IndisputableMonolith.Foundation.RS_FDN_Structural_006
Foundation certificate module packaging a domain-level cost functional, its non-negativity, and a strictly positive canonical threshold. Structural item 006 in the RS FDN series. Supplies an inhabited certificate record so downstream forcing and ledger arguments can cite a single named bundle rather than raw Cost lemmas.
claimDefine a domain cost $C$ (built from the RS $J$-cost), prove $C \ge 0$ and $C$ agrees with the pointwise evaluation identity, and fix a canonical threshold $\theta > 0$. Package these facts as an inhabited structural certificate $\mathrm{RSFDNStructural006Cert}$.
background
Recognition Science measures mismatch by the unique symmetric cost $J(x) = (x + x^{-1})/2 - 1$ (equivalently $\cosh(\log x) - 1$), forced at T5 of the unified chain. The Cost import exposes that functional and its elementary identities; Constants supplies the RS-native tick $\tau_0 = 1$.
This module lifts the pointwise $J$-cost to a domain-level cost functional and isolates a positive numerical threshold used as a structural cutoff. The local setting is pure foundation: no dynamics, no mass ladder, only the algebraic skeleton needed by later ledger and forcing certificates.
Sibling definitions introduce domainCost, the evaluation identity domainCost_at_eq, non-negativity, canonicalThreshold with positivity, and the certificate record itself.
proof idea
Definition-plus-certificate module. The cost and threshold are defined from the imported Cost primitives; non-negativity and positivity are short algebraic consequences of $J \ge 0$ and the fixed-point structure of $\varphi$. The certificate record simply bundles those lemmas; inhabitation is by direct construction of the record fields. No deep tactic proof.
why it matters in Recognition Science
Structural FDN item 006 sits in the foundation layer that feeds the unified forcing chain (T0–T8) and later ledger arguments. By packaging domain cost non-negativity and a positive canonical threshold into one inhabited cert, it gives downstream modules a single named hypothesis interface rather than ad-hoc Cost citations. No direct used_by edges are recorded on this page; the cert is intended for structural closure checks and for any argument that needs a uniform positive cutoff built from $J$.
scope and limits
- Does not derive uniqueness of $J$; that is T5 upstream.
- Does not fix numerical values of $c$, $\hbar$, $G$, or $\alpha$.
- Does not address mass ladder, eight-tick dynamics, or $D=3$.
- Does not claim the threshold is optimal, only that it is positive.
- Does not discharge later forcing or ledger theorems by itself.