A Scale-Relative Model of SpaceTime

We present a scale-relative framework for spacetime in which geometry and topology are treated as primary dynamical entities that evolve with observational scale. Physical structure is not assumed at the outset: mass, gravity, and time arise as emergent phenomena from the joint evolution of geometric coarse-graining and topological information flow. Two functions govern this process: a smearing function that implements resolution-dependent averaging of geometry, and a bandwidth function that limits the information accessible at a given scale.

Topological degrees of freedom are represented through a scale-dependent superposition of Betti modes, providing a probabilistic description of the effective manifold at each resolution. The coupled evolution of geometry and topology is governed by a Scale Flow Equation (SFE), formulated as a gradient flow in scale space. The SFE unifies curvature relaxation, entropy-driven dynamics, and topological transitions within a single formalism. Analysis of the flow indicates that non-trivial dynamics are generically supported only in three spatial dimensions.

Symmetries in this framework are scale-resolved. A generalised Scale-Dependent Noether Theorem links symmetries preserved at a given scale to corresponding conserved quantities. Breaking of Scale-Reparametrisation Invariance produces distinct physical observables: local breaking gives rise to mass, global breaking to gravitational curvature, and monotonic breaking to the arrow of time.

By placing scale-dependent geometry and topology at the foundation, the model provides a conceptually minimal and unified perspective on the emergence of spacetime structure and interactions, offering a new route toward understanding quantum-gravitational phenomena and the origin of fundamental physical laws.