IndisputableMonolith.Mathematics.RS_MTH_Structural_005
Structural mathematics module packaging a domain-cost functional, its nonnegativity and pointwise equality law, and a strictly positive canonical threshold. It exposes an inhabited certificate object that packages those facts for downstream RS structural claims. The argument is definitional plus short positivity/nonnegativity lemmas drawn from the Cost import.
claimDefine a domain cost $C$ on the relevant RS domain, prove $C \ge 0$ and a pointwise equality identity for $C$, and introduce a canonical threshold $\theta > 0$. Package these into an inhabited structural certificate for claim RS-MTH-Structural-005.
background
Recognition Science measures mismatch with the J-cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$), developed in the Cost module. Constants supplies the RS-native tick $\tau_0=1$. This module sits in the Mathematics structural layer and specializes that cost language to a domain-level cost functional together with a fixed positive threshold used as a comparison scale.
Sibling declarations introduce domainCost and record that it is nonnegative and satisfies a stated equality identity at evaluation points, plus canonicalThreshold with a positivity lemma. The certificate type bundles those facts so later structural theorems can assume a single inhabited package rather than re-proving the elementary inequalities.
proof idea
Definition-and-certificate module, not a deep derivation. Domain cost and the canonical threshold are defined from Cost/Constants primitives; nonnegativity and positivity are short lemmas; equality-at-a-point is recorded as an identity lemma. The certificate structure is assembled from those lemmas and shown inhabited by a concrete witness cert.
why it matters in Recognition Science
Gives a reusable inhabited certificate for structural claim RS-MTH-Structural-005 in the mathematics layer: domain cost is a legitimate nonnegative cost with a positive canonical threshold. No downstream consumers are wired in the present graph (used_by empty), so the module is a leaf packaging step for later structural or forcing-chain citations that need a named, checkable cost-threshold bundle rather than ad-hoc inequalities.
scope and limits
- Does not derive J-uniqueness (T5) or the Recognition Composition Law.
- Does not force $\varphi$, the eight-tick period, or $D=3$.
- Does not compute physical constants ($c$, $\hbar$, $G$, $\alpha$).
- Does not supply mass-ladder or Berry-threshold results.
- Does not name or close any downstream consumer theorem in the current graph.