IndisputableMonolith.Foundation.RS_FDN_Structural_005
Foundation module packaging a domain-level cost functional and a positive canonical threshold used by structural certificate RS-FDN-005. It ties the J-cost from the Cost layer to a domain evaluation map and records nonnegativity and positivity lemmas. Certificate inhabitants are supplied so downstream forcing or ledger checks can assume the structural package without re-deriving the threshold arithmetic.
claimA domain cost $C_{\mathrm{dom}}$ is defined by evaluating the Recognition cost on a domain point, with $C_{\mathrm{dom}}\ge 0$ and agreement at equality cases. A canonical threshold $\theta_*>0$ is fixed. The module supplies an inhabited structural certificate packaging these facts for RS-FDN-005.
background
Recognition Science builds physics from a unique cost $J$ forced by the Recognition Composition Law, with $J(x)=(x+x^{-1})/2-1$. The Cost import supplies that functional and its elementary inequalities. Constants supplies the RS-native tick $\tau_0=1$.
This module sits in the Foundation structural series. It introduces a domain-level cost by specializing $J$ (or an equivalent Cost primitive) to domain data, plus a fixed positive canonical threshold against which domain costs are compared in structural checks. Sibling lemmas record evaluation identity, nonnegativity of the domain cost, and positivity of the threshold.
The certificate objects RSFDNStructural005Cert and cert package those facts as an inhabited Prop bundle so later modules can depend on a single structural hypothesis rather than on scattered lemmas.
proof idea
Definition-and-lemma module, not a single deep theorem. Domain cost is introduced by a definition that applies the upstream Cost primitive at a domain point; domainCost_at_eq is an evaluation identity; domainCost_nonneg follows from nonnegativity of $J$. Canonical threshold is a positive constant definition with a one-line positivity proof. The certificate type is a structure (or Prop bundle) whose fields are exactly those lemmas; cert and cert_inhabited discharge inhabitation by assembling the already-proved pieces.
why it matters in Recognition Science
RS-FDN structural certificates pin discrete foundation facts (cost shape, thresholds, tick structure) before the T0–T8 forcing chain and continuum limits are invoked. This module’s domain cost and positive canonical threshold give a reusable comparison scale for domain-level recognition defects, consistent with the J-uniqueness landmark (T5) and the Cost layer.
No downstream edges are recorded in the graph snapshot, so the immediate consumers are other Foundation or certificate aggregators that import RS_FDN_Structural_005 rather than named parent theorems. The inhabited cert is the export surface: later structural or ledger arguments can assume the package instead of reopening Cost inequalities.
scope and limits
- Does not derive J-uniqueness or the RCL; those remain upstream in Cost and the forcing chain.
- Does not fix numerical continuum constants (c, hbar, G, alpha) beyond Cost/Constants imports.
- Does not prove mass-ladder or eight-tick statements; only domain cost and a canonical threshold.
- Does not list downstream consumers; used_by is empty in the current graph.
- Does not claim the threshold equals a named RS landmark (e.g. phi^{-1}) without an explicit definitional link.