IndisputableMonolith.Cosmology.TensorToScalarRatioFromRS
Module that defines the RS tensor-to-scalar ratio r from the golden-ratio constant and certifies elementary bounds (positivity, r<1, a numerical band). Cosmologists comparing RS inflation predictions to CMB limits would cite it. Structure is definitional plus short inequality lemmas and a bundled certificate record.
claimDefines the Recognition Science tensor-to-scalar ratio $r$ in terms of $\varphi$, records the identity linking $r$ to a power of $\varphi$, proves $r>0$ and $r<1$, places $r$ in a concrete numerical band, and packages the claims in a certificate structure.
background
Recognition Science forces dimensionless cosmology observables from the same self-similar fixed point $\varphi$ that appears in the forcing chain (T6) and in the RS-native constants module. The tensor-to-scalar ratio $r$ is the standard inflation observable comparing primordial gravitational-wave power to scalar curvature perturbations; observational upper bounds come from CMB B-mode searches.
This module sits in the Cosmology domain and imports only Mathlib and Constants (where the RS time quantum $\tau_0=1$ tick and the derived constants built from $\varphi$ live). Sibling declarations introduce the raw ratio, an equality to a $\varphi$-expression, positivity and strict upper-bound lemmas, a band statement, and a certificate record that bundles the proved facts.
proof idea
Definition-plus-lemmas module rather than a single deep theorem. The ratio is introduced as a definition in terms of $\varphi$; a short algebraic identity (phi2_eq) rewrites it; positivity and $r<1$ are immediate from $\varphi>1$; the band lemma assembles decimal bounds; the certificate is a structure packing those facts for downstream discharge. No long tactic scripts or external analytic number theory.
why it matters in Recognition Science
Supplies the RS-native value and certified bounds for the inflationary tensor-to-scalar ratio, the quantity that sets the expected amplitude of primordial B-modes. It sits downstream of the Constants module and of the $\varphi$-forcing step (T6) in the unified chain. The graph currently lists no further used-by edges, so the module is a leaf that future CMB-comparison or slow-roll theorems can import. It does not itself derive the scalar spectral index or the full slow-roll hierarchy.
scope and limits
- Does not derive the scalar spectral index $n_s$ or slow-roll parameters $\epsilon,\eta$.
- Does not claim observational detection of $r$; only an RS-side band.
- Does not re-derive $\varphi$ or the forcing chain; imports Constants.
- Does not model reheating, late-time cosmology, or foregrounds.
- Certificate is local; does not auto-discharge external CMB likelihood codes.