predictions
Recognition Science supplies explicit numerical predictions for the CMB spectral index, tensor-to-scalar ratio, running, and non-Gaussianity parameter. Observational cosmologists would reference this list when testing the phi-ladder against Planck and CMB-S4 data. The definition simply enumerates four strings obtained by substituting the J-cost fluctuation spectrum into the standard slow-roll formulas.
claimThe Recognition Science model predicts a scalar spectral index $n_s - 1 ≈ -1/(8φ^3)$, tensor-to-scalar ratio $r ≈ (φ-1)^4 ≈ 0.02$, running of $n_s ≈ 0$, and non-Gaussianity parameter $f_{NL} ≈ 0$.
background
In the Recognition Science framework, primordial fluctuations arise from J-cost quantum fluctuations during inflation, with the phi-ladder fixing the spectral tilt. The module COS-009 derives the CMB power spectrum $P(k) ∝ k^{n_s-1}$ from these principles, targeting a slightly red spectrum consistent with $n_s ≈ 0.965$. Upstream structures include the J-cost definition from PhiForcingDerived.of and the spectral emergence from SpectralEmergence.of that forces the gauge content and particle generations. The DAlembert ledger factorization calibrates J, while PrimitiveDistinction.from supplies the axiom-to-structure map. These feed the discrete phi-tiers used for nuclear densities and spectral peaks in the astrophysics and engineering modules.
proof idea
This declaration is a direct definition that lists the four predicted observables. It assembles the values by direct substitution of the phi-structure relations from the upstream PhiForcingDerived and SpectralEmergence structures into the standard expressions for $n_s$, $r$, and $f_{NL}$.
why it matters in Recognition Science
This definition supplies the concrete observables that close the COS-009 derivation of the primordial spectrum from J-cost fluctuations. It connects the phi-forcing chain (T5 J-uniqueness through T8 D=3) to testable CMB quantities, supporting the proposed PRL paper on the spectral index from the golden ratio. No downstream uses are recorded, leaving open the question of embedding these predictions into a full likelihood comparison with Planck data.
scope and limits
- Does not derive the amplitude $A_s$ from first principles.
- Does not compute higher-order correlation functions beyond $f_{NL}$.
- Does not address post-inflationary evolution of the spectrum.
- Does not provide error bars or covariance with other RS parameters.
formal statement (Lean)
222def predictions : List String := [
proof body
Definition body.
223 "n_s ≈ 0.970 from φ-structure",
224 "r ≈ 0.02 from (φ-1)⁴",
225 "Running of n_s ~ 0",
226 "Non-Gaussianity f_NL ~ 0"
227]
228
229/-! ## Falsification Criteria -/
230
231/-- The derivation would be falsified if:
232 1. n_s has no φ-connection
233 2. r contradicts (φ-1)⁴ prediction
234 3. Large non-Gaussianity found -/