IndisputableMonolith.Cosmology.RS_Cosmo_Module_002
Cosmology support module that packages a domain-level cost functional, its nonnegativity and evaluation identities, and a positive canonical threshold into an inhabited certificate RSCosmo002Cert. Cosmology arguments that need a uniform cost-versus-threshold comparison cite this bundle. The module is mostly definitional: Cost and Constants are imported, then elementary lemmas discharge the certificate fields.
claimDefine a domain cost $C_{\mathrm{dom}}$ (from the RS $J$-cost), prove $C_{\mathrm{dom}}\ge 0$ and the pointwise evaluation identity, fix a canonical threshold $\theta>0$, and package these into an inhabited certificate for RS cosmology module 002.
background
Recognition Science measures mismatch with the $J$-cost from the Cost library; Constants supplies the RS-native tick $\tau_0=1$. In the cosmology layer one needs a domain-scoped cost (how expensive a cosmological domain configuration is relative to the recognition functional) and a fixed positive threshold against which that cost is compared.
This module sits in IndisputableMonolith.Cosmology and imports only Mathlib, Constants, and Cost. Sibling declarations introduce domainCost with equality-at-a-point and nonnegativity lemmas, a canonicalThreshold with a positivity proof, and the certificate type RSCosmo002Cert together with an inhabited instance. No forcing-chain step (T0–T8) is proved here; the module only freezes the cost/threshold interface those later cosmology claims will quote.
proof idea
Definition module with thin lemma wrappers, not a deep derivation. domainCost is defined from the imported Cost primitives; domainCost_at_eq and domainCost_nonneg are short algebraic or order facts. canonicalThreshold is a numeric/structural constant; canonicalThreshold_pos is a one-line positivity check. RSCosmo002Cert aggregates those fields, and cert / cert_inhabited build a concrete inhabitant so downstream files can assume the bundle without re-proving the side conditions.
why it matters in Recognition Science
Gives cosmology a single named certificate for “domain cost is nonnegative and comparable to a fixed positive threshold,” so later RS cosmology developments do not re-open Cost bookkeeping. The forcing landmarks (J-uniqueness, $\varphi$, eight-tick, $D=3$) are upstream of Cost/Constants; this module only consumes them. No downstream used_by edges are recorded yet, so the certificate is presently a leaf interface waiting on higher cosmology theorems (expansion, horizon, or ladder-matching claims) to import cert.
scope and limits
- Does not derive J-uniqueness, $\varphi$, eight-tick structure, or $D=3$.
- Does not prove observational cosmology (FLRW fits, CMB, or luminosity distances).
- Does not fix numerical values of $G$, $H_0$, or $\alpha$ beyond Cost/Constants imports.
- Does not assert which physical domain the cost is evaluated on outside the certificate fields.
- Does not supply downstream cosmology theorems; used_by is currently empty.