IndisputableMonolith.Cosmology.RS_Cosmo_Module_011
Cosmology module that packages a nonnegative domain cost, a positive canonical threshold, and an inhabited certificate bundle for RS cosmology item 011. Cosmologists working in the Recognition ladder would cite it when they need a certified cost-threshold pair rather than ad hoc cutoffs. The module is mostly definitions plus elementary positivity and equality lemmas over the imported J-cost.
claimDefine a domain cost $C_{\mathrm{dom}}$ built from the Recognition $J$-cost, prove $C_{\mathrm{dom}}\ge 0$ and an evaluation identity, introduce a canonical threshold $\theta_*>0$, and package these into an inhabited certificate $\mathrm{RSCosmo011Cert}$.
background
Recognition Science measures mismatch with the unique cost $J(x)=(x+x^{-1})/2-1$ forced by the Recognition Composition Law. The Cost import supplies that functional; Constants supplies the RS-native tick $\tau_0=1$.
In the cosmology layer one needs a domain-level cost (aggregate mismatch on a spatial or causal domain) and a fixed positive threshold against which that cost is compared. This module introduces those two objects and the elementary analytic facts (nonnegativity, positivity of the threshold, evaluation identity) required before any dynamical claim.
The certificate record simply bundles the definitions and the proved inequalities so downstream cosmology statements can assume a single inhabited witness rather than re-proving the same inequalities.
proof idea
Definition module with short supporting lemmas. domainCost is defined from the imported $J$-cost; domainCost_nonneg and domainCost_at_eq are direct consequences of $J\ge 0$ and the evaluation rule. canonicalThreshold is a positive constant (positivity is a one-line arithmetic check). RSCosmo011Cert packages the above; cert_inhabited builds a concrete inhabitant from those lemmas. No deep forcing or dynamical argument lives here.
why it matters in Recognition Science
Supplies the certified cost-threshold pair that later RS cosmology statements can import without re-deriving nonnegativity of domain mismatch or positivity of the cutoff. It sits under the Cosmology domain and depends only on Constants and Cost, so it is a thin interface layer rather than a forcing-chain step (T0–T8). With no recorded downstream users yet, its role is to close the scaffolding for item 011: any theorem that needs a nonnegative domain cost compared to a fixed positive threshold can cite the inhabited certificate instead of open hypotheses.
scope and limits
- Does not derive domainCost from the forcing chain T0–T8 or from RCL uniqueness.
- Does not identify canonicalThreshold with a physical constant (Hubble scale, recombination, etc.).
- Does not prove dynamical evolution, FLRW matching, or observational bounds.
- Does not claim uniqueness of the certificate beyond inhabitation.
- Does not connect the threshold to eight-tick cadence or $D=3$.