IndisputableMonolith.Mathematics.RS_MTH_Structural_009
Structural mathematics module packaging a domain cost functional, its nonnegativity and fixed-point identity, and a positive canonical threshold into an inhabited certificate. Analysts working the RS cost calculus or threshold arguments cite the cert bundle rather than the individual lemmas. The module is definition-plus-elementary-inequality work over the imported Cost and Constants layers.
claimDefine a domain cost $C$ built from the RS $J$-cost, prove $C \ge 0$ and $C(x)=C(x)$ at the equality locus, introduce a canonical threshold $\tau_*>0$, and package these facts as an inhabited structural certificate for claim RS-MTH-Structural-009.
background
Recognition Science measures mismatch by the unique cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$), forced by the Recognition Composition Law and the T5 uniqueness step. The Cost import supplies that functional and its elementary calculus; Constants supplies the RS-native time quantum and related scale factors.
This module sits in the mathematics track as a numbered structural claim. It introduces a domain-level cost (a specialization or lift of $J$ to the ambient domain under study), records that the cost is nonnegative and agrees with itself on the equality set, and fixes a positive canonical threshold used as a comparison scale. The certificate datatype is the standard RS packaging: a structure whose fields are the named lemmas, together with an inhabitation proof that the structure is realizable.
proof idea
Definition module with short supporting lemmas, not a deep derivation. The domain cost is defined from the imported $J$-cost; nonnegativity and the on-diagonal identity are immediate from Cost facts. The canonical threshold is a positive constant (positivity is a one-line arithmetic check). The certificate structure assembles those fields; inhabitation is by supplying the proved components. No multi-step tactic chain beyond elementary rewriting and positivity.
why it matters in Recognition Science
Gives a reusable, nameable bundle for structural claim 009 in the mathematics layer so downstream developments can depend on one cert rather than a scatter of lemmas. Used_by is empty in the current graph, so the module is a leaf packaging step: it closes a local structural obligation (domain cost well-behaved, threshold positive) that later mass, gap, or forcing arguments can import without re-proving $J$-calculus basics. It sits downstream of Cost and Constants only, consistent with the T5 $J$-uniqueness and RCL background rather than with the geometric forcing steps T7–T8.
scope and limits
- Does not derive uniqueness of $J$ or the Recognition Composition Law; those are upstream.
- Does not force $\varphi$, the eight-tick period, or $D=3$.
- Does not compute physical constants ($\alpha$, masses, $G$) or rung formulae.
- Does not assert any dynamical evolution or measurement protocol beyond the static cost and threshold.
- Does not supply downstream consumers; the current used_by list is empty.