IndisputableMonolith.Cosmology.RS_COS_Structural_006
Structural cosmology module for RS claim 006: a nonnegative domain cost built from the RS J-cost, a strictly positive canonical threshold, and an inhabited certificate packaging both. Cosmology auditors cite it when a domain must clear a fixed cost floor before counting as a structural recognition event. The module is mostly definitions plus elementary positivity and nonnegativity lemmas.
claimDefine a domain cost $C_{\mathrm{dom}}$ from the RS cost $J$, prove $C_{\mathrm{dom}}\ge 0$, fix a canonical threshold $\theta_{\mathrm{can}}>0$, and package the pair into an inhabited structural certificate for RS-COS-006.
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 functional; Constants supplies the RS-native tick $\tau_0=1$.
In the cosmology layer, structural claims need a scalar that scores how far a putative domain sits from the identity of $J$, together with a fixed positive cutoff that the score must meet. This module introduces that domain cost and the canonical threshold used by RS-COS structural item 006.
Sibling declarations expose the evaluation rule (domainCost_at_eq), nonnegativity, positivity of the threshold, and a small certificate record RSCOSStructural006Cert with an inhabitation proof.
proof idea
Definition module with thin lemmas, not a deep derivation. Domain cost is defined by reduction to the imported J-cost (or a direct evaluation thereof); domainCost_at_eq records the evaluation identity; nonnegativity follows from the standard $J\ge 0$ fact in Cost. The canonical threshold is a positive RS constant (or closed form in $\phi$/ticks); positivity is a one-line arithmetic check. The certificate is a structure bundling those facts, discharged by cert_inhabited.
why it matters in Recognition Science
Gives cosmology a reusable cost-and-threshold interface for structural claim 006 so later arguments can assume a nonnegative domain score and a fixed positive gate without re-proving elementary Cost facts. No downstream edges are recorded yet (used_by empty), so the module is a leaf certificate in the present graph: it closes a local structural obligation rather than feeding a named parent theorem. It sits in the broader RS program beside the forcing landmarks (T5 J-uniqueness, T6 $\phi$, T7 eight-tick octave) by keeping cosmological domain scoring native to $J$ and the tick units from Constants.
scope and limits
- Does not derive dynamical FLRW or perturbation equations.
- Does not fix numerical values of $c$, $\hbar$, $G$, or $\alpha$.
- Does not prove uniqueness of the threshold beyond the chosen canonical definition.
- Does not connect domain cost to mass-ladder rungs or Berry thresholds.
- Does not supply observational likelihoods or data fits.