IndisputableMonolith.Cosmology.Inflation_Parameters5
Module defining the fifth RS inflation-parameter certificate: a nonnegative domain cost, its pointwise evaluation identity, a positive canonical threshold, and an inhabited certificate record. Cosmologists locking RS slow-roll or e-fold windows would cite it. Content is definitional plus elementary nonnegativity and positivity lemmas from the Cost import.
claimThe module introduces a domain cost $C_{\mathrm{dom}}$, proves $C_{\mathrm{dom}}\ge 0$ and an evaluation identity, defines a positive canonical threshold $\theta_*>0$, and packages these data into an inhabited inflation-parameter certificate.
background
Recognition Science cosmology extracts inflationary scales from the same cost functional and self-similar fixed point that force the particle spectrum. The Cost import supplies $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$); Constants fixes the RS time quantum $\tau_0=1$ tick.
This Cosmology module layers a domain-level cost on that foundation, together with a canonical threshold used to gate inflationary parameter windows. Sibling objects record nonnegativity of the domain cost, positivity of the threshold, and a certificate structure that packages both.
No forcing-chain step (T5--T8) is re-proved here; the module assumes the upstream cost and constant infrastructure and only assembles the fifth inflation certificate.
proof idea
Definition module with short supporting lemmas. The domain cost is introduced from the Cost layer; nonnegativity and the evaluation identity are elementary; the canonical threshold is defined and shown positive; the certificate record is assembled and shown inhabited. No multi-step forcing or analytic estimates appear.
why it matters in Recognition Science
Supplies the fifth inflation-parameter certificate in the RS cosmology stack: domain cost plus positive threshold, wrapped so downstream cosmology developments can discharge numerical windows without re-deriving positivity. The import graph shows no consumers yet, but the certificate pattern matches other RS cert modules used to lock slow-roll or e-fold bounds. It sits downstream of J-cost uniqueness (T5) and the phi fixed point (T6) only indirectly, via the shared Cost and Constants infrastructure.
scope and limits
- Does not derive slow-roll parameters from the forcing chain.
- Does not prove observational bounds on $n_s$ or $r$.
- Does not link the threshold to the eight-tick octave or $D=3$.
- Does not compute e-fold counts or the inflationary energy scale.
- Does not introduce new physics beyond Cost and Constants.