IndisputableMonolith.Cosmology.RS_Cosmo_Module_003
Cosmology module that packages a domain cost functional, its nonnegativity and evaluation identities, and a positive canonical threshold into a single certificate type. Cosmologists working in the RS ledger would cite it when bounding domain-scale recognition costs against a fixed threshold. The module is mostly definitions plus short positivity and equality lemmas feeding an inhabited certificate record.
claimDefine a domain cost $C_{\mathrm{dom}}$ on the RS cost structure, prove $C_{\mathrm{dom}}\ge 0$ and the pointwise evaluation identity, introduce a canonical threshold $\theta>0$, and bundle these into an inhabited certificate $\mathrm{RSCosmo003Cert}$.
background
Recognition Science measures mismatch with the J-cost $J(x)=(x+x^{-1})/2-1$ from the Cost layer, forced uniquely by the Recognition Composition Law. Constants supplies the RS-native tick $\tau_0=1$. Cosmology modules lift that cost language to domain-scale bookkeeping: how expensive it is, in recognition units, for a spatial or causal domain to sit away from the self-similar fixed point.
This file sits in the Cosmology domain and imports only Mathlib, Constants, and Cost. It introduces a domain cost functional together with a canonical threshold meant as a comparison scale for that cost. Sibling names indicate elementary analytic facts (nonnegativity, evaluation at a point) rather than a full dynamical cosmology derivation.
The certificate record is the module's public face: a single inhabited structure that downstream cosmology arguments can assume instead of re-proving the local cost lemmas.
proof idea
Definition-heavy module. Domain cost is introduced as a def on the Cost infrastructure; domainCost_at_eq and domainCost_nonneg are short algebraic or order lemmas. Canonical threshold is a positive constant def with a one-line positivity proof. The certificate type packages those facts; inhabitation is by assembling the already-proved fields. No deep tactic scripts or multi-step forcing arguments appear at module scope.
why it matters in Recognition Science
Gives Cosmology a reusable cost-and-threshold certificate so later RS cosmology claims can cite one object instead of reopening Cost and Constants. No downstream edges are recorded yet (used_by is empty), so this is currently a leaf packaging layer rather than a step inside T0–T8. It does not itself force $D=3$, the eight-tick octave, or the $\phi$-ladder mass formula; it only standardizes domain-cost comparison language those later arguments may need. Lands as scaffolding for certificate-style cosmology lemmas that want a positive threshold against which domain cost is measured.
scope and limits
- Does not derive FLRW dynamics, expansion history, or observational cosmology fits.
- Does not prove uniqueness of the domain cost beyond the imported Cost axioms.
- Does not fix numerical values of $c$, $\hbar$, $G$, or $\alpha$ inside this module.
- Does not discharge any T0–T8 forcing step or the mass ladder.
- Does not yet feed recorded downstream theorems; used_by is empty.