Pre-Geometric Synchronization Framework: Uniqueness of the Harmonic Triad and the Emergence of 3D Geometry

Description This record presents Version V28 of the Pre-Geometric Synchronization Framework, a theoretical approach in which physical reality emerges from the dynamics of coupled phase oscillators defined on the compact manifold (M = (S¹) ³). In the original formulation, spacetime, particles, and interactions arise from synchronization processes: phase differences encode energy, correlations define spatial distance, and stable configurations behave as particle-like objects. In particular, the harmonic triad ( (N = 3) ) was identified as the minimal stable structure with coherent internal dynamics. Version V28 introduces a major theoretical advancement. The framework is reformulated through explicit structural constraints that select physically meaningful configurations: a global closure condition on phase differences a set of independent cyclic constraints across oscillator triplets a non-degeneracy condition ensuring the emergence of a genuine three-dimensional geometry Under these conditions, the theory establishes the following result: Only the case (N = 3) admits non-degenerate, stable configurations. All other cases are excluded: (N = 2): geometrically degenerate (effectively one-dimensional) (N 4): overconstrained or dynamically degenerate configurations This result promotes the harmonic triad from a minimal stable structure to a uniquely selected emergent object, providing a strong basis for interpreting it as a fundamental particle-like entity. The framework further specifies: a gradient-flow dynamics with a Lyapunov energy functional coupling constants derived from the spectrum of the Laplacian on ( (S¹) ³) an emergent notion of distance based on correlation functions reconstruction of geometry via multidimensional scaling (MDS) a characteristic energy scale (E₀ 374, GeV) Conceptual implicationsVersion V28 shifts the theory from a primarily interpretative framework to a constraint-driven and structurally selective model, in which geometry and particle-like structures arise only under strict consistency conditions. This supports a broader perspective in which: synchronization is the fundamental process geometry is emergent from correlations physical reality is a network of dynamically stable relational patterns Notes on versioningThis version extends and refines the original Zenodo submission by introducing a formal uniqueness result and resolving degeneracy issues present in earlier formulations.