phi_prediction_tilt
The definition supplies the Recognition Science prediction for the CMB spectral tilt deviation as one over eight phi cubed. Cosmologists testing RS inflation models against Planck data would cite this when checking the expected n_s near 0.965. It is obtained by direct numerical comparison of several phi-powered expressions to the observed |n_s - 1| ≈ 0.035, selecting the closest match.
claim$|n_s - 1| = 1/(8 phi^3)$
background
The COS-009 module derives the primordial power spectrum from J-cost quantum fluctuations during inflation, where the phi-ladder fixes the spectral tilt of P(k) ∝ k^(n_s - 1). Observed values are n_s ≈ 0.965 and A_s ≈ 2.1 × 10^{-9}. The supplied analysis lists five candidate phi expressions and identifies 1/(8 phi^3) as the best numerical fit to |n_s - 1| ≈ 0.035.
proof idea
One-line definition that directly encodes the selected numerical match 1/(8 phi^3) from the listed phi-powered candidates.
why it matters in Recognition Science
This supplies the concrete RS prediction for the scalar spectral index tilt inside the broader primordial spectrum derivation. It links the phi-ladder (T6) and J-uniqueness (T5) to CMB observables, supporting the paper proposition on the spectral index from the golden ratio. The 0.030 versus 0.035 residual leaves open the need for higher-order corrections or exact dynamical derivation from J-cost.
scope and limits
- Does not derive the tilt from explicit J-cost fluctuation dynamics.
- Does not compute the scalar amplitude or tensor-to-scalar ratio.
- Does not incorporate scale dependence beyond the constant tilt value.
formal statement (Lean)
94noncomputable def phi_prediction_tilt : ℝ := 1 / (8 * phi^3)
proof body
Definition body.
95