IndisputableMonolith.Mathematics.RS_MTH_Structural_002
Structural mathematics module for RS claim MTH-002: a domain cost functional, its nonnegativity, and a strictly positive canonical threshold, packaged as an inhabited certificate. Analysts working the cost side of Recognition would cite the threshold and nonnegativity facts. The module is mostly definitions plus short positivity/nonnegativity arguments over the imported J-cost infrastructure.
claimOn the Recognition cost side, a domain cost $C_D$ is defined so that $C_D$ agrees with the pointwise cost on the domain, $C_D \ge 0$, and a canonical threshold $\tau_{\mathrm{can}} > 0$ is fixed. These facts are bundled into an inhabited structural certificate for MTH-002.
background
Recognition Science measures mismatch with the unique cost $J(x) = \frac{x+x^{-1}}{2}-1$ (equivalently $\cosh(\log x)-1$), forced by the Recognition Composition Law and the T5 uniqueness step. The imported Cost layer supplies that $J$-calculus; Constants supplies the RS-native tick $\tau_0 = 1$.
This module sits in the mathematics track and introduces a domain-level cost (aggregate or restricted $J$-cost on a designated domain) together with a canonical numerical threshold used as a structural cutoff. Sibling facts record agreement of the domain cost with the underlying pointwise evaluation, nonnegativity of that cost, and positivity of the threshold.
The certificate pattern (RSMTHStructural002Cert, cert, cert_inhabited) is the monolith's standard way to package a small bundle of structural lemmas so downstream developments can assume the whole package at once.
proof idea
Definition-heavy module, not a deep derivation. Domain cost is introduced as a def; equality with the pointwise cost is a short unfolding or extensionality fact. Nonnegativity reduces to nonnegativity of the underlying $J$-cost (or a sum/integral of nonnegative terms). The canonical threshold is a concrete positive constant (or a def from Constants/Cost), and positivity is a one-line numeric or algebraic check. The certificate structure collects these fields; inhabitation is by supplying the proved components.
why it matters in Recognition Science
Gives the mathematics track a named structural package (MTH-002) for domain-level cost and a positive cutoff, so later mass, gap, or forcing arguments can cite one certificate rather than re-prove nonnegativity and threshold positivity. No downstream edges are recorded yet in the graph, so this is a leaf package waiting on consumers in the structural or physics layers. It touches the cost side of the forcing chain (T5 $J$-uniqueness and the RCL) only by packaging consequences, not by re-deriving $J$.
scope and limits
- Does not derive J-uniqueness or the Recognition Composition Law; those stay upstream in Cost/forcing.
- Does not fix a physical mass, coupling, or dimension; only a domain cost and threshold package.
- Does not claim the threshold equals any named RS constant (phi, hbar, etc.) unless a sibling def says so.
- Does not supply downstream consumers; used_by is empty in the current graph.
- Does not replace full analytic estimates; nonnegativity/positivity are structural, not sharp bounds.