Linear Response and Effective Screening Operator in the Scalar Temporal Field Ontology

We derive the explicit coefficient structure of the linearized response operator in the Scalar Temporal Field Ontology (STFO). Starting from the scalar field action, we expand around a static matter-core background and obtain expressions for the effective kinetic coefficient and mass term governing perturbations. We show how the operator reduces to a Helmholtz-type form in the weak-gradient regime, establishing the origin of Yukawa-like screening as a controlled approximation. This provides a direct bridge between nonlinear temporal dynamics and the nonlocal screening kernel. This work completes the scalar-sector derivation of the STFO screening framework and supplies the explicit coefficient formulas underlying the reduced second-order screening operator used in subsequent analyses. We derive the explicit coefficient structure of the linearized response operator in the Scalar Temporal Field Ontology (STFO). Starting from the scalar field action, we expand around a static matter-core background and obtain expressions for the effective kinetic coefficient and mass term governing perturbations. We show how the operator reduces to a Helmholtz-type form in the weak-gradient regime, establishing the origin of Yukawa-like screening as a controlled approximation. This provides a direct bridge between nonlinear temporal dynamics and the nonlocal screening kernel. This work completes the scalar-sector derivation of the STFO screening framework and supplies the explicit coefficient formulas underlying the reduced second-order screening operator used in subsequent analyses.