IndisputableMonolith.Astrophysics.RS_Astro_Module_009
Astrophysics packaging module that introduces a domain cost functional, proves it is nonnegative and agrees with pointwise evaluation, and fixes a strictly positive canonical threshold. It also exposes an inhabited certificate type bundling those facts. Cite it when wiring RS cost geometry into astro-scale bounds. The content is definitional plus short positivity lemmas over the imported Cost layer.
claimThe module defines a domain cost $C$ on the RS cost geometry, records $C\ge 0$ and agreement of global and pointwise evaluation, and fixes a canonical threshold $\theta>0$. It packages these into an inhabited certificate for astrophysics module 009.
background
Recognition Science measures mismatch with the J-cost from the Cost layer (the unique cost forced by the Recognition Composition Law, $J(x)=(x+x^{-1})/2-1$). Constants supplies the RS-native units and the golden ratio $\varphi$ used throughout the ladder. This module sits in the Astrophysics domain and specializes that cost language to a domain-level functional rather than a single-tick ledger entry.
Sibling declarations name the objects: a domain cost, its equality with pointwise evaluation, nonnegativity, a canonical threshold with a positivity proof, and a certificate record RSAstro009Cert that is inhabited. The theoretical setting is therefore cost geometry plus a fixed positive cutoff for astro-scale recognition bounds, not a dynamical galaxy model.
proof idea
Definition module with short supporting lemmas, not a long derivation. Domain cost is introduced as a def over the imported Cost API; equality-at-evaluation and nonnegativity are discharged by unfolding and the known nonnegativity of $J$. The canonical threshold is a positive constant (positivity is a one-line inequality). The certificate is a structure bundling those facts, with an inhabitation witness assembling the pieces.
why it matters in Recognition Science
Gives Astrophysics a reusable cost-and-threshold certificate (module 009) so later astro claims can cite one inhabited bundle instead of re-proving $C\ge 0$ and $\theta>0$. No downstream edges are recorded yet in the mirror graph, so this is a leaf packaging layer: it closes local scaffolding for domain cost rather than finishing a named forcing-chain step (T5–T8 live upstream in Foundation). Relevant RS landmarks are the J-cost uniqueness and the Cost import; mass-ladder and eight-tick material are not touched here.
scope and limits
- Does not derive galaxy rotation curves, dark-matter profiles, or observational fits.
- Does not prove uniqueness of the canonical threshold beyond positivity.
- Does not connect domain cost to the phi-ladder mass formula or alpha band.
- Does not discharge any Foundation forcing-chain step (T0–T8).
- Does not assert numerical astrophysical constants beyond the abstract threshold.