A Scale-Relative Model of SpaceTime

We propose that spacetime is not a fixed entity but a scale-dependent object whose geometry and topology evolve under a deterministic flow in observational scale. Unlike conventional frameworks that treat geometry on a fixed background, the present approach describes effective spacetime as a family of scale-dependent manifolds whose observable properties emerge through finite-resolution smearing and bandwidth limitation. Two complementary functions govern this evolution: a smearing function describing geometric coarse graining and a bandwidth function characterising the finite information accessible to an observer. Topological structure is represented through scale-dependent persistence distributions and Betti spectra, from which intrinsic entropy, topological superposition, and observable quantities are constructed.

The resulting Scale Flow Equation provides a unified dynamical framework for the coupled evolution of geometry and topology across scale. Scale-Reparametrisation Invariance and its breaking lead naturally to scale-dependent symmetries and conservation laws, while observable irreversibility emerges through coarse graining of the underlying topological structure. Within this framework, gravity, temporal ordering, and matter-like structures may be interpreted as emergent consequences of scale-dependent topology and its evolution.

The framework combines ideas from geometric flows, persistent homology, information theory, and renormalisation into a common scale-dependent description of effective spacetime and provides a mathematical foundation for further investigations of scale-dependent physical observables and phenomenology. By placing scale-dependent geometry and topology at the foundation, this framework provides a conceptually minimal yet physically rich perspective, offering new avenues to understand the origin of spacetime, quantum gravitational phenomena, and the emergence of fundamental interactions.