IndisputableMonolith.Astrophysics.RS_Astro_Module_010
Astrophysics certificate module that packages a domain cost functional and a positive canonical threshold for RS-native astrophysical bounds. Researchers citing RS ladder or J-cost constraints on astrophysical scales would use it. The module defines the cost, proves nonnegativity and a pointwise identity, then wraps them in an inhabited certificate record.
claimThe module introduces a domain cost $C_{\mathrm{dom}}$ (built from the RS $J$-cost), proves $C_{\mathrm{dom}}\ge 0$ and a pointwise evaluation identity, and fixes a canonical threshold $\theta>0$. These are bundled as an inhabited certificate $\mathrm{RSAstro010Cert}$.
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 time quantum $\tau_0=1$ tick and related native units.
This module sits in the Astrophysics domain of the monolith. It specializes the global cost language to a domain-level cost $C_{\mathrm{dom}}$ and a single positive threshold used as a comparison scale for astrophysical statements (mass ladders, binding, or recognition bounds), without yet stating a full dynamical theorem.
Sibling declarations name the pieces: domainCost and its nonnegativity and evaluation lemmas, canonicalThreshold with positivity, and the certificate record RSAstro010Cert with an inhabitation proof.
proof idea
Definition-and-certificate module rather than a deep derivation. Domain cost is defined from the imported $J$-cost; nonnegativity and the pointwise identity are short lemmas off Cost. The canonical threshold is a positive constant definition. The certificate record packages those facts; inhabitation is by constructing a witness from the proved fields. No multi-step forcing argument lives here.
why it matters in Recognition Science
Gives Astrophysics a small, checkable RS-native cost/threshold interface so later astro claims can cite a single certificate instead of re-opening Cost and Constants. Downstream use is not yet wired in this graph (used_by empty), so the module is infrastructure for forthcoming ladder or binding results rather than a closed paper proposition. It aligns with the RS pattern of packaging local hypotheses as inhabited cert records before they feed mass-formula or eight-tick scale arguments. Does not itself force $\phi$, $D=3$, or the $\alpha$ band.
scope and limits
- Does not derive domain cost from the full T0–T8 forcing chain.
- Does not prove a specific astrophysical mass, radius, or luminosity law.
- Does not fix numerical values of $c$, $\hbar$, $G$, or $\alpha$ beyond imports.
- Does not assert uniqueness of the canonical threshold among all possible cutoffs.
- Does not yet feed a named parent theorem in the current dependency graph.