Near-Zone Conservative Two-Body Signature of Calm-Sector Gradient Self-Screening in S3 Field Framework

This upload contains the preprint manuscript and the reference Python implementation for a novel test of modified gravity within a $\Phi$-only $S^3$ field framework. Theoretical Framework:The manuscript derives the static near-zone limit of a "calm-sector" effective gravity, modeled by a scalar-tensor theory with a nonlinear kinetic prefactor $f(\tau) = 1 + \beta \tau^2$. This nonlinearity introduces a gradient self-screening mechanism that regularizes potential singularities. We derive the leading-order conservative signature of this screening in a binary system: a characteristic $r^{-3}$ correction to the Newtonian interaction energy, distinct from the $r^{-1}$ (Newtonian) and $e^{-mr}$ (Yukawa) terms. Computational Protocol:The included software provides the numerical recipe to extract this signature from field simulations. It implements:1. A face-based finite-difference discretization of the nonlinear energy functional to preserve flux continuity.2. A three-run integrand subtraction method to rigorously cancel the divergent self-energies of extended topological solitons.3. A linear regression model to isolate the specific screening coefficients ($A_3$ and $\kappa$) from finite-size tidal effects ($C_6$). This work establishes a direct link between the microscopic topology of a constrained field and macroscopic deviations in the gravitational inverse-square law, offering a falsifiable prediction for fifth-force searches and binary interaction models.