IndisputableMonolith.Astrophysics.RS_Astro_Module_006
Astrophysics packaging module for a domain-level cost built from the Recognition cost, with equality and nonnegativity lemmas, a strictly positive canonical threshold, and an inhabited certificate bundling those facts. Workers on RS structure or ladder bounds cite the certificate when attaching domain cost floors. Content is definitional plus short positivity arguments from the imported cost layer.
claimIntroduces a domain cost $C_{\mathrm{dom}}$ on the astrophysical side, the identity $C_{\mathrm{dom}}(x)=C_{\mathrm{dom}}(x)$ at equal arguments, nonnegativity $C_{\mathrm{dom}}\ge 0$, a canonical threshold $\theta>0$, and an inhabited certificate that packages these properties for downstream use.
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 module lives in the Astrophysics domain and lifts the scalar cost into a named domain cost together with a fixed positive threshold used as a comparison scale. Sibling declarations record equality at matching arguments, nonnegativity of the domain cost, positivity of the threshold, and a certificate type whose inhabitant witnesses the package.
No new dynamical law is claimed here: the setting is bookkeeping and certification so later astro lemmas can quote a single inhabited object rather than re-prove cost sign facts.
proof idea
Definition module, not a deep proof development. Domain cost is introduced as a thin wrapper on the imported cost; equality at equal arguments is definitional or by congruence. Nonnegativity is inherited from the Cost layer's nonnegativity of $J$. The canonical threshold is a positive constant (positivity by a short arithmetic or constant lemma). The certificate is a structure packing those props, inhabited by assembling the prior lemmas.
why it matters in Recognition Science
Gives Astrophysics a numbered, citable cost-and-threshold certificate (module 006) so domain-level bounds do not re-open the Cost import each time. Downstream used-by edges are empty in the current graph, so the module is a leaf cert rather than a parent of named theorems yet; it is meant to be consumed by later RS astro structure, ladder, or threshold comparisons that need $C_{\mathrm{dom}}\ge 0$ and $\theta>0$ in one place.
It sits downstream of T5 J-uniqueness only indirectly, via the Cost import, and does not itself touch T6--T8, the eight-tick octave, or the $\alpha$ band. Its role is scaffolding hygiene: one inhabited cert instead of scattered positivity facts.
scope and limits
- Does not derive a new physical cost law beyond the imported J-cost.
- Does not prove mass formulae, galaxy scaling, or ladder rungs.
- Does not fix numerical astrophysical observables or fit data.
- Does not establish uniqueness of the canonical threshold choice.
- Does not feed any recorded downstream theorem in the current graph.