Explicit Evaluation of Mode-Dependent Screening Coefficients in the Scalar Temporal Field Ontology

This work presents an explicit evaluation of mode-dependent screening coefficients within the Scalar Temporal Field Ontology (STFO). Building on previous developments introducing integral-based screening functionals, we compute full spatial integrals for representative bound-state modes using hydrogen-like approximations and a short-range interaction kernel. Analytic expressions are derived for 1s, 2s, and 2p modes, allowing direct comparison of screening strength across different spatial structures. The results demonstrate that radial and angular features produce quantitatively distinct contributions, with explicit coefficients governing mode-dependent screening behavior. These results replace previous scaling arguments with concrete expressions and establish a predictive mathematical framework for mode ordering in the STFO.