IndisputableMonolith.Gravity.RunningGDerivation
This module defines the voxel density scaling N(r) as the effective number of recognition voxels at radius r, providing the foundation for deriving scale-dependent gravitational effects in Recognition Science. Physicists modeling nanometer-scale gravity or running constants would cite it when connecting the J-cost to macroscopic limits. The module structures its argument through supporting definitions drawn from the phi-ladder and voxel counting without internal proofs.
claimThe effective number of recognition voxels is given by the scaling function $N(r)$ at radius $r$.
background
Recognition Science sets the fundamental time quantum as τ₀ = 1 tick. The Cost module supplies the J-cost and Recognition Composition Law, while the RunningG module states that G(r) approaches G_∞ as r → ∞ and strengthens at nanometer scales. This derivation module assembles those elements to introduce N(r) for gravity applications.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module feeds the sibling definitions voxel_density_scaling, beta_running_derived, and running_g_scaling, advancing the C51 claim that G runs at small scales. It links the eight-tick octave and phi-ladder to gravitational voxel counting, closing part of the path from the UnifiedForcingChain to observable running-G predictions.
scope and limits
- Does not compute explicit values of N(r) at given radii.
- Does not prove the running of G from the J-cost alone.
- Does not extend beyond three spatial dimensions.
- Does not include direct experimental mappings or falsification tests.