IndisputableMonolith.Cosmology.RS_Cosmo_Module_005
Cosmology module that packages a domain cost functional, its nonnegativity, and a positive canonical threshold into a single certificate bundle. Cosmologists working in the RS ledger would cite it when a domain-scale cost bound or threshold comparison is needed. The module is mostly definitional: cost is specialized from the global J-cost, then wrapped with elementary positivity lemmas and an inhabited certificate record.
claimDefine a domain cost $C_{\mathrm{dom}}$ on the RS cost structure, prove $C_{\mathrm{dom}}\ge 0$ and an evaluation identity at equality points, introduce a canonical threshold $\theta_*>0$, and package these facts as an inhabited certificate $\mathrm{RSCosmo005Cert}$.
background
Recognition Science measures ledger mismatch by the J-cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$), forced unique by the Recognition Composition Law. The Cost import supplies that global cost apparatus; Constants supplies the RS-native tick $\tau_0=1$.
This module sits in the Cosmology domain and specializes cost to a domain-scale functional $C_{\mathrm{dom}}$. Sibling declarations record the evaluation identity at equality, nonnegativity of the domain cost, and a strictly positive canonical threshold. Those pieces are then bundled into the certificate record used by later cosmology claims that need a single named witness rather than a scatter of lemmas.
proof idea
Definition-first module, not a deep proof development. Domain cost is introduced by specializing the imported Cost primitives; the equality and nonnegativity facts are short algebraic or order lemmas on that specialization. The canonical threshold is a positive constant (or closed-form expression) with a one-line positivity proof. The certificate type collects these fields, and inhabitation is witnessed by assembling the already-proved components into the record constructor.
why it matters in Recognition Science
Gives Cosmology a reusable, named cost-and-threshold package so downstream RS cosmology arguments can cite one certificate instead of re-deriving domain nonnegativity and threshold positivity. No downstream edges are recorded yet in the mirror graph, so the module presently acts as a local scaffolding node inside Cosmology rather than a feeder of a named parent theorem. It ties the global J-cost apparatus (Cost, Constants) to domain-scale cosmology bookkeeping, consistent with the RS program of deriving macroscopic structure from the same cost functional that forces $\phi$, the eight-tick octave, and $D=3$.
scope and limits
- Does not derive domain cost from the forcing chain T0–T8.
- Does not prove observational cosmology bounds or fit data.
- Does not identify the canonical threshold with a measured cosmological constant.
- Does not establish uniqueness of the domain cost among all possible specializations.
- Does not feed any recorded downstream theorem in the current mirror graph.