IndisputableMonolith.Cosmology.RS_Cosmo_Module_006
Cosmology certificate module that packages a nonnegative domain cost functional together with a strictly positive canonical threshold. It exposes a small inhabited certificate record so downstream cosmology arguments can cite a single named bundle rather than reassemble the inequalities. The content is definitional plus elementary positivity and evaluation lemmas over the RS cost layer.
claimThe module defines a domain cost $C_{\mathrm{dom}}$ (built from the RS $J$-cost), proves $C_{\mathrm{dom}}\ge 0$ and an evaluation identity at equality cases, and introduces a canonical threshold $\theta>0$. It packages these into an inhabited certificate record $\mathrm{RSCosmo006Cert}$.
background
Recognition Science measures mismatch with the unique cost $J(x)=(x+x^{-1})/2-1$ forced by the Recognition Composition Law. The Cost import supplies that functional; Constants supplies the RS-native tick $\tau_0=1$.
In the cosmology layer one often needs a domain-level cost (an aggregate or restriction of $J$ over a spatial or causal domain) and a fixed positive threshold against which that cost is compared. This module isolates those two objects and the elementary facts that the domain cost is nonnegative and that the threshold is strictly positive.
The local setting is certificate-style packaging: small named records that later cosmology theorems can assume or inhabit without reopening the Cost API.
proof idea
Definition module with thin lemmas. Domain cost is introduced as a def over the imported $J$-cost; nonnegativity and the on-equality evaluation identity are short algebraic or order facts. The canonical threshold is a positive constant (or closed-form expression) with a one-line positivity proof. The certificate record simply bundles these fields; inhabitation is by constructing the record from the preceding defs and lemmas.
why it matters in Recognition Science
Gives cosmology proofs a single citeable certificate for domain-cost nonnegativity and a positive comparison threshold, rather than ad-hoc Cost lemmas. No downstream edges are recorded yet in the mirror graph, so the module is a leaf packaging step inside the Cosmology domain. It sits under the broader RS cost and forcing chain (T5 $J$-uniqueness) without claiming new physics constants or the mass ladder.
scope and limits
- Does not derive Friedmann or FLRW dynamics from the domain cost.
- Does not fix numerical values of cosmological parameters or dark-energy density.
- Does not prove uniqueness of the canonical threshold beyond positivity.
- Does not connect domain cost to the eight-tick octave or $D=3$ forcing steps.
- Does not supply observational falsifiers or data fits.