IndisputableMonolith.Cosmology.RS_Cosmo_Module_012
Cosmology packaging module that defines a domain cost functional, proves its nonnegativity and evaluation identity, and introduces a positive canonical threshold together with an inhabited certificate type RSCosmo012Cert. Cosmologists working in the RS ledger cite it when they need a checked cost-threshold pair rather than raw constants. The module is mostly definitional with short positivity and equality lemmas over the imported Cost and Constants layers.
claimThe module introduces a domain cost $C_{\mathrm{dom}}$, shows $C_{\mathrm{dom}}\ge 0$ and an evaluation identity at a distinguished point, defines a canonical threshold $\theta>0$, and packages these into an inhabited certificate record for RS cosmology item 012.
background
Recognition Science measures mismatch with the J-cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$), developed in the Cost import. Constants supplies the RS-native tick $\tau_0=1$. Cosmology modules lift those primitives to domain-level bookkeeping: a cost assigned to a cosmological domain and a threshold that marks when the ledger treats the domain as resolved or open.
Module 012 sits in that layer. It does not re-derive $J$ or the forcing chain (T5–T8); it assumes Cost and Constants and builds a thin certificate surface: domain cost, its elementary analytic properties, a positive canonical threshold, and a record type that witnesses the package is inhabited. Downstream cosmology arguments can then depend on the certificate rather than on ad-hoc numeric cuts.
proof idea
Definition-first module. domainCost and canonicalThreshold are introduced as defs; domainCost_nonneg and canonicalThreshold_pos are short nonnegativity/positivity arguments over the Cost primitives; domainCost_at_eq is an evaluation identity. RSCosmo012Cert bundles the pieces, with cert and cert_inhabited supplying a concrete inhabitant. No deep tactic scripts: the logical content is packaging plus elementary inequalities.
why it matters in Recognition Science
Gives cosmology a named, checkable cost-threshold certificate (item 012) instead of scattered constants. Parent use sites are not yet wired in this graph (used_by empty), so the module is a leaf certificate surface for later RS cosmology theorems that need a nonnegative domain cost and a positive threshold in one place. It sits downstream of Cost/Constants only and does not itself close forcing-chain steps T0–T8, the RCL, or the mass ladder; it is infrastructure for cosmological ledger claims that will cite those landmarks elsewhere.
scope and limits
- Does not derive J-uniqueness, phi, eight-tick structure, or D=3.
- Does not prove observational cosmology fits or FLRW dynamics.
- Does not fix numerical values of alpha, G, or particle masses.
- Does not assert the certificate is used by any downstream theorem yet.
- Does not replace the full Cost theory; only packages domain-level facts.