confined into localized bonded structures. Transverse defects generate electromagnetic-like interactions through circulation of the phase gradient. The interaction between localized excitations is shown to originate exclusively from cross-term momentum flux in the stress tensor, providing a unified mechanical mechanism for all three behaviors without invoking spacetime curvature, gauge fields, or additional potentials. The resulting framework yields a coherent hierarchy of ranges, strengths, and information-theoretic signatures, and leads to concrete observational consequences at cosmological scales. This substrate thus offers a single, flat, static medium whose geometric and mechanical properties account for the full spectrum of interaction phenomena.