IndisputableMonolith.Cosmology.RS_Cosmo_Module_005
Cosmology module that packages a domain cost functional, its nonnegativity and evaluation identities, and a positive canonical threshold into an inhabitable certificate RSCosmo005Cert. Cosmologists working in the RS ledger cite it when they need a certified cost bound or threshold for domain-scale recognition events. The module is mostly definitional with short positivity and equality lemmas over the imported J-cost.
claimDefine a domain cost $C_{\mathrm{dom}}$ built from the RS cost $J$, prove $C_{\mathrm{dom}}\ge 0$ and the pointwise evaluation identity, introduce a canonical threshold $\theta>0$, and package these into an inhabited certificate record for cosmology module 005.
background
Recognition Science measures mismatch with the cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$), forced unique by the T5 step of the unified forcing chain and obeying the Recognition Composition Law. The Cost import supplies that $J$ and its elementary inequalities; Constants supplies the RS-native tick $\tau_0=1$.
In the cosmology layer one lifts $J$ from single ratios to domain-scale aggregates. This module introduces that lift as domain cost, records that it stays nonnegative, and fixes a positive canonical threshold against which domain costs are compared when deciding whether a cosmological recognition event has crossed the ledger bound.
The local setting is a thin certificate module: definitions plus the minimal positivity and evaluation facts needed so downstream cosmology developments can assume a single inhabited cert rather than re-proving cost hygiene.
proof idea
Definition module with short supporting lemmas, not a deep derivation. domainCost is introduced from the imported Cost primitives; domainCost_at_eq is an evaluation identity; domainCost_nonneg follows from nonnegativity of $J$. canonicalThreshold is a positive constant (canonicalThreshold_pos). RSCosmo005Cert bundles these fields; cert and cert_inhabited discharge inhabitation so callers obtain a concrete certificate term.
why it matters in Recognition Science
Gives the cosmology stack a named, inhabitable cost-and-threshold certificate (module 005) so later RS cosmology arguments can cite one object instead of rebuilding domain-cost hygiene. It sits on Constants and Cost, tying domain-scale bookkeeping to the same $J$ forced at T5 and to the RS tick. No downstream edges are recorded in the graph yet; the module is infrastructure for certified cosmological bounds rather than a closed forcing-chain step (T0–T8). Open use is any cosmology lemma that needs $C_{\mathrm{dom}}\ge 0$ or comparison to the canonical threshold inside an RS ledger argument.
scope and limits
- Does not derive Friedmann equations or observational cosmology fits.
- Does not prove uniqueness of the canonical threshold beyond positivity.
- Does not connect domain cost to the mass ladder or alpha band.
- Does not discharge global forcing-chain goals T0–T8.
- Does not assert numerical values for cosmological parameters.