IndisputableMonolith.Astrophysics.RS_Astro_Module_007
Astrophysics module packaging a domain cost functional, its non-negativity, and a positive canonical threshold, together with an inhabited certificate type RSAstro007Cert. Observers of RS galactic or halo scaling arguments would cite the cost and threshold lemmas. The module is mostly definitional plus short positivity and equality facts built on the global J-cost and RS constants.
claimThe module introduces a domain cost $C_{\mathrm{dom}}$ on the RS cost structure, proves $C_{\mathrm{dom}}\ge 0$ and an evaluation identity, defines a canonical threshold $\theta_*>0$, and packages these into an inhabited certificate $\mathrm{RSAstro007Cert}$.
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 related units.
In the astrophysics layer one often needs a domain-restricted cost (integrated or projected mismatch over a spatial or kinematic domain) and a fixed positive threshold against which that cost is compared. This module names those objects: domainCost, the pointwise identity domainCost_at_eq, non-negativity domainCost_nonneg, and the positive scalar canonicalThreshold.
The certificate bundle RSAstro007Cert (with cert and cert_inhabited) is the standard RS pattern that freezes the above facts into a single inhabitable record for downstream astrophysics lemmas.
proof idea
Definition module with short supporting lemmas rather than a long derivation. domainCost is introduced from the Cost import; domainCost_at_eq is an evaluation or unfolding identity; domainCost_nonneg follows from non-negativity of J (or of the assembled domain functional). canonicalThreshold is a named positive constant; canonicalThreshold_pos is the positivity proof. RSAstro007Cert packages the pieces; cert_inhabited supplies a witness. No deep tactic script is required beyond applying Cost and Constants facts.
why it matters in Recognition Science
Gives the astrophysics tree a reusable domain-cost and threshold interface so later halo, rotation-curve, or structure-formation claims can cite a single certificate instead of re-proving non-negativity and positivity. Used_by is presently empty, so this module is a leaf provider rather than a consumer. It sits downstream of the global Cost and Constants layers and upstream of any RS astrophysics argument that compares a domain mismatch to a fixed positive scale. It does not itself invoke the forcing chain T5–T8, the mass ladder, or the alpha band; those enter only if a parent theorem imports this certificate.
scope and limits
- Does not derive domainCost from first principles beyond the imported J-cost.
- Does not prove uniqueness of the canonical threshold.
- Does not connect the threshold to observational galaxy data.
- Does not establish eight-tick, D=3, or mass-ladder claims.
- Does not supply numerical bounds on alpha or G.