Benchmark Nonlocal Screening Functionals for Bound States in the Scalar Temporal Field Ontology

We construct benchmark screening functionals for bound-state configurations in the Scalar Temporal Field Ontology (STFO). Using the previously derived nonlocal screening kernel, we express the self-interaction strength of a density distribution as a double integral over a Yukawa-type interaction. We reduce the functional to a radial form for spherically symmetric configurations and define explicit benchmark expressions for the 1s, 2s, and 2p states. This provides a concrete framework for quantitative evaluation of mode-dependent screening. This work establishes the explicit functional to be evaluated in subsequent quantitative analysis of the STFO screening framework.