Energy–Momentum and Field Structures in Scale-Relative Spacetime

We develop a scale-relative formulation of spacetime in which energy and momentum emerge from the evolution of topological configurations under the Scale Flow Equation. Spatial dimensionality arises naturally as the minimal number of independent parameters required to resolve persistent topological features, providing the coordinates of the emergent manifold. A scale-dependent action functional yields an energy–momentum tensor driven by gradients of topological entropy, linking scale dynamics to conserved quantities. Extending the entropy functional to include explicit spacetime dependence generates entropy-driven currents, while the introduction of non-integrable components produces a minimal antisymmetric field tensor, capturing the first indications of field-like behaviour. Observable quantities, including effective energy density and coarse-grained fields, are obtained via scale-conditioned expectation and coarse-graining, connecting microscopic topological structure to physically interpretable macroscopic quantities. This framework provides a coherent route from topological evolution to emergent energy–momentum structures and associated currents in a scale-relative spacetime.