growth_factor_cert
Growth factor certificate asserts that the ILG density contrast D remains positive and at least as large as the scale factor a for positive inputs, while the growth rate f exceeds one and the prefactor β stays positive. Cosmologists modeling dark energy or Hubble tension via ILG modifications cite it to confirm the growth equations obey the required inequalities. The proof is a direct term construction that assembles the four component lemmas into the GrowthFactorCert structure.
claimFor all real $a, k, τ_0 > 0$, the ILG density contrast satisfies $D(a,k,τ_0) > 0$ and $D(a,k,τ_0) ≥ a$, the growth rate satisfies $f(a,k,τ_0) > 1$, and the prefactor satisfies $β(k,τ_0) > 0$, where $D(a,k,τ_0) = a (1 + β(k,τ_0) a^α)^{1/(1+α)}$ with $β(k,τ_0) = (2/3) φ^{-3/2} (k τ_0)^{-α}$ and $f$ the corresponding growth rate, $α = α_{Lock}$.
background
The ILG Growth Factor module encodes the modified growth equations from the dark-energy and Hubble-tension papers. Core definitions are $D(a,k) = a (1 + β a^α)^{1/(1+α)}$ and $f(a,k) = 1 + [α/(1+α)] β a^α / (1 + β a^α)$, with $β(k) = (2/3) φ^{-3/2} (k τ_0)^{-α}$ and $α ≈ 0.191$. These recover the GR limits $D = a$ and $f = 1$ when β vanishes.
proof idea
Term proof that constructs the GrowthFactorCert instance by field assignment. It maps D_positive to D_growth_pos, D_ge_a to D_growth_ge_a, f_gt_one to f_growth_gt_one, and beta_positive to beta_growth_pos. The construction relies on the structure definition and the four component theorems.
why it matters in Recognition Science
Supplies a single certified object for the ILG growth properties used in dark-energy modeling. It aggregates the module's positivity lemmas, enabling direct reference in cosmological calculations. Within Recognition Science this supports consistency checks for the ILG extension, though it does not invoke the forcing chain T0-T8 or the Recognition Composition Law directly.
scope and limits
- Does not derive the explicit forms of D or f.
- Does not prove the GR limit f → 1 as β → 0.
- Does not quantify the size of growth enhancement.
- Does not link to the phi ladder or eight-tick octave.
formal statement (Lean)
136theorem growth_factor_cert : GrowthFactorCert where
137 D_positive := D_growth_pos
proof body
Term-mode proof.
138 D_ge_a := D_growth_ge_a
139 f_gt_one := f_growth_gt_one
140 beta_positive := beta_growth_pos
141
142end
143
144end ILGGrowthFactor
145end Gravity
146end IndisputableMonolith