Emergent Spacetime from Relational Fundamental Dynamics

Abstract Spacetime is treated as a presupposed structure in most physical theories. In an approach in which spacetime itself is not fundamentally given, but is instead expected to emerge as a macroscopic form of order from deeper degrees of freedom, such a presupposition is methodologically inadmissible. This work formulates a relational framework in which the possible emergence of spacetime-like structure can be investigated without any prior assumption of space, time, coordinates, or a metric. The starting point is a finite set of abstract elementary units whose physically relevant information is encoded exclusively in their relations. These relations are described by a weighted coupling structure whose labels have no geometric meaning. On this basis, a trace-invariant action formulation is introduced that enables stability, statistical homogeneity, and the dynamical suppression of uncontrolled long-range couplings without introducing an external background geometry. For stable homogeneous configurations, collective fluctuations are analyzed, and effective geometric diagnostics are obtained from the linear response structure. In particular, emergent notions of distance, an effective Laplace operator, spectral dimension, and dispersion relations are introduced as operational quantities. These quantities do not serve as a retrospective interpretation of an already presupposed geometry; rather, they constitute the criteria by which it is determined whether a relational dynamics develops spacetime-like properties in the macroscopic regime. This work thereby develops a testable minimal framework for emergent spacetime. Its aim is not to formulate a complete theory of quantum gravity, but to determine precisely the conditions under which relational fundamental dynamics can give rise to stable, homogeneous, and low-dimensional spacetime phases. Within this framework, spacetime appears not as a postulate, but as a possible outcome of a dynamically selected macroscopic structure.