IndisputableMonolith.Astrophysics.RS_Astro_Module_006
Astrophysics module that packages a domain cost functional, its nonnegativity and evaluation identities, and a positive canonical threshold into a single certificate type. Observers of RS galactic or stellar scaling arguments would import the certificate rather than rebuild the cost bookkeeping. The module is mostly definitional: lemmas are short algebraic consequences of the imported J-cost and RS constants.
claimDefine a domain cost $C$ on the astrophysical configuration space, prove $C \ge 0$ and the pointwise evaluation identity, introduce a canonical threshold $\theta > 0$, and bundle these into an inhabited certificate $\mathrm{RSAstro006Cert}$.
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 Recognition Composition Law. The Cost import supplies that functional and its elementary inequalities; Constants supplies the RS-native tick $\tau_0 = 1$ and the golden-ratio ladder used for mass and length yardsticks.
In the astrophysics layer one often needs a single nonnegative scalar that scores how far a galactic or stellar configuration sits from the RS fixed-point geometry, together with a fixed positive cutoff against which that score is compared. This module names that scalar the domain cost and that cutoff the canonical threshold.
The certificate record simply freezes the cost, its nonnegativity proof, the evaluation identity, and positivity of the threshold so downstream astrophysics lemmas can assume one inhabited package rather than four separate hypotheses.
proof idea
Definition module with thin lemma layer. domainCost is introduced as a Cost-derived functional on the module's domain; domainCost_nonneg and domainCost_at_eq are one-line appeals to nonnegativity and evaluation lemmas already in Cost. canonicalThreshold is a positive RS-constant expression; canonicalThreshold_pos is immediate positivity. RSAstro006Cert is a structure packing those four pieces; cert and cert_inhabited supply a canonical inhabitant.
why it matters in Recognition Science
Gives the astrophysics development a reusable cost-and-threshold certificate so later galactic scaling, rotation-curve, or threshold-crossing arguments need not re-derive J-cost bookkeeping. No downstream consumers are wired in the current graph (used_by is empty), so the module is presently a leaf package waiting on higher RS-astro theorems. It sits downstream of the Cost and Constants foundations (T5 J-uniqueness and the phi ladder) and does not itself touch the forcing chain T0-T8.
scope and limits
- Does not derive any galactic rotation curve or mass model.
- Does not fix numerical values of astrophysical observables.
- Does not prove uniqueness of the domain cost among all possible scorings.
- Does not connect the canonical threshold to observational survey cuts.
- Does not discharge any open forcing-chain or particle-mass obligation.