Particle Mass Emergence from Scalar-Time Coherence in Time-Scalar Field Theory

Time-Scalar Field Theory (TSFT) promotes time to a fundamental scalar field Θ (x, t) governing information flow and coherence structure. Previous work established the existence of stable scalar-time excitations, rivet structures, and discrete spectral families arising from viabilityconstrained scalar-time dynamics. However, particle mass prediction remained incomplete due to the absence of a first-principles derivation linking scalar-time coherence spectra to relativistic mass. In this work, we derive particle masses directly from scalar-time coherence dynamics. Starting from the scalar-time field Θ (x, t), we construct a viability-constrained coherence functional whose stationary points define a self-adjoint scalar-time operator. The resulting eigenvalue spectrum generates a discrete scalar-time frequency ladder. We then derive the relativistic propagation equation for scalar-time excitations and demonstrate that the coherence eigenvalues appear as Klein–Gordon mass-shell terms. This yields the fundamental relationm sub n ² = λ sub nwhere λ sub n are scalar-time coherence eigenvalues. Particle families, spins, and charges emerge from previously derived rivet closure and holonomy structures, while masses arise directly from the spectral ladder. Importantly, particle identities are not assumed. Instead, the theory produces an unlabeled catalog of admissible scalar-time coherence states. Known particles are identified only through post hoc comparison, while unmatched states constitute predictions. This establishes a noncircular, first-principles pathway from scalar-time dynamics to particle spectrum emergence.