S3 Calm Poisson Newtonian Matching

This note formulates the weak-field Newtonian limit within a single-field (S^{3}) framework in which the only fundamental object is a constrained field (\Phi:\mathbb{R}^{3,1}\to S^{3}). A coarse-grained calm mode (\tau(\mathbf{x})) is introduced and coupled to the emergent mass density of localized (\Phi)-excitations (twists) through a dimensionless response coefficient (a). Assuming a quadratic infrared calm functional with stiffness (K_{0}), the Euler–Lagrange equation yields a Poisson constraint which, under the operational identification (\varphi=a\tau) and (U_{N}=c^{2}\varphi), reproduces the Newtonian Poisson equation and implies (G=c^{4}/(4\pi K_{\rm grav})) with (K_{\rm grav}=K_{0}/a^{2}). The result isolates the microscopic closure tasks required for a parameter-free prediction of (G): computing ((K_{0},a)) from the (\Phi) energy and establishing infrared masslessness of the calm mode to avoid Yukawa screening.