inflation_cert
plain-language theorem explainer
The declaration certifies that the Recognition Science inflationary model satisfies the attractor relation alpha equals phi plus one, its positivity, the spectral index interval at 55 e-folds, positive modulation frequency Omega_0, and unit curvature bound at the recognition scale. Cosmologists modeling phi-driven inflation would cite it to confirm consistency with CMB slow-roll constraints. The proof constructs the InflationCert structure by direct substitution of five supporting lemmas.
Claim. InflationCert holds: the attractor parameter satisfies $alpha = phi + 1 > 0$, the spectral index at 55 e-folds lies in $(0.96, 0.97)$, the modulation frequency satisfies $Omega_0 > 0$, and the curvature bound at the recognition scale is $1/ell_0^2 = 1$.
background
The Gravity.Inflation module formalizes RS inflationary predictions from the phi-ladder and J-cost self-similarity. The alpha-attractor parameter is defined via the equality alpha_attractor = phi + 1. The spectral index n_s(N) follows the standard slow-roll form evaluated at N = 55. Omega_0 is the log-periodic modulation frequency 2 pi / ln(1/X_opt) with X_opt = phi/pi. ell0 is the recognition length scale fixed to 1 in RS-native units, yielding the curvature bound |R| <= 1 at R0.
proof idea
The proof is a term-mode record construction of the InflationCert structure. It supplies the five fields by direct reference to the lemmas alpha_attractor_eq_phi_plus_one for the attractor equality, alpha_attractor_pos for positivity, n_s_at_55 for the spectral interval, Omega_0_pos for modulation positivity, and curvature_bounded_at_R0 for the curvature equality.
why it matters
This theorem completes the inflationary certification inside the Gravity.Inflation module and assembles the core predictions listed in the module documentation: alpha derived as phi plus one, n_s near 0.964, positive Omega_0 approximately 9.47, and curvature bound at R0. It links the upstream J-uniqueness and phi fixed-point results to observable cosmology. No downstream uses are recorded.
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