tensor_to_scalar
The definition supplies the RS-specific tensor-to-scalar ratio r(N) = 12 φ² / N² for N e-foldings. Cosmologists modeling primordial spectra in Recognition Science would cite it to replace the free α parameter in α-attractor models with the fixed value φ². It is realized as a direct substitution of the alpha_attractor constant into the standard slow-roll expression.
claimThe tensor-to-scalar ratio is given by $r(N) = 12 φ^2 / N^2$ for $N > 0$, where $φ$ denotes the golden ratio and the attractor parameter $α$ is fixed at $φ^2$ rather than left free.
background
In the Inflation from phi module the α-attractor parameter is fixed to φ² by the self-similarity condition of the cost functional. The upstream definition in Cosmology.PrimordialSpectrum supplies the general ratio r = A_T / A_s. This declaration specializes that ratio by inserting the RS value α = φ², producing the explicit dependence r(N) = 12 φ² / N².
proof idea
One-line definition that substitutes the value of alpha_attractor (itself equal to phi squared) into the standard expression 12 α / N².
why it matters in Recognition Science
It supplies the concrete formula used by r_at_55_bounds and r_in_detectable_range to bound r for N near 55 and by r_from_jcost to recover the explicit φ dependence. The result fills the RS-specific prediction for the tensor-to-scalar ratio in the Universe-Origin Paper, replacing arbitrary α with φ² from J-cost self-similarity. It connects to the phi-ladder and the eight-tick octave in the forcing chain.
scope and limits
- Does not derive A_T or A_s from the underlying cost functional.
- Does not specify the range of N beyond the positivity assumption used downstream.
- Does not include higher-order slow-roll corrections or running of the spectral index.
- Does not incorporate the log-periodic modulation term Ω₀.
Lean usage
theorem r_from_jcost (N : ℝ) (hN : 0 < N) : tensor_to_scalar N = 12 * phi ^ 2 / N ^ 2 := by unfold tensor_to_scalar alpha_attractor; rflformal statement (Lean)
52noncomputable def tensor_to_scalar (N : ℝ) : ℝ := 12 * alpha_attractor / N ^ 2
proof body
Definition body.
53
54/-- For N = 55 e-foldings: r ≈ 12 * 2.618 / 3025 ≈ 0.0104. -/