IndisputableMonolith.Foundation.RS_FDN_Structural_001
Foundation certificate module packaging a domain-level cost functional, its nonnegativity, and a strictly positive canonical recognition threshold. Structural auditors and forcing-chain consumers cite it as the RS_FDN_Structural_001 bundle. Contents are definitional bindings plus short positivity and equality lemmas over the imported J-cost and RS constants.
claimThe module introduces a domain cost $C_{\mathrm{dom}}$, proves $C_{\mathrm{dom}}\ge 0$ and an evaluation identity, fixes a canonical threshold $\theta_*>0$, and packages these into an inhabited structural certificate $\mathrm{RS\_FDN\_Structural\_001}$.
background
Recognition Science measures mismatch with a nonnegative cost built from the unique J-functional $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$), forced at T5 of the unified chain. The Cost import supplies that J-cost; Constants supplies the RS-native tick $\tau_0=1$ and related units.
This module lifts cost from single ratios to a domain-level functional domainCost, records that it agrees with pointwise evaluation where expected, and isolates a strictly positive canonicalThreshold used as a recognition gate. Those facts are then wrapped as the named structural certificate RSFDNStructural001Cert.
The local setting is pure foundation bookkeeping: no dynamics, no mass ladder, only the structural positivity and identity facts needed by later FDN certificates.
proof idea
Definition module with thin lemma layer. domainCost and canonicalThreshold are defs over Cost/Constants. Nonnegativity and positivity are short appeals to the corresponding properties of J-cost and of the chosen positive constant. domainCost_at_eq is an evaluation identity. The certificate record RSFDNStructural001Cert is assembled from those fields; cert_inhabited supplies a concrete inhabitant.
why it matters in Recognition Science
RS_FDN_Structural_001 is the first structural foundation certificate in the FDN series: it freezes domain-cost nonnegativity and a positive canonical threshold as a single inhabited bundle. Downstream FDN or forcing consumers that need a named, citable structural gate land here rather than re-proving J-nonnegativity ad hoc.
No reverse edges are recorded yet (used_by empty), so the module presently serves as an export surface for the foundation layer. It sits upstream of any argument that treats recognition as a thresholded nonnegative cost, consistent with T5 J-uniqueness and the RCL algebra, without yet invoking T6–T8 (phi, eight-tick, D=3).
scope and limits
- Does not derive J-uniqueness or the Recognition Composition Law; those stay in the forcing chain.
- Does not fix numerical values of c, hbar, G, or alpha; only structural cost and threshold facts.
- Does not prove mass-ladder, eight-tick, or D=3 results.
- Does not assert dynamical stability or existence of physical spectra.
- Does not record downstream consumers; used_by is currently empty.