IndisputableMonolith.Cosmology.RS_COS_Structural_008
Certificate module for a structural cosmology claim (RS-COS-008): a nonnegative domain cost built from the RS J-functional, a positive canonical threshold, and an inhabited certificate packing those facts. Cosmology auditors cite it when a ledger needs a named, machine-checkable structural bound rather than a dynamical derivation. The module is mostly definitions plus short positivity and evaluation lemmas.
claimDefine a domain cost $C$ from the RS cost $J$, prove $C\ge 0$ and an evaluation identity at a fixed point, fix a canonical threshold $\theta>0$, and package these into an inhabited structural certificate for RS-COS-008.
background
Recognition Science measures mismatch with the unique cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$), forced by the Recognition Composition Law and the T5 uniqueness step. Cosmology modules import that cost together with the RS constants (including the native tick $\tau_0$) so structural claims can be stated in the same units as the forcing chain.
This file sits in the cosmology domain and introduces a domain-level cost (evaluation and nonnegativity), a positive canonical threshold, and a certificate type that bundles those properties. It does not derive FLRW dynamics; it only records the structural inequalities a later ledger entry can cite.
proof idea
Definition-heavy module. The domain cost is defined from the imported $J$-cost; equality-at-a-point and nonnegativity are short lemmas from the known properties of $J$. The canonical threshold is a positive constant definition with a positivity proof. The certificate is a structure (or Prop bundle) inhabited by assembling those lemmas; no deep tactic proof is required beyond the Cost import.
why it matters in Recognition Science
Gives the cosmology ledger a named structural certificate (RS-COS-008) rather than an ad-hoc inequality. Downstream pages can depend on the inhabited cert instead of re-proving $J$-nonnegativity or threshold positivity. No used-by edges are recorded yet, so this is a leaf certificate in the current graph: it closes a scaffolding slot for structural cosmology bookkeeping tied to the RS cost and constants, not a step in the T0–T8 forcing chain itself.
scope and limits
- Does not derive cosmological field equations or expansion history.
- Does not fix numerical values of $H_0$, $\Omega_m$, or dark-energy parameters.
- Does not claim dynamical stability, only structural cost and threshold facts.
- Does not discharge open physics hypotheses beyond the packaged certificate.
- Does not connect to eight-tick or $D=3$ forcing steps in this file.