S3 Field Framework Calm-Sector Poisson Constraint and the Newtonian 1r Tail

This paper derives the static Newtonian gravitational potential () from a fundamental, single-field theory based on a constrained map . Using effective field theory techniques, the work bridges microscopic Skyrme-like dynamics with macroscopic static forces. Key contributions include: Emergent Macroscopic Variable: Gravity is described by a scalar , defined as the potential of the coarse-grained gradient of the fundamental field. Mechanism for Masslessness: A symmetry-based argument (shift symmetry) is used to rigorously justify why the emergent scalar field remains massless, ensuring a long-range inverse-square force rather than a short-range Yukawa potential. Poisson Constraint: The derivation yields a nonlinear constraint equation that reduces to the standard Poisson equation in the weak-field limit. Gauss-Law Normalization: The model successfully recovers the correct Newtonian tail and Gauss-law normalization for localized sources. This work establishes a proof-of-concept for recovering Newtonian gravity within the static limit of the S3-Field Framework program, offering a geometric origin for the interaction without postulating independent gravitational fields.