S3 Field Framework: Gauge-Fixed Metric Dilation and a Microscopic Determination of the Gravitational Source Coefficient a

This note works within an (S^{3})-valued single-field framework and addresses a key obstruction in defining a nonzero Newtonian source coupling from localized defects. It shows that “global phase–type” calm deformations are symmetry-removable and therefore force the linear response coefficient (a) to vanish. A symmetry-resistant, gauge-fixed alternative is then implemented via an anchored conformal metric dilation (\Omega(\mathbf{x})=1+\tau(\mathbf{x})), leading to a well-posed microscopic definition of (a) from the centered (\pm\varepsilon) energy response of a fixed-sector twist/soliton. A reproducible 3D relaxation protocol with topological-sector control is provided, including a linear-response plateau check and a resolution study over (N=19)–25 yielding a stable continuum estimate for (a) (and the matching combination (a^{2}/K_{0})). The package includes code, configuration, results, and figure scripts to reproduce all reported numbers.