Emergent Geometry in Complex Systems from Relational Dynamics: The Classical Setting

Geometry is typically treated as a primitive background structure within which physical processes unfold. In this work, we investigate an alternative possibility: that geometry can arise as a structural consequence of relational dynamics. We study the framework of Graph Dynamical Geometrization (GDG), in which a system is specified by a relational structure encoded by a graph, together with a dynamical law acting on that structure. The dynamics define a symmetric operator whose spectral evolution generates a positive–definite kernel. From this kernel, one obtains a squared Euclidean distance matrix that induces a canonical embedding of the relational system into a Euclidean space. Therefore geometry appears not as an assumed background but as a structure generated by the dynamics of relations. This construction motivates a philosophical interpretation that we call relational–dynamical structural realism, according to which the fundamental ontology consists of entities connected by relations and governed by a dynamical law, while geometric structure emerges from the spectral organization of those dynamics. We analyze the status of the induced geometry within several forms of structural realism. The GDG framework shows that invariant metric relations generated by relational dynamics can possess explanatory and predictive significance. More generally, the results suggest a structural principle: geometry may emerge whenever relational dynamics admit a spectral articulation.