IndisputableMonolith.Cosmology.RS_COS_Structural_009
Structural cosmology certificate module defining a domain cost functional and a positive canonical threshold, then packaging them as an inhabited RS-COS-009 certificate. Cosmologists working in the Recognition framework cite it when they need a nonnegativity-checked cost gate before applying structural selection rules. The module is mostly definitions plus elementary positivity lemmas over the imported J-cost.
claimThe module introduces a domain cost $C_{\mathrm{dom}}$ built from the Recognition $J$-cost, proves $C_{\mathrm{dom}}\ge 0$ and an evaluation identity at equality points, fixes a canonical threshold $\theta>0$, and assembles an inhabited certificate record $\mathrm{RSCOSStructural009Cert}$ asserting these structural facts.
background
Recognition Science measures mismatch with the unique cost $J(x)=(x+x^{-1})/2-1$ forced by the Recognition Composition Law (T5). Cosmology modules import that cost together with the RS-native constants (including the tick $\tau_0$) so structural claims can be stated in dimensionless ladder units.
This file sits in the cosmology structural series. It isolates a domain-level cost $C_{\mathrm{dom}}$ (a non-negative functional of the underlying $J$-cost) and a fixed positive threshold against which that cost is compared. The threshold is the gate that later selection or exclusion arguments use; the certificate record simply bundles the definitions and the elementary positivity facts so downstream cosmology lemmas can depend on a single inhabited witness rather than on scattered lemmas.
proof idea
Definition-heavy module. domainCost is introduced from the imported Cost layer; domainCost_at_eq and domainCost_nonneg are short algebraic or order lemmas (nonnegativity of $J$ and evaluation at the identity). canonicalThreshold is a positive constant definition with a one-line positivity proof. The certificate type RSCOSStructural009Cert packages these fields; cert and cert_inhabited supply a concrete inhabitant so the structural claim is available as a Prop/Type witness rather than as open goals.
why it matters in Recognition Science
RS-COS structural certificates are the bookkeeping layer that lets later cosmology theorems assume a checked cost gate without re-proving $J\ge 0$ or threshold positivity each time. No downstream edges are recorded yet for this module, so it currently stands as a leaf certificate in the cosmology package: a reusable witness for claim 009 rather than a step inside a named parent theorem. In the broader forcing picture it only touches the cost side (T5 $J$-uniqueness and nonnegativity), not the eight-tick, dimension, or mass-ladder landmarks. Its value is organizational: once inhabited, any structural selection rule that needs "domain cost below canonical threshold" can import the certificate instead of rebuilding the inequalities.
scope and limits
- Does not derive dynamical cosmological equations or Friedmann-type evolution.
- Does not fix numerical values of $c$, $\hbar$, $G$, or $\alpha$ beyond imported Constants.
- Does not prove uniqueness of the canonical threshold, only positivity of the chosen one.
- Does not connect domain cost to observational datasets or parameter fits.
- Does not discharge later RS-COS claims; it only supplies the 009 certificate witness.