IndisputableMonolith.Cosmology.InflationSpectralIndexFromJCost
The module derives the inflationary spectral index from the J-cost function in Recognition Science cosmology. It defines gap45 as the body-plan ceiling of 45 and builds nsRS with value, bounds, and a certification near the Planck measurement. Cosmologists working on RS-derived inflation models would cite these results to connect J-uniqueness to observable n_s. The module proceeds via a chain of definitions and elementary inequalities.
claim$gap_{45}=45$, $n_s^{RS}$ with $0<n_s^{RS}<1$ and $n_s^{RS}$ in the band near the Planck value, together with the certificate $SpectralIndexCert$ that $n_s^{RS}$ matches the observed spectral index.
background
The module sits in the Cosmology domain and imports the base constants module whose fundamental time quantum is given by τ₀ = 1 tick. It introduces gap45 as the body-plan ceiling equal to 45, a parameter that enters the phi-ladder mass formula. From this it constructs nsRS as the RS-native spectral index obtained from the J-cost, together with its numerical value, strict inequalities, and proximity to the Planck result.
proof idea
This is a definition module, no proofs. It consists of a linear sequence of definitions (gap45, nsRS, nsRS_val) followed by elementary lemmas establishing the bounds nsRS_lt_one, nsRS_gt_zero, nsRS_band and the final certification SpectralIndexCert.
why it matters in Recognition Science
The module supplies the J-cost derivation of the inflationary spectral index to the Recognition Science cosmology framework. It grounds the observable n_s in the forcing chain landmarks T5 (J-uniqueness) and T6 (phi fixed point) and provides the concrete certification that feeds into later cosmological matching arguments.
scope and limits
- Does not derive the inflationary potential or slow-roll parameters from the J-cost.
- Does not compute the tensor-to-scalar ratio or higher-order spectral running.
- Does not address reheating or post-inflationary evolution.
- Does not connect to the alpha band or fine-structure constant.