fluctuations_from_jcost
plain-language theorem explainer
Primordial density fluctuations arise in Recognition Science as quantum variations in J-cost during inflation, with relative perturbations satisfying δρ/ρ proportional to δJ/J and the power spectrum P(k) proportional to the variance of those J-cost deviations. Cosmologists deriving the CMB spectrum from the Recognition functional equation would cite this when linking inflation to the observed amplitude A_s ≈ 2×10^{-9} and spectral tilt. The proof is a one-line triviality that accepts the statement without reduction, deferring substance to the J
Claim. In Recognition Science the relative density perturbation satisfies $δρ/ρ ∝ δJ/J$ and the primordial power spectrum obeys $P(k) ∝ ⟨(δJ)^2⟩$.
background
Recognition Science derives physics from the J-cost functional obeying the Recognition Composition Law. The shifted cost H(x) = J(x) + 1 satisfies the d'Alembert equation H(xy) + H(x/y) = 2 H(x) H(y), as defined in CostAlgebra. The inflaton potential on the recognition manifold is V(φ_inf) = J(1 + φ_inf), with vacuum value zero. Module COS-009 targets derivation of the primordial power spectrum from J-cost quantum fluctuations during inflation, where δJ ~ √(ℏ/τ₀) freeze out to seed density perturbations; the φ-ladder supplies the spectral tilt n_s - 1.
proof idea
The proof is a one-line wrapper that applies the trivial tactic. It accepts the claim as an axiom without invoking any lemmas from the nine upstream dependencies (CostAlgebra.H, InflatonPotentialFromJCost.V, FunctionalEquation.H, PrimitiveDistinction.from, SpectralEmergence.V, SMatrixUnitarity.amplitude, DoubleSlit.amplitude, Glossary.V).
why it matters
This declaration fills the COS-009 target stated in the module doc-comment of deriving the primordial spectrum from RS principles, aimed at a PRL paper on the CMB spectral index from the golden ratio. It connects J-cost fluctuations to the forcing chain landmarks T5 (J-uniqueness) and T6 (phi fixed point) and supplies the entry point for amplitude derivations such as A_s ~ (H/m_P)^2 ~ (V/m_P^4). Although currently unused downstream, it anchors the link between the eight-tick octave and observed scalar amplitude.
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