IndisputableMonolith.Gravity.Inflation
Defines the RS α-attractor scale α = φ² and the derived slow-roll observables (spectral index n_s, tensor-to-scalar ratio r) evaluated near 55 e-folds. Cosmologists working the RS inflation sector cite it for the forced curvature scale and the numerical windows on n_s and r. Content is definitional plus elementary algebraic identities from the golden-ratio relation φ² = φ + 1.
claimThe module fixes the α-attractor parameter $\alpha = \varphi^2$ (equivalently $\alpha = \varphi + 1$), the optimal field coordinate $X_{\mathrm{opt}}$, the background density parameter $\Omega_0$, and the slow-roll observables $n_s$ and $r$ (with explicit bounds at $N = 55$ e-folds) that follow from the quadratic curvature of the J-cost near the identity.
background
Recognition Science forces a unique cost $J(x) = \frac12(x + x^{-1}) - 1$ (T5) whose self-similar fixed point is the golden ratio $\varphi$ (T6). Near $x = 1$, $J$ is quadratic; the curvature scale inherited by any inflaton potential built from $J$ is therefore set by the algebraic identity $\varphi^2 = \varphi + 1$.
The present module lives in the Gravity domain and imports only the RS constants (including the native tick $\tau_0$). It introduces the α-attractor parameter $\alpha = \varphi^2$ together with the derived slow-roll quantities: spectral index $n_s$, tensor-to-scalar ratio $r$, an optimal coordinate $X_{\mathrm{opt}}$, and a background density $\Omega_0$. These objects supply the numerical targets that the subsequent J-cost-as-inflaton development must hit.
proof idea
Definition module with short algebraic lemmas. The core definition sets $\alpha = \varphi^2$; a one-line rewrite uses $\varphi^2 = \varphi + 1$ to obtain the equivalent form. Positivity and elementary bounds on $\alpha$ are immediate from $\varphi > 1$. Spectral index and tensor-to-scalar ratio are defined by the standard slow-roll formulae evaluated on the J-derived potential; the $N = 55$ specializations and the detectable-range claim for $r$ are direct numerical evaluations of those closed forms. No deep tactic proofs appear.
why it matters in Recognition Science
Supplies the forced curvature scale and the concrete $n_s$, $r$ windows that the downstream module Gravity.JCostInflaton consumes. That module proves the Recognition Composition Law forces the inflaton potential to be exactly $J$ and then extracts slow-roll parameters $\varepsilon$ and $\eta$ from the curvature of $J$ in log coordinates; the α-attractor value and the 55-efold bounds defined here are the numerical anchors for those derivations. In the broader forcing chain the construction realises T5–T6 (unique $J$, unique $\varphi$) inside early-universe cosmology, giving a parameter-free prediction for the tensor-to-scalar ratio inside the observationally accessible band.
scope and limits
- Does not derive the inflaton potential from the Recognition Composition Law (that is JCostInflaton).
- Does not compute slow-roll parameters ε, η from first principles.
- Does not claim a full CMB power-spectrum fit beyond n_s and r windows.
- Does not address reheating, multi-field dynamics, or non-Gaussianity.
- Does not prove observational detection of r; only places it in a detectable numerical range.
used by (1)
depends on (1)
declarations in this module (17)
-
def
alpha_attractor -
theorem
alpha_attractor_eq_phi_plus_one -
theorem
alpha_attractor_pos -
theorem
alpha_attractor_bounds -
def
spectral_index -
def
tensor_to_scalar -
theorem
r_at_55_bounds -
theorem
n_s_at_55 -
theorem
r_in_detectable_range -
def
X_opt -
theorem
X_opt_pos -
def
Omega_0 -
theorem
Omega_0_pos -
def
k_rec_com -
theorem
curvature_bounded_at_R0 -
structure
InflationCert -
theorem
inflation_cert