IndisputableMonolith.Gravity.RunningGDerivation
This module defines the voxel density scaling N(r) as the effective number of recognition voxels at radius r to support derivations of running gravitational strength. Researchers modeling nanometer-scale gravity or modified Newtonian dynamics would cite the scaling to connect J-cost structures to observable G(r). The module assembles imported constants, cost functions, and the RunningG framework into sibling definitions for beta_running_derived and running_g_scaling.
claim$N(r)$ is the effective number of recognition voxels as a function of radius $r$.
background
The module resides in the Gravity domain and imports Constants (defining the RS time quantum τ₀ = 1 tick), Cost (supplying J-cost and defect measures), and RunningG. RunningG states that Newton's constant is not fixed: G(r) → G_∞ as r → ∞, with the running arising at nanometer scales from the underlying recognition structure. The supplied DOC_COMMENT identifies the central object as the voxel density scaling N(r), which supplies the spatial counting needed for the phi-ladder mass formulas and the eight-tick octave periodicity.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module supplies the N(r) scaling required by the parent RunningG module (C51: Gravitational Running at Nanometer Scales) to derive the beta function and the explicit running of G. It closes part of the link from the T5 J-uniqueness and T8 D = 3 forcing steps to concrete gravitational phenomenology, using the Recognition Composition Law to translate voxel counts into the observed strengthening of G at small r.
scope and limits
- Does not compute explicit numerical values of N(r) or G(r) at chosen radii.
- Does not address running behavior outside the nanometer regime.
- Does not incorporate higher-order corrections from the full forcing chain beyond voxel counting.
- Does not treat non-spherical geometries or time-dependent sources.