We evaluate the benchmark nonlocal screening functionals in the Scalar Temporal Field Ontology (STFO) across all regimes of the dimensionless screening parameter x = μa. Building on the previously derived semi-analytic formulation and asymptotic analysis, we analyze the behavior of the screening functional in the weak, strong, and intermediate regimes. Using controlled approximations, we determine the relative ordering of screening strength for the 1s, 2s, and 2p bound states. We find that the hierarchy 1s > 2s > 2p is preserved across all regimes examined, indicating a robust dependence of screening on spatial compactness. This work provides a complete evaluation of benchmark screening behavior within the controlled approximation framework and establishes a stable prediction for mode-dependent screening in STFO.
Cale Scott Howe (Sun,) studied this question.