r_at_55_bounds
plain-language theorem explainer
The theorem establishes that the tensor-to-scalar ratio at 55 e-foldings is strictly positive under the Recognition Science inflationary model. Cosmologists comparing RS forecasts to CMB data would cite this bound when confirming that the predicted r lies above zero and inside the detectable window. The proof is a short term-mode argument that unfolds the definition of the ratio and applies the div_pos lemma together with mul_pos and alpha_attractor_pos.
Claim. For $N=55$, the tensor-to-scalar ratio $r(N) := 12 phi^2 / N^2 > 0$, where $phi$ is the golden-ratio fixed point and the formula follows from the RS-specific alpha-attractor with alpha = phi^2.
background
In the Recognition Science treatment of inflation the alpha-attractor parameter is identified with phi squared, where phi is the self-similar fixed point forced by the T5-T6 steps of the unified forcing chain. The tensor-to-scalar ratio is then specialized to the parameter-free expression r(N) = 12 phi^2 / N^2. The module derives the standard slow-roll relations n_s approx 1-2/N together with this RS-specific r formula and the log-periodic frequency Omega_0 = 2 pi / ln(1/X_opt).
proof idea
The proof unfolds the local definition of tensor_to_scalar to obtain the explicit quotient 12 * alpha_attractor / 55^2. It applies the div_pos lemma, reducing the claim to positivity of numerator and denominator. The numerator is discharged by mul_pos applied to a norm_num fact and the upstream theorem alpha_attractor_pos; the denominator is discharged by positivity.
why it matters
This result supplies the concrete positivity needed to place the RS prediction r approx 0.0104 inside the detectable window for future CMB experiments. It directly supports the module claim that the tensor-to-scalar ratio follows from the alpha-attractor with alpha = phi^2. The parent framework step is the derivation of the eight-tick octave and D=3 from the forcing chain, which fixes phi and thereby fixes the amplitude of r.
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