Hyperbolic Scale Relativity

Hyperbolic Scale Relativity proposes that the effective temporal resolution of internal dynamics depends on the ratio between the characteristic spatial scales 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 Ω(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, the late-time Hubble-scale mapping provides the most direct background-level observational implication of the minimal scalar-conformal realization.The least-coupling conformal vacuum yields a definite present-epoch relation between the Einstein and physical expansion rates while allowing an approximately standard early-time regime. At galactic scales, the present construction establishes quantitative weak-field requirements associated with rotation curves, baryonic Tully–Fisher scaling, and gravitational lensing, although their full dynamical realization lies beyond the minimal radial sector. The framework therefore provides a controlled separation between consequences of the hyperbolic fixed-point geometry and conformal dynamics, and additional scale-response effects to be developed and tested phenomenologically. For completeness, an explicitly conditional microscopic EFT application to the muon anomalous magnetic moment is also considered. It uses the sevenfold fixed-point constant only as a candidate normalization of a standard dipole Wilson coefficient and remains separate from the core scale-relational and scalar-conformal construction. Revision note. Section 3 has been strengthened by deriving negative curvature under the stated fixed-point and geometric assumptions rather than taking it as an independent input. Appendix C has also been revised to derive the massive-particle conformal response directly from the universal HSR matter action.