IndisputableMonolith.Foundation.RS_FDN_Structural_002
Foundation module packaging a domain-level cost functional, its nonnegativity, and a positive canonical threshold, then wrapping them in an inhabitated structural certificate. Foundation authors cite it when they need a named, checkable bundle for RS structural claim 002 rather than ad-hoc cost inequalities. The argument is definitional plus short positivity lemmas over the imported J-cost.
claimDefine a domain cost $C$ built from the Recognition cost $J$, prove $C \ge 0$ and $C$ agrees with evaluation at a distinguished point, introduce a canonical threshold $\theta > 0$, and package $(C,\theta)$ into an inhabited structural certificate for foundation claim 002.
background
Recognition Science measures mismatch with the J-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 functional; Constants supplies the RS-native tick $\tau_0=1$.
This module lifts pointwise $J$ to a domain-level cost and fixes a positive canonical threshold against which domain costs are compared. The local setting is structural foundation bookkeeping: named, reusable positivity and threshold facts rather than a new dynamical law.
Sibling objects include the domain cost, its evaluation identity, nonnegativity, the canonical threshold and its positivity, and the certificate type with an inhabited instance.
proof idea
Definition module with short supporting lemmas. Domain cost is introduced as a def over the imported cost; equality-at-evaluation is a one-line unfolding; nonnegativity reduces to nonnegativity of $J$. Canonical threshold is a positive constant def; positivity is immediate from the constant's construction. The certificate is a structure bundling these facts, discharged by an inhabited instance that assembles the lemmas.
why it matters in Recognition Science
Gives the foundation layer a single named certificate for structural claim 002 (domain cost plus threshold) instead of scattered inequalities. Downstream foundation developments that need a nonnegative domain cost bounded relative to a fixed positive threshold can import the certificate rather than re-prove positivity. No used-by edges are recorded yet; the module sits as reusable scaffolding under the Foundation domain, upstream of later forcing-chain and measurement packaging that consume cost nonnegativity and threshold comparisons. It does not itself advance T5--T8; it only packages cost infrastructure those steps already rely on.
scope and limits
- Does not derive uniqueness of J or invoke the Recognition Composition Law.
- Does not force phi, the eight-tick octave, or D=3.
- Does not prove mass-ladder or alpha-band numerical claims.
- Does not assert dynamical evolution, only static cost and threshold facts.
- Does not record downstream consumers; used_by is empty.