IndisputableMonolith.Gravity.ILGGrowthFactor
The module defines the ILG growth kernel parameter β(k) = (2/3) φ^{-3/2} (k τ_0)^{-α} with α = (1-1/φ)/2 in the Recognition Science gravity setting. Cosmologists computing structure growth within RS models would cite these definitions and their supporting lemmas. The module supplies core definitions for β, D, and f growth factors plus lemmas on positivity and limiting behavior.
claimThe ILG growth kernel parameter satisfies $β(k) = (2/3) φ^{-3/2} (k τ_0)^{-α}$ where $α = (1 - φ^{-1})/2$ and $τ_0$ denotes the RS time quantum.
background
Recognition Science builds gravity from the unified forcing chain with φ as self-similar fixed point and τ_0 = 1 tick as fundamental time. This module imports the Constants module for τ_0 and introduces the growth kernel β(k) whose prefactor φ^{-3/2} matches the CPM paper constant C. It further defines D_growth as the integrated growth factor, f_growth as its logarithmic derivative, and GrowthFactorCert as the certifying structure.
proof idea
This is a definition module. It states the explicit formula for β(k) and the auxiliary growth functions, then proves positivity via beta_growth_pos and D_growth_pos together with limit statements via D_growth_gr_limit and f_growth_gr_limit.
why it matters in Recognition Science
The module supplies the growth kernel β(k) required for perturbation evolution in ILG gravity. It directly supports GrowthFactorCert and links to the phi-ladder constants with α derived from φ. The construction ties the prefactor to earlier CPM results and places the growth calculation inside the RS framework.
scope and limits
- Does not derive the formula for β(k) from the RCL or forcing chain.
- Does not perform numerical evaluations for specific wavenumbers.
- Does not match the kernel to observational growth data.
- Does not extend the definitions beyond three spatial dimensions.