module module high

`IndisputableMonolith.Gravity.Inflation`

The Gravity.Inflation module defines the α-attractor parameter α = φ² arising from self-similarity of the J-cost functional near x=1. It supplies supporting definitions for spectral index and tensor-to-scalar ratio bounds used in RS inflation models. Cosmologists working in the Recognition framework cite these when linking the φ² = φ + 1 identity to inflaton curvature. The module consists entirely of definitions and bounds with no proofs.

claimThe α-attractor parameter satisfies $α = φ²$, where $φ$ is the self-similar fixed point of the Recognition Composition Law and the inflaton potential inherits the quadratic character of $J(x)$ near $x=1$.

background

Recognition Science derives physics from the J-cost functional satisfying the Recognition Composition Law, with $J(x) = ½(x + x^{-1}) - 1$. The imported Constants module supplies the fundamental RS time quantum $τ₀ = 1$ tick. This module introduces the α-attractor in the Gravity domain, where the inflaton potential inherits the quadratic character of $J(x)$ near $x=1$ and the identity $φ² = φ + 1$ sets the curvature scale.

proof idea

this is a definition module, no proofs

why it matters in Recognition Science

This module supplies the α-attractor parameter that feeds the JCostInflaton module, which proves the Recognition Composition Law forces the inflaton potential to be $J(x) = ½(x + x^{-1}) - 1$ and derives slow-roll parameters ε and η. It fills the step connecting T5 J-uniqueness and T6 phi fixed point to inflationary observables. The downstream module uses these definitions to obtain the curvature of J in log coordinates.

scope and limits

Does not derive the full slow-roll parameters ε and η.
Does not address reheating or post-inflationary evolution.
Does not connect definitions to specific observational datasets.
Does not prove the value of the spectral index from first principles.

used by (1)

From the project-wide theorem graph. These declarations reference this one in their body.

IndisputableMonolith.Gravity.JCostInflaton module_import

depends on (1)

Lean names referenced from this declaration's body.

IndisputableMonolith.Constants module_import

declarations in this module (17)

def alpha_attractor L35
theorem alpha_attractor_eq_phi_plus_one L37
theorem alpha_attractor_pos L39
theorem alpha_attractor_bounds L41
def spectral_index L47
def tensor_to_scalar L52
theorem r_at_55_bounds L55
theorem n_s_at_55 L62
theorem r_in_detectable_range L67
def X_opt L78
theorem X_opt_pos L80
def Omega_0 L88
theorem Omega_0_pos L91
def k_rec_com L103
theorem curvature_bounded_at_R0 L107
structure InflationCert L112
theorem inflation_cert L119