IndisputableMonolith.Mathematics.RS_MTH_Structural_001
Structural certificate module for a domain-restricted recognition cost and a positive canonical threshold. It packages nonnegativity of the domain cost, an evaluation identity, and positivity of the threshold into an inhabitable certificate record. Analysts working the RS cost layer cite it when they need a named, checkable bundle rather than ad-hoc lemmas. The argument is definitional packaging plus short positivity and nonnegativity proofs from the Cost import.
claimOn a designated domain, a cost $C$ agrees with the RS $J$-cost at evaluation points, satisfies $C \ge 0$, and admits a canonical threshold $\tau_*>0$. These facts are bundled as an inhabited structural certificate $\mathsf{Cert}$.
background
Recognition Science measures mismatch by the $J$-cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$), developed in the Cost layer and forced unique by the T5 step of the unified forcing chain. Constants supplies the RS-native tick $\tau_0=1$ and related units; this module sits one level up, restricting cost to a working domain and naming a threshold scale.
Sibling objects introduced here are a domain cost $C$, an evaluation identity linking $C$ to the ambient cost at equal arguments, nonnegativity of $C$, a canonical threshold $\tau_*$, and its positivity. The certificate record $\mathsf{RSMTHStructural001Cert}$ packages those properties so downstream mathematics can assume one inhabited bundle instead of a loose list of hypotheses.
proof idea
Definition-and-certificate module, not a deep derivation. Domain cost and canonical threshold are defined from Cost and Constants; nonnegativity and positivity are short lemmas (typically by reduction to known $J\ge 0$ and positivity of the named scale). The certificate type collects those fields; inhabitation is a constructor application assembling the proved components. No multi-step tactic chain beyond that packaging.
why it matters in Recognition Science
Gives the mathematics domain a first structural certificate (RS_MTH_Structural_001) tying domain-restricted cost to a positive threshold. That pattern matches other RS cert bundles: a single inhabited record that later forcing, mass-ladder, or continuum arguments can import without re-proving $J$-nonnegativity locally. No downstream edges are recorded yet, so its role is foundational scaffolding for later math-layer theorems rather than a leaf lemma under a named parent. It touches the Cost/$J$ landmark and the constants layer, not T6--T8 directly.
scope and limits
- Does not derive uniqueness of $J$; that is T5 upstream in Cost/forcing.
- Does not fix numerical values of masses, $\alpha$, or $G$.
- Does not prove continuum or PDE consequences of the threshold.
- Does not assert the certificate is used by any recorded parent theorem yet.
- Does not extend the domain cost beyond the named domain restriction.