voxel_density_scaling
plain-language theorem explainer
The definition sets the effective number of recognition voxels N(r) to scale as a power law in radius using the pre-derived running exponent beta. Physicists modeling scale-dependent gravity or voxel resolution effects in Recognition Science cite it when connecting density variations to the phi^{-5} lag. It is a direct one-line power-law definition with no additional steps.
Claim. The voxel density scaling function is defined by $N(r) = r^beta$, where beta is the gravitational running exponent fixed by the ratio of the recognition lag $C_{lag} = varphi^{-5}$ to the self-similarity scaling factor.
background
In the Gravity.RunningGDerivation module this definition supplies the radial dependence of effective voxel count in RS-native units. The voxel is the fundamental length quantum normalized to 1, while the gravitational constant G takes the RS form lambda_rec^2 c^3 / (pi hbar). Upstream results include the structure of nuclear densities on phi-tiers from NucleosynthesisTiers and the log-coordinate reparametrization G_F t = F(exp t) from the Cost functional equation.
proof idea
This is a one-line definition that applies the power-law scaling directly to the input radius r using the beta_running exponent already obtained in the same module.
why it matters
The definition supplies the density factor needed to make effective G proportional to local resolution, completing the beta-running derivation step that links the phi^{-5} lag to the phi-ladder. It feeds the running_g_scaling sibling and supports the T5 J-uniqueness and T6 phi fixed-point landmarks in the forcing chain. It touches the open question of empirical confirmation of the running exponent at laboratory scales.
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