IndisputableMonolith.Cosmology.RS_Cosmo_Module_007
Cosmology module 007 packages a domain cost functional, its nonnegativity and evaluation identities, a positive canonical threshold, and an inhabited certificate bundle RSCosmo007Cert. Cosmologists working in the RS ledger cite it when they need a named cost-and-threshold interface rather than raw J. The file is mostly definitions plus short positivity and equality lemmas over the imported Cost layer.
claimDefine a domain cost $C_{\mathrm{dom}}$ on the RS cost layer, prove $C_{\mathrm{dom}}\ge 0$ and the pointwise evaluation identity, fix a canonical threshold $\theta_*>0$, and package these into an inhabited certificate $\mathrm{RSCosmo007Cert}$.
background
Recognition Science measures mismatch with the J-cost $J(x)=(x+x^{-1})/2-1$ from the Cost module; Constants supplies the native tick $\tau_0=1$. Cosmology modules lift that cost to domain-scale bookkeeping (horizon patches, causal cells, or ledger domains) rather than single-link edges.
This file sits in the Cosmology domain and imports only Mathlib, Constants, and Cost. Sibling names indicate a domain cost map, an evaluation identity, nonnegativity, a canonical threshold with positivity, and a certificate record RSCosmo007Cert with an inhabited instance. No forcing-chain (T0–T8) material is re-proved here; the module assumes the Cost API and exposes a thin cosmology-facing wrapper.
proof idea
Definition-heavy module. Domain cost is introduced as a named functional on the Cost layer; domainCost_at_eq is an evaluation identity and domainCost_nonneg is nonnegativity, both short lemmas. canonicalThreshold is a positive constant (canonicalThreshold_pos). The certificate record bundles these facts; cert_inhabited supplies a witness. No deep tactic scripts or multi-hop forcing arguments.
why it matters in Recognition Science
Gives cosmology code a stable cost-and-threshold handle instead of calling raw J at every call site. Downstream used-by edges are empty in the current graph, so the module is an interface seed rather than a proved parent theorem. It aligns with RS cost bookkeeping (J-uniqueness T5, RCL) and prepares threshold comparisons of the same family as Berry-scale cutoffs, without claiming a specific observational cosmology theorem yet.
scope and limits
- Does not derive spatial dimension D=3 or the eight-tick octave.
- Does not prove uniqueness of J or the Recognition Composition Law.
- Does not fix numerical cosmological parameters (H_0, Omega_m, etc.).
- Does not connect domain cost to the mass ladder or alpha band.
- Does not assert observational falsification criteria beyond the certificate interface.