r_from_jcost
plain-language theorem explainer
The theorem establishes that the tensor-to-scalar ratio equals twelve times phi squared over N squared for any positive real N. Cosmologists working from the J-cost inflaton potential would cite this identity when computing inflationary observables. The proof is a one-line wrapper that unfolds the definitions of the tensor-to-scalar ratio and the alpha-attractor parameter then applies reflexivity.
Claim. For any positive real number $N$, the tensor-to-scalar ratio $r$ satisfies $r = 12 phi^2 / N^2$, where $phi$ is the golden ratio and $r$ is obtained from the alpha-attractor model with alpha equal to $phi^2$.
background
The J-Cost as the Inflaton Potential module shows that the Recognition Composition Law forces the inflaton potential to J(x) = ½(x + x^{-1}) - 1. In log coordinates t = ln(x) this becomes G(t) = cosh(t) - 1, a Starobinsky-style plateau with G(0) = 0, G'(0) = 0 and G''(0) = 1. Slow-roll parameters follow from the curvature of G in these coordinates. The upstream tensor_to_scalar definition from PrimordialSpectrum sets r equal to the ratio of tensor to scalar power spectra. The alpha_attractor definition fixes alpha = phi^2, and the tensor_to_scalar definition in Inflation supplies the general formula r = 12 alpha / N^2.
proof idea
The proof is a one-line wrapper that unfolds tensor_to_scalar and alpha_attractor then applies reflexivity.
why it matters
This identity supplies the RS-specific value of the tensor-to-scalar ratio once the J-cost is adopted as the inflaton potential. It completes the chain from the self-similar fixed point phi (T6) through the alpha-attractor to concrete observables such as r approximately 0.0104 at N = 55. The result supports the master certificate InflationFromJCostCert and the spectral-index prediction 1 - 2/N derived from the same curvature.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.