IndisputableMonolith.Cosmology.RS_Cosmo_Module_009
Cosmology module packaging a domain-level recognition cost and a positive canonical threshold, together with a small certificate that the cost is nonnegative and the threshold is positive. Cosmologists working in the RS ledger would cite it when fixing the cost scale of a spatial domain before comparing to expansion or horizon data. The module is mostly definitions plus elementary positivity lemmas, closed by an inhabited certificate record.
claimDefine a domain cost $C_{\mathrm{dom}}$ built from the RS cost functional $J$, prove $C_{\mathrm{dom}}\ge 0$ and an evaluation identity at equality of arguments, and fix a canonical threshold $\theta>0$. Package these facts as an inhabited certificate $\mathrm{RSCosmo009Cert}$.
background
Recognition Science measures mismatch by the J-cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$), forced uniquely by the Recognition Composition Law. The Cost import supplies that functional and its elementary inequalities; Constants supplies the RS-native tick $\tau_0=1$.
In the cosmology layer one lifts pointwise $J$ to a domain cost $C_{\mathrm{dom}}$ that scores an entire spatial region (or its scale factor) rather than a single ratio. A canonical threshold $\theta$ then marks the minimal cost at which a domain is treated as cosmologically resolved. The module sits after the forcing chain (T5 J-uniqueness, T6 $\phi$, T7 eight-tick octave, T8 $D=3$) and before any comparison with expansion histories or horizon counts.
proof idea
Definition module with short supporting lemmas. domainCost is introduced from the imported Cost functional; domainCost_at_eq records the evaluation identity when the two arguments coincide; domainCost_nonneg follows from nonnegativity of $J$. canonicalThreshold is a fixed positive real, with canonicalThreshold_pos a one-line positivity proof. RSCosmo009Cert bundles these facts; cert and cert_inhabited exhibit a concrete inhabitant so downstream code can assume the package without re-proving the elementary inequalities.
why it matters in Recognition Science
Gives the cosmology stack a named, certified cost-and-threshold pair so later modules can quote a single certificate rather than re-open Cost. No downstream edges are recorded yet; the natural consumers are horizon, expansion, or domain-counting arguments that need $C_{\mathrm{dom}}\ge 0$ and a fixed positive cutoff. Within the broader RS program it is scaffolding for translating the T5–T8 ledger into cosmological observables (domain scales measured in ticks, thresholds near Berry or $\phi$-ladder rungs) without smuggling classical FRW cost functions.
scope and limits
- Does not derive Friedmann or acceleration equations from $C_{\mathrm{dom}}$.
- Does not fix numerical values of $H_0$, $\Omega_m$, or dark-energy density.
- Does not prove uniqueness of the canonical threshold among all positive cutoffs.
- Does not connect domain cost to observed CMB multipoles or BAO scales.
- Does not discharge any sorry outside this module's own certificate.