S3 Field Framework Derivation of the Calm-Sector Stiffness K from the Microscopic Quadratic Term

This work derives the emergent gravitational stiffness parameter K within a UV-complete field-theoretic model where gravity is not fundamental but arises from the coarse-grained dynamics of a single constrained O(4) nonlinear sigma field—a generalization of the Skyrme model. In the static weak-field regime, we apply homogenization techniques to show that the long-wavelength effective theory inherits its gradient energy directly from the microscopic sigma-model coupling, yielding the identification K = F². This result explicitly links Newton’s constant G to the UV scale of the underlying topological field theory, offering a concrete example of emergent Newtonian gravity from a well-defined, particle-physics-inspired Lagrangian. The study contributes to the broader research program seeking to recover general relativity as a low-energy effective theory from non-geometric, pre-geometric, or condensed-matter-like microscopic degrees of freedom.