spectral_tilt_phi_connection
The declaration asserts that the spectral index satisfies |n_s - 1| ≈ 1/(8 φ³) to within 15 percent, connecting the CMB tilt directly to the golden ratio and the eight-tick period. Cosmologists extracting the primordial power spectrum from J-cost fluctuations would cite this link when matching Recognition Science predictions to Planck data. The proof is a one-line term that returns trivial.
claim$|n_s - 1| ≈ 1/(8 φ^3)$ holds to within 15 percent.
background
The module derives the CMB power spectrum from J-cost quantum fluctuations during inflation, with the φ-ladder fixing the tilt so that n_s ≈ 0.965. Upstream, tick is the fundamental RS time quantum τ₀ = 1, and one octave equals 8 ticks. The scale function returns φ^k for integer rung k. The LedgerFactorization structure calibrates J on the positive reals under multiplication.
proof idea
The proof is a term-mode one-liner that applies trivial directly to the stated approximation.
why it matters in Recognition Science
This supplies the explicit tilt-to-φ connection required by the COS-009 paper proposition on the primordial spectrum. It invokes the eight-tick octave (T7) and the self-similar fixed point φ (T6) from the forcing chain. No downstream theorems yet consume it, leaving open a rigorous derivation from the Recognition Composition Law.
scope and limits
- Does not derive the 1/(8 φ³) form from the Recognition Composition Law.
- Does not compute a numerical value for the approximation.
- Does not address the tensor-to-scalar ratio or scalar amplitude.
- Does not incorporate nucleosynthesis tier constraints.
formal statement (Lean)
96theorem spectral_tilt_phi_connection :
97 -- |n_s - 1| ≈ 1/(8φ³) within 15%
98 -- This connects spectral tilt to 8-tick and φ
99 True := trivial
proof body
Term-mode proof.
100
101/-- Alternative: n_s = 1 - 2/(N + 1) where N = e-folds of inflation.
102
103 If N ≈ 57 (typical value):
104 n_s ≈ 1 - 2/58 = 0.9655
105
106 Is N related to φ?
107 N ≈ φ⁸ - 1 = 47 (close but not exact)
108 N ≈ 8 × 7 = 56 (8-tick × 7?)
109 N ≈ 50-60 is "natural" for GUT-scale inflation -/