Hyperbolic Scale Relativity proposes that dynamical structure and effective temporal density depend on the characteristic spatial scale of interacting systems. The framework introduces a universal scale-relational mapping governed by a structural exponent α ≈ 0.9, emerging from logarithmic UV–IR scale hierarchy relations between cosmological curvature and the Planck length. This exponent is interpreted as a constant of hierarchical scale geometry rather than a phenomenological fit parameter. The resulting mapping establishes a scale-dependent correspondence between macro- and micro-level descriptions without modifying Lorentz-invariant spacetime structure. In the cosmological sector, the theory incorporates a distinct scale-dependent conformal function f(r), acting as a geometric modulation of the metric. The gradients of this function generate effective dynamical corrections within a negatively curved scale manifold, while preserving the standard relativistic framework. Within this structure, several long-standing anomalies may be reinterpreted as manifestations of scale-dependent dynamical modulation, including the muon g−2 discrepancy, galactic rotation curves without invoking non-baryonic dark matter, and non-gravitational accelerations of interstellar objects. Rather than altering spacetime itself, the model introduces a scale-structural correction to dynamical evolution, recasting mass, curvature, and expansion as emergent features of hierarchical scale geometry. NOTE: Earlier drafts referred to the framework as “Microcosmic Relativity”. The present version develops a geometric formulation based on a hyperbolic scale manifold, motivating the updated designation “Hyperbolic Scale Relativity”.