IndisputableMonolith.Cosmology.RS_COS_Structural_004
Structural certificate module for RS cosmology claim 004: a nonnegative domain cost built from the RS J-cost, together with a positive canonical threshold and an inhabited certificate packing those facts. Cosmology auditors cite it when wiring cost-threshold comparisons into larger RS structural chains. The module is mostly definitions plus short nonnegativity and positivity 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 $\mathrm{RS\text{-}COS\text{-}Structural\text{-}004}$.
background
Recognition Science derives physics from a single cost functional $J$ on positive reals, with $J(x)=(x+x^{-1})/2-1$ forced by the Recognition Composition Law and the T5 uniqueness step. The Cost import supplies that $J$-machinery; Constants supplies RS-native units (including the tick $\tau_0$).
This module sits in the Cosmology domain and introduces a domain-level cost built from $J$, plus a canonical positive threshold against which that cost is compared. Sibling names indicate evaluation identities, nonnegativity, threshold positivity, and a certificate record that bundles the structural claim for downstream cosmology wiring.
No external paper proposition text is attached beyond the module name RS_COS_Structural_004; the local setting is certificate-style packaging of cost and threshold facts rather than a full dynamical cosmology derivation.
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 on that definition. canonicalThreshold is a fixed positive RS-scale quantity, with canonicalThreshold_pos recording positivity. RSCOSStructural004Cert (and cert / cert_inhabited) assemble those pieces into an inhabited certificate structure. No deep tactic proof is indicated; the argument is definition plus elementary nonnegativity and positivity.
why it matters in Recognition Science
Gives Cosmology a reusable structural certificate (claim 004) that a $J$-derived domain cost is nonnegative and that a canonical threshold is positive, so later RS cosmology results can assume a clean cost-threshold interface without re-proving basics. used_by is empty in the current graph, so this module is a leaf certificate rather than an intermediate lemma already consumed upstream. It touches the RS cost backbone (T5 $J$-uniqueness, RCL) only through the Cost import, not through forcing-chain steps T6–T8 directly. Open wiring: any parent cosmology theorem that needs a named Structural-004 cert can inhabit or project this record.
scope and limits
- Does not derive dynamical FLRW or expansion equations.
- Does not fix numerical cosmological parameters beyond the canonical threshold definition.
- Does not prove uniqueness of the domain cost among all possible costs.
- Does not connect to eight-tick, $D=3$, or mass-ladder results in this module.
- Does not list downstream consumers; used_by is currently empty.