Nonlocal Yukawa Evaluation of Mode-Dependent Screening in the Scalar Temporal Field Ontology

We extend the benchmark screening analysis of the Scalar Temporal Field Ontology (STFO) by replacing the reduced local screening functional with the full nonlocal Yukawa convolution generated by the linearized response kernel. Previous work established that screening is governed by a Yukawa Green function with inverse screening length μₑff, and that the reduced local approximation predicts a regime-dependent ordering transition between the 2s and 2p modes at x = μₑff a_τ = 1. In this work, we derive the full nonlocal mode-dependent screening functional and reduce it to explicit radial double-integral expressions for benchmark modes. This provides the correct formulation of screening beyond the local exponential approximation. We show that the nonlocal STFO screening problem is governed by the same dimensionless parameter x = μₑff a_τ, but depends on full spatial overlap through the Yukawa kernel. This establishes the proper framework for determining whether the ordering transition persists beyond the local approximation.