rs_prediction_r
plain-language theorem explainer
RS supplies the tensor-to-scalar ratio prediction r = (phi - 1)^4 for comparison with CMB observations. Cosmologists deriving primordial spectra from J-cost fluctuations would cite this when forecasting signals for CMB-S4. The definition is a direct noncomputable assignment of the algebraic expression in the golden ratio.
Claim. The Recognition Science prediction for the tensor-to-scalar ratio is given by $r = (φ - 1)^4$, where $φ$ is the golden ratio.
background
The Cosmology.PrimordialSpectrum module derives the primordial power spectrum from J-cost quantum fluctuations during inflation, with the phi-ladder fixing the spectral tilt n_s. This definition isolates the RS expression for the tensor-to-scalar ratio r, which enters amplitude and tilt calculations and is listed among observable predictions in the module. The local setting targets a nearly scale-invariant spectrum P(k) ∝ k^(n_s - 1) seeded by these fluctuations, as stated in the module goals for a PRL paper on the CMB spectral index from the golden ratio.
proof idea
The declaration is a direct definition that assigns the closed-form expression (phi - 1)^4. No lemmas are applied and no tactics are invoked; the value follows immediately from algebraic substitution of the golden ratio conjugate.
why it matters
This definition supplies the concrete value referenced by the theorem r_prediction, which proves the bound 0.1 < (phi-1)^4 < 0.2 while noting the numerical discrepancy with the claimed 0.021. It realizes the COS-009 paper proposition on deriving the CMB spectral index from the golden ratio. Within the framework it connects to T6 where phi is forced as the self-similar fixed point, producing an r value inside the observable window for CMB-S4.
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