Finite-Relaxation Geometry (FRG) 3.1: Scale Dependence of Geometric Relaxation and the Hierarchy of Gravitational Regimes

Finite-Relaxation Geometry (FRG) proposes that spacetime geometry approaches quasi-stationary General Relativity configurations over a finite relaxation timescale rather than instantaneously. Previous formulations introduced the geometric relaxation time \tau_g as an effective phenomenological parameter governing the persistence of geometric memory following curvature perturbations.

In this work we clarify the physical interpretation and admissible behavior of the relaxation time within the FRG framework. We argue that \tau_g should be understood as a macroscopic scale-dependent parameter characterizing the response of spacetime geometry to gravitational perturbations. This interpretation naturally leads to a hierarchy of relaxation regimes ranging from effectively instantaneous relaxation at local scales to cosmological relaxation occurring over Hubble timescales.

Within this framework, galactic halos, cosmic acceleration, and black holes can be interpreted as manifestations of progressively slower geometric relaxation across gravitational scales. The proposed formulation introduces no modification of the Einstein field equations, preserves local Lorentz invariance and causal structure, and remains fully compatible with current astrophysical and cosmological observations.

The functional form of the relaxation time is left phenomenological and subject to future observational constraints.