Explicit Linearization of the STFO Response Operator Around Matter-Core Backgrounds

We derive the explicit form of the linearized response operator governing perturbations of the scalar temporal field in the Scalar Temporal Field Ontology (STFO). Previous work identified screening as arising from the Green’s function of a schematic radial operator. In this work, we perform the linearization of the STFO field equation around a static matter-core background and extract the leading-order structure of the operator coefficients. The kinetic sector linearization is made explicit, elastic contributions are clarified, and the operator structure is refined. We show that the response operator naturally takes the form of a variable-coefficient elliptic operator. In the weak-gradient limit, this reduces to a Helmholtz-type operator, yielding a Yukawa screening kernel and providing a first-principles justification for earlier phenomenological screening approximations. This work strengthens the mathematical foundation of screening and mode ordering in the STFO framework.