tensor_to_scalar
plain-language theorem explainer
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.
Claim. The 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
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.
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