IndisputableMonolith.Foundation.RS_FDN_Structural_009
Foundation module packaging a domain-level cost functional, its nonnegativity and evaluation identity, and a strictly positive canonical threshold, together with an inhabited structural certificate. Recognition auditors cite it when a later forcing or ledger step needs a named, nonnegative cost on a domain and a fixed positive cutoff. The file is mostly definitions plus short positivity and equality lemmas feeding the certificate record.
claimOn the RS cost layer one introduces a domain cost $C_{\mathrm{dom}}$, proves $C_{\mathrm{dom}}\ge 0$ and an evaluation identity at distinguished points, and fixes a canonical threshold $\theta>0$. An inhabited certificate packages these facts as structural item FDN-009.
background
Recognition Science measures mismatch with a nonnegative cost built from the unique $J$-functional forced by the Recognition Composition Law, $J(x)=(x+x^{-1})/2-1$. The Cost import supplies that layer; Constants fixes the RS time quantum $\tau_0=1$ tick so thresholds sit in native units.
This module lifts cost from pointwise $J$ to a domain-level functional domainCost, records that it is nonnegative, and names a strictly positive canonicalThreshold. Those three ingredients are the structural payload of FDN-009: a cost that cannot go negative, a concrete positive cutoff, and a certificate that both exist and cohere.
The local setting is pure Foundation scaffolding: no dynamics, no mass ladder, only the algebraic and order facts later ledger or forcing steps may quote when they need a domain cost bound.
proof idea
Definition-first module. domainCost and canonicalThreshold are introduced as defs; domainCost_nonneg and canonicalThreshold_pos are short nonnegativity/positivity lemmas; domainCost_at_eq is an evaluation identity. The certificate type RSFDNStructural009Cert bundles those facts, and cert / cert_inhabited exhibit an inhabitant. No deep tactic proof: algebraic identities and order facts from Cost/Constants close the goals.
why it matters in Recognition Science
FDN-009 is a named structural gate in the Foundation layer: later Recognition arguments that need a nonnegative domain cost and a fixed positive threshold can depend on the certificate rather than re-proving positivity. The module sits under the Cost/$J$ stack that ultimately supports T5 $J$-uniqueness and the RCL, and under Constants ($\tau_0=1$ tick). No downstream edges are recorded yet in the mirror graph, so its role is supply-side: close a structural obligation and keep the certificate inhabited for any parent that imports FDN-009.
scope and limits
- Does not derive $J$-uniqueness or the RCL; those live upstream in Cost/forcing.
- Does not fix numerical values of masses, $\alpha$, or $G$; only a domain cost and threshold shape.
- Does not prove dynamical stability or convergence of any ledger process.
- Does not assert uniqueness of the canonical threshold beyond the named def.
- Does not supply downstream consumers; used_by is empty in the graph.