IndisputableMonolith.Foundation.RS_FDN_Structural_004
Foundation certificate module packaging a domain-level cost functional and a positive canonical threshold used in structural Recognition Science arguments. A reader checking FDN structural claims would cite the nonnegativity of the domain cost and the positivity of the threshold. The module is mostly definitions plus short positivity and equality lemmas, closed by an inhabited certificate record.
claimDefine a domain cost $C_{\mathrm{dom}}$ built from the RS cost $J$, record $C_{\mathrm{dom}}\ge 0$ and its value on equality cases, and fix a canonical threshold $\theta_*>0$. Package these facts as the structural certificate $\mathrm{RS\text{-}FDN\text{-}Structural\text{-}004}$.
background
Recognition Science measures mismatch with the cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$), forced unique by the Recognition Composition Law and the T5 step of the unified forcing chain. The Cost import supplies that $J$-calculus; Constants supplies the RS-native tick $\tau_0=1$.
This module lifts pointwise $J$-cost to a domain-level cost $C_{\mathrm{dom}}$ and isolates a single positive scale $\theta_*$ (the canonical threshold) against which structural comparisons are made. The local setting is Foundation structural bookkeeping: small, named certificates that freeze elementary inequalities so later forcing and mass-ladder arguments can cite them by certificate id rather than re-proving nonnegativity each time.
proof idea
Definition-first module. domainCost is introduced as a functional of the imported $J$-cost; domainCost_at_eq and domainCost_nonneg are short algebraic or order lemmas (nonnegativity of $J$ and equality on the fixed point $x=1$). canonicalThreshold is a concrete positive real; canonicalThreshold_pos is its positivity proof. The record RSFDNStructural004Cert (with cert / cert_inhabited) bundles those facts into one inhabited certificate object. No deep tactic development: wrappers and elementary real inequalities.
why it matters in Recognition Science
FDN structural certificates sit under the forcing chain (T0–T8) and the cost layer that feeds mass-ladder and coupling arguments. This module freezes two primitives later structural and quantitative claims need: a nonnegative domain cost compatible with $J$, and a fixed positive threshold scale. Downstream pages can import the inhabited certificate instead of re-opening the Cost API. No used_by edges are recorded yet, so the module is presently a leaf certificate in the graph; its role is audit packaging for Foundation structural claim 004 rather than a step inside T5–T8 themselves.
scope and limits
- Does not derive uniqueness of $J$ or the Recognition Composition Law (those live upstream in Cost / forcing).
- Does not fix numerical values of $c$, $\hbar$, $G$, or $\alpha$; only a structural threshold and domain cost.
- Does not prove dimensional forcing ($D=3$), eight-tick periodicity, or mass-ladder formulae.
- Does not assert physical units for the threshold beyond RS-native positivity.
- Does not currently feed recorded downstream theorems (used_by is empty).