Energy in Scale-Relative SpaceTime

Energy is traditionally introduced as a fundamental conserved quantity associated with temporal symmetry. In this work, within the Scale-Relative SpaceTime framework, energy is instead constructed as a property of scale-dependent geometric evolution. Physical quantities are not assumed a priori, but arise from the dynamics of a scale-dependent manifold governed by the Scale Flow Equation and its renormalised form. Intrinsic energy is defined as the quadratic magnitude of the scale flow in the configuration space of metrics, inducing a natural geometric structure analogous to information and renormalisation-group metrics. This formulation yields a decomposition of energy into geometric, entropic, and coupling contributions, reflecting the interplay between curvature and entropy gradients, with topology entering through a multiplicative renormalisation factor. Beyond local dynamics, the framework admits global topological invariants associated with closed loops in configuration space. These quantities capture non-integrable structure in the scale flow and can take quantised values, providing a geometric origin for discrete energy contributions analogous to holonomy and phase effects. Together, these results establish a unified picture in which energy emerges as a measure of resistance to scale-dependent geometric evolution, combining continuous and topological aspects within a single formal structure. The relation to observable energy, spacetime dynamics, and energy–momentum is left for future work.