IndisputableMonolith.Cosmology.RS_COS_Structural_008
Structural certificate module for RS cosmology claim 008: a nonnegative domain cost built from the RS J-cost, a positive canonical threshold, and an inhabited certificate packaging those facts. Cosmology auditors cite it when wiring threshold comparisons against domain costs. The module is mostly definitions plus elementary positivity and evaluation lemmas.
claimDefine a domain cost $C$ from the RS cost functional $J$, prove $C \ge 0$ and an evaluation identity at equality cases, fix a canonical threshold $\theta > 0$, and package these into an inhabited structural certificate $\mathrm{RS\_COS\_Structural\_008}$.
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. The Cost import supplies that $J$; Constants supplies the RS-native tick $\tau_0 = 1$.
In the cosmology layer, structural claim 008 needs a scalar that scores a domain configuration and a fixed positive cutoff against which that score is compared. This module introduces domainCost as that scalar (tied to $J$), records nonnegativity and an on-equality evaluation lemma, and names canonicalThreshold with a positivity proof.
The certificate type RSCOSStructural008Cert (with witness cert and cert_inhabited) is the Lean packaging of those structural facts so downstream cosmology theorems can assume a single inhabited record rather than re-proving the elementary cost properties.
proof idea
Definition-first module, not a deep derivation. domainCost is introduced from the imported Cost/J apparatus; domainCost_nonneg and domainCost_at_eq are short positivity/evaluation lemmas. canonicalThreshold is a named positive constant with canonicalThreshold_pos. The certificate record and cert_inhabited assemble those pieces into one inhabited structural object. No multi-step forcing argument lives here.
why it matters in Recognition Science
Gives cosmology claim RS_COS_Structural_008 a single Lean certificate surface: nonnegative domain cost, positive canonical threshold, and an inhabited pack. Downstream cosmology developments that compare domain scores to a fixed cutoff can import this module instead of rebuilding $J$-cost nonnegativity. No used_by edges are recorded yet, so the module is presently a leaf structural interface rather than a proved bridge into a named parent theorem. It sits in the cosmology domain of the RS mirror and inherits the global $J$-cost uniqueness story (T5 / RCL) only through the Cost import.
scope and limits
- Does not derive observational cosmology (Hubble law, CMB peaks, dark energy).
- Does not prove uniqueness of the canonical threshold from first principles.
- Does not connect domainCost to the mass ladder or alpha band.
- Does not discharge any sorry outside this structural packaging.
- Does not assert dynamical evolution equations for the domain cost.