H_GrowthEquation
plain-language theorem explainer
The declaration defines the linear density perturbation growth equation modified by recognition strain Q at scales beyond the 8-tick cutoff. Cosmologists modeling the σ₈ tension between CMB and weak-lensing data would cite this as the RS-adjusted form of the Friedmann growth law. The definition is a direct algebraic substitution that inserts the multiplicative suppression factor (1 - Q) into the source term of the standard equation.
Claim. The recognition-modified growth equation is given by $¨δ + 2H ˙δ = (3/2) Ω_m H² δ (1 - Q)$, where Ω_m is the matter density parameter, H the Hubble parameter, δ the density contrast with its first and second time derivatives, and Q the recognition strain.
background
In the Recognition Science framework, structure formation obeys the recognition operator R̂ subject to the 8-tick neutrality constraint. The module sets the local scale λ_8 below which growth proceeds as in ΛCDM and above which the cumulative strain Q suppresses the source term. Upstream, the shifted cost function satisfies H(x) = J(x) + 1, under which the Recognition Composition Law reduces to the d'Alembert equation H(xy) + H(x/y) = 2 H(x) H(y). The fundamental time quantum is the tick τ₀ = 1, and the scale ladder is defined by rung k mapping to φ^k.
proof idea
This is a direct definition that encodes the modified equation as a Prop. No lemmas are invoked; the body is simply the algebraic statement of the growth law with the (1 - Q) factor inserted.
why it matters
The definition supplies the governing equation used by the sibling results sigma8_predicted and suppressionFactor to compute the late-time suppression that reconciles Planck CMB σ₈ ≈ 0.811 with weak-lensing values ≈ 0.76. It implements the 8-tick octave (T7) and the recognition strain accumulation described in the module, providing the RS-native modification that matches observations to within 2σ. The declaration closes the interface between the CostAlgebra H and the cosmological growth dynamics.
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