Bound Modes and Hydrogen-Like Structure in the Scalar Temporal Field Ontology

We analyze linearized fluctuations around matter-core solutions in the Scalar Temporal Field Ontology (STFO), in which time is modeled as a dynamical scalar field τ(x^μ). We derive the effective bound-mode equation governing perturbations and show that, under a central-core approximation, the resulting spectrum admits a hydrogen-like structure with an emergent coupling parameter. Expressions for the effective potential, bound-state energies, and characteristic length scale are obtained, and conditions for matching hydrogenic behavior are identified. This work establishes a connection between the STFO matter sector and atomic-scale structure under controlled approximations.