IndisputableMonolith.Mathematics.RS_MTH_Structural_006
Structural mathematics module that packages a domain cost functional, its nonnegativity and evaluation identity, and a strictly positive canonical threshold into an inhabitable certificate. Analysts working the RS cost layer cite it when they need a named threshold against which domain-level cost is compared. The module is mostly definitional: lemmas discharge elementary positivity and evaluation facts from the Cost import.
claimDefine a domain cost $C$ on the RS cost layer, prove $C \ge 0$ and the pointwise evaluation identity for $C$, and fix a canonical threshold $\tau_*>0$. Package these into an inhabited certificate recording the structural claim.
background
The module sits in the Mathematics track of IndisputableMonolith and imports the RS Constants layer (fundamental time quantum $\tau_0=1$ tick in RS-native units) together with the Cost layer. In Recognition Science the cost is the unique $J$-functional forced by the Recognition Composition Law, $J(x)=(x+x^{-1})/2-1$, nonnegative and minimized at the identity.
Here the objects of interest are a domain-level cost (a lift or specialization of that $J$-cost to a named domain) and a canonical numerical threshold against which that cost is measured. Sibling declarations expose the cost value, its evaluation identity, nonnegativity, the threshold, and positivity of the threshold.
The certificate bundle RSMTHStructural006Cert is the module's public interface: a Prop-carrying record that downstream developments can assume or inhabit rather than re-proving the elementary cost facts.
proof idea
Definition-heavy module, not a single deep theorem. The cost and threshold are introduced as defs; nonnegativity and positivity are short lemmas discharging against the imported Cost/Constants facts; the evaluation identity is an equality lemma at the defining point. The certificate structure assembles those fields, and inhabitation is a constructor application once the component lemmas are in hand.
why it matters in Recognition Science
Gives the Mathematics track a named structural certificate (claim 006) tying domain cost to a positive canonical threshold. No downstream edges are recorded yet, so the module presently serves as a leaf package: a stable interface for later forcing-chain or mass-ladder arguments that need a certified cost-versus-threshold comparison. It does not itself advance T5–T8; it only freezes the cost/threshold vocabulary those steps may consume.
scope and limits
- Does not derive uniqueness of the domain cost from the Recognition Composition Law.
- Does not identify the canonical threshold with a named RS constant such as $\varphi^{-1}$ or $\varphi^{-5}$.
- Does not prove any forcing-chain step (T5–T8) or mass-ladder identity.
- Does not supply numerical bounds beyond positivity and nonnegativity.
- Does not record downstream consumers; the certificate may still be unused.