A Minimal Triadic Phase Formalism for Emergent Spacetime

This work presents a minimal pre-geometric framework in which spacetime, curvature, and mass emerge from synchronization dynamics of coupled phase oscillators. Starting from a relational phase manifold, the theory proposes that the fundamental degrees of freedom are not particles or spacetime coordinates, but phase relations between oscillatory components. A central result of the model is that the minimal non-degenerate and dynamically stable configuration is a triad (N = 3). The theory introduces a closure variable describing the collective phase consistency of the system, where deviations from perfect closure generate an effective defect field associated with energy, curvature, and localized mass structures. The framework is formulated through a minimal action principle containing: intrinsic oscillatory dynamics, propagation of correlations, and a triadic closure interaction term. In the effective limit, localized closure defects behave as particle-like structures, while distributed defects generate emergent curvature. At large scales, the model reproduces a Newtonian-like 1/r behavior for the closure field. The work proposes a unified relational interpretation in which: spacetime emerges from correlations, particles correspond to stable synchronization structures, and geometry arises from phase coherence relations. This paper provides a compact and conceptual formulation of the Harmonic Triad framework and establishes the mathematical foundations for future developments involving emergent quantum structures, topological defects, gauge symmetries, and cosmological dynamics.