IndisputableMonolith.Astrophysics.RS_Astro_Module_012
Astrophysics module 012 packages a domain cost functional and a positive canonical threshold, together with a small certificate that the threshold is inhabited. Observers of RS galactic or stellar scaling would cite the nonnegativity and evaluation lemmas. The module is mostly definitional: cost is pulled from the shared Cost layer and the certificate is a trivial inhabitant.
claimDefine a domain cost $C_{\mathrm{dom}}$ on the RS cost layer, prove $C_{\mathrm{dom}}\ge 0$ and an evaluation identity at equality points, fix a canonical threshold $\theta>0$, and exhibit an inhabited certificate that these objects are well-formed for astrophysics module 012.
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 shared Cost import exposes that functional and related defect distances; Constants supplies the RS tick $\tau_0=1$.
This module sits in the Astrophysics domain and specializes those primitives to a domain-level cost $C_{\mathrm{dom}}$ and a single positive canonical threshold $\theta$. Sibling lemmas record nonnegativity of the cost, an on-equality evaluation identity, and positivity of $\theta$. A thin certificate type bundles the package so downstream astro claims can assume a single well-formed witness rather than re-proving the local arithmetic.
proof idea
Definition-heavy module. The domain cost is introduced as a thin wrapper on the imported Cost primitives; nonnegativity and the evaluation identity are short algebraic or rewriting arguments. The canonical threshold is a positive real constant; positivity is immediate from the defining expression. The certificate type is inhabited by assembling those facts, so the inhabitant proof is essentially constructive packaging rather than deep analysis.
why it matters in Recognition Science
Gives Astrophysics a named, certifiable cost-and-threshold pair so later RS astro results can cite one local witness instead of re-deriving Cost facts. No downstream edges are recorded yet, so the module is presently a leaf package. It touches the J-cost layer (T5) only by import; it does not itself advance the forcing chain T0–T8, the eight-tick octave, or the mass ladder. Its role is infrastructural: keep domain cost, threshold positivity, and certificate inhabitance in one place for galactic or stellar scaling arguments that need a fixed positive cutoff.
scope and limits
- Does not derive galactic rotation curves or mass-to-light ratios.
- Does not fix numerical values of $G$, $\alpha$, or rung gaps.
- Does not prove uniqueness of the canonical threshold beyond positivity.
- Does not advance T0–T8 forcing or the eight-tick octave.
- Does not supply downstream consumers; used_by is presently empty.