IndisputableMonolith.Foundation.RS_FDN_Structural_009
Structural certificate module for a domain-level cost and its canonical positivity threshold in the Recognition foundation layer. It packages nonnegativity of a domain cost built from the RS J-cost, a positive threshold cut, and an inhabited certificate record. Foundation auditors cite it when wiring cost-gated structural claims. The argument is definitional plus short positivity and equality lemmas over the imported Cost and Constants layers.
claimDefine a domain cost $C_{\mathrm{dom}}$ from the RS cost functional, prove $C_{\mathrm{dom}}\ge 0$ and an evaluation identity, introduce a canonical threshold $\theta>0$, and package these into an inhabited structural certificate $\mathsf{RSFDNStructural009Cert}$.
background
Recognition Science measures mismatch with a unique cost $J$ forced by the Recognition Composition Law; the Cost import supplies that functional and its elementary calculus. Constants supplies the RS-native tick $\tau_0=1$ and related units. This module sits in the Foundation structural series: it lifts pointwise cost data to a domain-level cost and a fixed positive threshold used as a gate.
Sibling objects named in the module are the domain cost, its evaluation identity and nonnegativity, the canonical threshold and its positivity, and a certificate record with an inhabitation proof. No external physics model is assumed beyond the Cost/Constants interface.
proof idea
Definition module with short supporting lemmas, not a deep derivation. Domain cost is defined from the imported cost layer; equality-at-evaluation and nonnegativity are discharged by unfolding and the known nonnegativity of $J$. The canonical threshold is a positive constant (positivity is a one-line arithmetic fact). The certificate record bundles these facts; inhabitation is by constructing the record from the proved fields.
why it matters in Recognition Science
Closes a numbered structural obligation (RS-FDN-Structural-009) in the foundation ledger: cost nonnegativity and a positive threshold available as a reusable certificate. Downstream used-by edges are empty in the current graph, so this module is a leaf certificate rather than a step inside T5–T8 forcing. It keeps later recognition or gating arguments from re-proving elementary cost positivity. It does not itself force $\phi$, the eight-tick octave, or $D=3$; those live in the UnifiedForcingChain.
scope and limits
- Does not derive uniqueness of $J$ or the RCL; those are upstream Cost/forcing facts.
- Does not fix numerical continuum constants ($c$, $\hbar$, $G$, $\alpha$) beyond Cost/Constants imports.
- Does not prove mass-ladder or Berry-threshold claims.
- Does not supply downstream consumers in the current dependency graph.
- Does not address dimensional forcing ($D=3$) or eight-tick periodicity.