Hyperbolic Scale Relativity

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 exponent α ≃ 0.9, interpreted not as a phenomenological fitting parameter but as a structural constant of hierarchical scale geometry. Geometrically, this exponent is associated with the fundamental angular periodicity 2π/7 of a minimal hyperbolic scale structure; dynamically, it appears as the logarithmic slope relating inverse spatial scale separation to internal temporal-density scaling. The resulting mapping establishes a correspondence between macro- and micro-level descriptions without modifying local Lorentz-invariant spacetime structure. Microscopic systems are not assumed to inhabit a separate physical time; rather, they are described as possessing a higher internal dynamical density relative to macroscopic observers. Through the relation , this unresolved microscopic frequency structure may appear macroscopically as energetic and probabilistic behavior. In the cosmological sector, the theory incorporates a distinct scale-dependent conformal function f(r), acting as a geometric modulation of the physical metric. Gradients of this conformal factor generate effective dynamical corrections within a negatively curved scale manifold, while standard relativistic behavior is recovered in the local equilibrium limit. Within this structure, several long-standing anomalies may be reinterpreted as manifestations of scale-dependent dynamical modulation, including the Hubble tension, the muon g−2 discrepancy, galactic rotation curves, and non-gravitational accelerations of interstellar objects. These applications are presented as phenomenological consequences of the conformal scale sector rather than as completed replacements for the standard cosmological and particle-physics frameworks. The model therefore recasts mass, curvature, cosmic expansion, and apparent probabilistic behavior as possible emergent features of hierarchical hyperbolic scale geometry. Update note. This revision refines Section 5 by clarifying that the cosmological scale factor is defined in the Einstein frame and that the logarithmic scale field is written with its kinetic normalization absorbed, leaving any dimensional normalization implicit.