Emergent Oscillatory Structure from Minimal Null Geometry

This work develops a minimal geometric framework in which intrinsic oscillatory dynamics emerge from a system of three null directions subject to a global closure constraint. The resulting configuration space is a two-dimensional simplex with nontrivial internal degrees of freedom, supporting a coupled evolution between geometric redistribution and an emergent phase variable. An invariant structure of the formℐ = (x₁x₂x₃) cos θorganizes the dynamics and enforces consistency between geometry and phase. The system does not admit a static configuration; instead, it exhibits intrinsic oscillatory behavior arising from its closed feedback structure. The amplitude of the phase oscillation is determined by the statistical properties of the simplex, while time-averaged quantities provide effective dimensionless parameters. A universal normalization factor π arises from the symmetrization of the realization operator and its Gaussian structure in the boundary-dominated regime. A global phase modulation introduces a stability condition, selecting physically realized configurations as fixed points under slow perturbations. In this framework, dimensionless constants are interpreted not as fundamental inputs, but as emergent quantities arising from the interplay of constrained geometry, statistical fluctuations, and dynamical consistency.