growth_eds_ilg
plain-language theorem explainer
This definition supplies the closed-form ILG growth factor D(a, k) in an Einstein-de Sitter background to first order in the correction amplitude C. Workers deriving reciprocity identities or comparing analytic growth to N-body runs cite it directly. The expression is assembled by substituting the ansatz D = a(1 + B a^alpha) into the growth ODE and inserting the solved prefactor B.
Claim. Let $D(a,k)$ be the growth factor. Then $D(a,k)=a(1+B(a/(k tau_0))^alpha)$, where $B=3C/(2alpha^2+5alpha)$, $alpha$ is the fine-structure constant, $tau_0$ the fundamental tick duration, and $C$ the kernel correction amplitude.
background
The ILG module works in an EdS background where the growth ODE is solved by the ansatz D = a(1 + B a^alpha). The prefactor B is obtained by direct substitution, giving the explicit algebraic form shown in growth_prefactor. KernelParams packages the inputs alpha, C and tau_0; tau_0 is the RS-native tick duration and alpha is the fine-structure constant taken from the constants derivation.
proof idea
Direct definition that multiplies the scale factor a by one plus the product of growth_prefactor(P.alpha, P.C) and the scaled term (a/(k P.tau0))^P.alpha. No tactics; the body is the closed expression obtained from the ODE substitution.
why it matters
Supplies the explicit D(a,k) required by the downstream reciprocity theorem growth_x_reciprocity. It closes the first-order ILG correction inside the EdS sector and feeds the constants tau_0 and alpha that descend from the forcing chain. The definition therefore anchors analytic checks of the Recognition Composition Law for growth observables.
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