Viability-Constrained Spectral Selection in Time-Scalar Field Theory: Derivation of a Finite Scalar Ladder

Time-Scalar Field Theory (TSFT) provides a framework in which discrete structure emerges from scalar-time coherence. Previous work has established the existence of a covariant scalartime coherence operator and demonstrated that boundary compatibility conditions produce a discrete spectral structure. However, these results do not determine which discrete modes correspond to physically persistent structures. In this work, we introduce a viability-constrained spectral selection principle based on temporal persistence requirements. We define a spectral functional that penalizes eigenmodes according to susceptibility to temporal thinning and oscillatory instability. Minimization of this functional selects a sparse subset of physically admissible scalar modes. We show that in the weak-variation regime the survival cost grows polynomially with mode index. Under mild regularity conditions on the scalar coherence operator, this transforms an infinite discrete spectrum into a finite viable branch. The resulting ladder provides a structural foundation for particle family emergence without empirical fitting. Finally, we demonstrate that relativistic extensions of the viability-selected scalar ladder provide a consistent pathway toward spinor particle structure. This establishes a unified framework connecting scalar-time coherence, spectral selection, and particle family structure.