spectral_tilt_phi_connection
plain-language theorem explainer
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
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.
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