IndisputableMonolith.Astrophysics.RS_Astro_Module_002
Astrophysics module defining a domain cost functional and a positive canonical threshold for RS-scale structure cuts. Supplies nonnegativity and evaluation lemmas, then packages them in an inhabitation certificate (RSAstro002Cert). Cited by anyone wiring cost-based selection rules into galactic or halo models. Argument is definitional plus short positivity/equality proofs over the imported J-cost.
claimIntroduce a domain cost $C_{\mathrm{dom}}$ built from the Recognition cost $J$, prove $C_{\mathrm{dom}}\ge 0$ and a pointwise evaluation identity, fix a canonical threshold $\theta>0$, and package these facts as an inhabited certificate for RS astrophysics module 002.
background
Recognition Science measures mismatch with the 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 exposes that functional; Constants supplies the RS-native tick $\tau_0=1$.
This module sits in the astrophysics layer. It lifts $J$ to a domain-level cost $C_{\mathrm{dom}}$ used to score spatial or mass domains (halos, disks, voids) and fixes a single positive canonical threshold $\theta$ against which those scores are compared. Sibling lemmas record evaluation at a point, nonnegativity of the domain cost, and positivity of $\theta$.
The certificate bundle RSAstro002Cert (with cert / cert_inhabited) is the standard RS pattern: a Prop-carrying structure that downstream astrophysics developments can assume inhabited rather than re-proving the local arithmetic each time.
proof idea
Definition module with short supporting lemmas, not a deep derivation. domainCost is defined from the imported Cost/J machinery; domainCost_at_eq is an evaluation identity; domainCost_nonneg follows from nonnegativity of $J$ on the positive reals. canonicalThreshold is a numeric (or Constants-linked) cutoff; canonicalThreshold_pos is a one-line positivity check. RSAstro002Cert packages the above; cert_inhabited builds a concrete witness. No multi-step tactic chain beyond unfolding and applying Cost facts.
why it matters in Recognition Science
Gives the astrophysics stack a reusable cost-and-threshold interface so selection or truncation rules (which domains count as bound, luminous, or void) stay tied to the same $J$ that forces $\phi$, the eight-tick octave, and $D=3$ upstream. The certificate pattern matches other RS_*_Cert modules: downstream halo, rotation-curve, or mass-ladder developments can depend on cert_inhabited instead of re-importing Cost lemmas. used_by is currently empty, so this is a leaf provider for later astro theorems rather than a step inside T0–T8. It does not itself derive masses or $\alpha$; it only standardizes the cost cut those derivations may call.
scope and limits
- Does not derive galactic rotation curves or dark-matter profiles.
- Does not fix numerical values of stellar or halo masses on the phi-ladder.
- Does not prove uniqueness of the canonical threshold beyond positivity.
- Does not connect domain cost to observational catalogs or survey pipelines.
- Does not discharge forcing-chain steps T0–T8 or the RCL identity.