n_s_44_vs_55
plain-language theorem explainer
The theorem shows that the spectral index at 44 e-folds is strictly smaller than at 55 e-folds under the slow-roll formula n_s = 1 - 2/N derived from the J-cost inflaton potential. Cosmologists comparing e-fold counts against Planck constraints would cite this to confirm the N-dependence in the Recognition Science derivation. The proof reduces directly to expanding the spectral index definition and applying a numerical comparison.
Claim. Let $n_s(N) = 1 - 2/N$. Then $n_s(44) < n_s(55)$.
background
The module identifies the inflaton potential with the J-cost in logarithmic coordinates, yielding G(t) = cosh(t) - 1. This plateau form produces the slow-roll parameters epsilon and eta, from which the spectral index follows as n_s = 1 - 2/N for N e-folds. The definition of the spectral index is the standard slow-roll result 1 - 2/N. The upstream band definition supplies arithmetic conjunction on stable states but is not invoked in this comparison.
proof idea
The term-mode proof unfolds the spectral index definition to the explicit form 1 - 2/N and applies norm_num to compare the concrete values at N = 44 and N = 55.
why it matters
This inequality supports the N-dependence of the spectral index in the J-cost inflation model and aligns with the module's main result n_s_from_jcost. It contributes to the chain from the Recognition Composition Law through the phi-ladder to observable cosmology, confirming that larger e-fold counts move n_s closer to 1 in the alpha-attractor setting with alpha = phi^2. No downstream uses are recorded yet.
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