Particle Solutions of the Realization Equation: Mass, Fine-Structure Constant, and Atomic Scale from a Single Constraint

This work develops a unified realization framework in which mass, the fine-structure constant, and atomic structure emerge from a single geometric constraint on phase and density. Starting from a conserved realization current and a nonlinear constraint on the phase gradient, we show that stable localized solutions give rise to timelike invariants identified with mass, a small realization depth corresponding to the fine-structure constant, and a characteristic orbital scale equivalent to the Bohr radius. The framework provides a direct bridge between null geometry and particle structure. Mass is shown to arise from non-collinear closure of null orientations, while the fine-structure constant reflects the fraction of realized correlations in near-null configurations. Quantization emerges as a phase-closure condition, and the Schrödinger equation appears as a non-relativistic limit of the same underlying system. A central result is the prediction of a universal low-frequency modulation of phase at Ω ≈ 31 nHz (T ≈ 373.36 days), interpreted as a global realization mode. This modulation is predicted to appear in realized systems (e.g. atomic clocks, pulsar timing arrays) and to be absent in completed structures such as black-hole boundaries, providing a clear route for experimental verification. The theory requires no additional fields or particles and instead reinterprets known physical quantities as consequences of a single consistency condition on realization.