Semi-Analytic Evaluation of the STFO Screening Kernel and Mode-Dependent Coefficients

We present a semi-analytic evaluation of screening in the Scalar Temporal Field Ontology (STFO) using a concrete benchmark model for the temporal-field action. Building on prior work establishing the Green’s-function structure of screening and the linearized response operator, we adopt explicit model choices P(X)=X+αX² and U(τ)=½m²τ² to derive the weak-gradient response kernel. The resulting operator reduces to Helmholtz form with screening scale μ_eff = m / √(1+2αX₀), yielding a Yukawa kernel for the induced temporal response. Using hydrogen-like benchmark mode profiles, we compute screening coefficients for the 1s, 2s, and 2p modes and confirm the STFO mode-ordering hierarchy. This work provides the first concrete benchmark model linking STFO action parameters to mode-dependent screening behavior.