Space as Separation: A Relational Account of Extension and Dimensionality

This paper develops a relation‑first account of space in which spatial extension and dimensionality are not primitive features of reality but emergent consequences of stabilized separation within a generative relational field. The central claim is that separation—the persistence of non‑coincidence across generative updates—is the fundamental operator underlying spatial structure. Extension arises when newly generated relational differences stabilize under boundary constraints, and dimensionality reflects the number of independent modes in which such separation can persist. The relational hypersphere is introduced as a representation of generative bandwidth, with its radius corresponding to the system’s capacity to sustain stable relational differences. Geometry and curvature are interpreted as patterns of constraint on allowable transformations rather than as properties of a background manifold. The resulting framework dissolves the container model and grounds spatial structure in the internal dynamics of relation itself.