A Scale-Relative Model of Spacetime

We propose a scale-relative framework in which spacetime is described not as a fixed geometric manifold but as a family of effective manifolds whose geometry and topology evolve with observational scale. The formulation is motivated by the observation that finite observational resolution naturally suppresses fine-scale topological structure, suggesting that effective geometry may emerge progressively from an underlying scale-dependent topological substrate.

The framework introduces a persistence distribution that characterises topological structure across observational scales and provides the common mathematical origin of two complementary quantities: an intrinsic topological entropy and a topological superposition function. Bandwidth-limited observation and persistent homology provide, the operational and mathematical foundations for describing how topological structure is filtered across scale, while geometric smearing provides the corresponding scale-dependent regularisation of effective geometry.

A deterministic Scale Flow Equation is developed to describe the evolution of the scale-relative metric. The equation combines geometric regularisation, entropy-driven evolution, and topological contributions within a unified covariant formulation. An equivalent Renormalised Scale Flow Equation further reveals topology as a cumulative renormalising influence that modulates geometric evolution across observational scale.

Rather than treating matter, fields, gravitation, or macroscopic temporal ordering as fundamental, the framework suggests that these phenomena may emerge as stable structures associated with scale-dependent dynamics and departures from underlying scale symmetry. The present work establishes the mathematical and conceptual foundations of this approach while leaving detailed phenomenological applications and quantitative solutions of the flow equations to future investigation.