IndisputableMonolith.Cosmology.RS_COS_Structural_001
Structural certificate module for the first RS cosmology claim: a nonnegative domain cost built from the J-cost, a positive canonical threshold, and an inhabited certificate packing those facts. Cosmologists and RS auditors cite it when wiring early-universe or horizon-scale inequalities to the cost layer. The module is mostly definitions plus elementary positivity and evaluation lemmas, not a deep existence proof.
claimDefine a domain cost $C_{\mathrm{dom}}$ from the Recognition $J$-cost, prove $C_{\mathrm{dom}}\ge 0$ and an evaluation identity at equality cases, fix a canonical threshold $\theta>0$, and package these into an inhabited structural certificate $\mathrm{RS\_COS\_Structural\_001}$.
background
Recognition Science measures mismatch with the J-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 primitive; Constants supplies the RS-native tick $\tau_0=1$.
This cosmology module lifts $J$ to a domain-level cost suitable for coarse-grained cosmological regions (horizons, causal patches, or ladder bins). The canonical threshold is a fixed positive scale against which that cost is compared when stating structural inequalities. The local setting is certificate-first: facts are bundled so downstream cosmology claims can assume a single inhabited record rather than re-proving nonnegativity each time.
proof idea
Definition-heavy module. domainCost is introduced from the Cost layer; domainCost_at_eq is an evaluation identity; domainCost_nonneg follows from nonnegativity of $J$. canonicalThreshold is a numeric/def constant with canonicalThreshold_pos as a one-line positivity fact. RSCOSStructural001Cert is a structure bundling those properties; cert and cert_inhabited discharge inhabitation by assembling the lemmas. No deep tactic scripts: algebraic inheritance from $J\ge 0$ plus packaging.
why it matters in Recognition Science
Gives Cosmology a minimal structural certificate (RS_COS_Structural_001) so later horizon, expansion, or ladder-gap arguments can cite one inhabited object instead of raw Cost lemmas. It sits on Constants and Cost only; the page lists no downstream used_by edges yet, so it is an entry stub for the cosmology certificate stack rather than a leaf of T0–T8. Ties the J-cost and $\phi$-native units into cosmological bookkeeping without claiming dynamical Einstein equations or observational fits. Auditors use it to confirm the cost-threshold interface is closed before heavier RS cosmology claims are attached.
scope and limits
- Does not derive Friedmann equations, FLRW metrics, or Einstein dynamics.
- Does not fix observational cosmological parameters or fit data.
- Does not prove uniqueness of the canonical threshold beyond positivity.
- Does not connect domain cost to mass-ladder rungs or alpha bounds.
- Does not assert eight-tick or D=3 forcing inside this module.