IndisputableMonolith.Materials.RS_Matl_Module_012
Materials module 012 packages a domain-level cost functional, its nonnegativity, and a positive canonical threshold into a single inhabited certificate. Condensed-matter and materials workers in the RS stack cite it when they need a reusable cost-threshold pair rather than ad-hoc constants. The module is mostly definitional: equalities and inequalities are discharged by direct appeal to the imported J-cost and constant infrastructure.
claimA materials domain cost $C_{\mathrm{dom}}$ is introduced together with the identity $C_{\mathrm{dom}}(x)=C_{\mathrm{dom}}(x)$ at equal arguments, the inequality $C_{\mathrm{dom}}\ge 0$, a canonical threshold $\theta>0$, and an inhabited certificate bundling these facts for RS materials module 012.
background
Recognition Science measures mismatch by the unique J-cost $J(x)=(x+x^{-1})/2-1$ forced at T5 of the unified forcing chain. The Cost import supplies that functional and its elementary calculus; Constants supplies the RS-native tick $\tau_0=1$.
In the materials layer one lifts the scalar J-cost to a domain cost $C_{\mathrm{dom}}$ that scores an entire material configuration or grain. A canonical threshold $\theta$ then marks the scale at which a domain is regarded as recognized or stable. Module 012 freezes one concrete choice of that pair and packages the supporting lemmas (reflexivity at equal arguments, nonnegativity, positivity of $\theta$) so downstream materials arguments can import a single certificate rather than re-prove the arithmetic.
proof idea
Definition module with thin lemma wrappers. domainCost and canonicalThreshold are introduced by definition; domainCost_at_eq is reflexivity; domainCost_nonneg and canonicalThreshold_pos are short nonnegativity/positivity appeals to the imported Cost and Constants facts. RSMatl012Cert is a structure bundling those fields; cert and cert_inhabited supply a canonical inhabitant. No deep tactic scripts or multi-step reductions appear.
why it matters in Recognition Science
Gives the materials domain a named, reusable cost-threshold certificate so later RS materials results need not re-derive nonnegativity or pick ad-hoc cutoffs. Sits downstream of the Cost module (J-uniqueness, T5) and Constants ($\tau_0$). No further used-by edges are recorded yet; the certificate is the hand-off point for subsequent grain, defect, or phase-boundary arguments in the materials layer. It does not itself touch the forcing chain T0–T8, the eight-tick octave, or the mass ladder.
scope and limits
- Does not derive the explicit formula for domainCost from J; it assumes the Cost import.
- Does not prove uniqueness of the canonical threshold.
- Does not connect domain cost to the phi-ladder mass formula or alpha band.
- Does not establish any materials phase diagram or experimental bound.
- Does not feed recorded downstream theorems yet (used_by is empty).