IndisputableMonolith.Materials.RS_Matl_Module_007
Materials module 007 packages a domain cost functional, its nonnegativity and evaluation identity, and a strictly positive canonical threshold into a single inhabited certificate. Condensed-matter and materials workers in the RS stack cite it when they need a certified cost-threshold pair rather than ad-hoc cutoffs. The module is mostly definitional: cost is wired to the global J-cost, and the certificate is discharged by elementary positivity lemmas.
claimThe module introduces a domain cost $C_{\mathrm{dom}}$ (built from the RS $J$-cost), proves $C_{\mathrm{dom}}\ge 0$ and an evaluation identity at equality cases, fixes a canonical threshold $\theta>0$, and packages these facts as an inhabited materials certificate $\mathrm{RSMatl007Cert}$.
background
Recognition Science measures mismatch with the unique cost $J(x)=(x+x^{-1})/2-1$ forced by the Recognition Composition Law. The Cost import exposes that functional; Constants supplies the RS-native tick $\tau_0=1$ and related units. Materials modules specialize this cost to condensed-matter domains (lattices, defects, phase boundaries) rather than particle rungs.
Module 007 defines a domain cost $C_{\mathrm{dom}}$ on the relevant configuration space, records that it is nonnegative, and records how it evaluates when the two arguments coincide (the zero-defect case). Alongside the cost it fixes a canonical threshold $\theta>0$ used as a recognition or nucleation cutoff in the materials layer.
These pieces are bundled into a certificate type whose inhabitation is the module's export: downstream materials arguments can assume a single cert object rather than re-proving cost nonnegativity and threshold positivity in each lemma.
proof idea
Definition-heavy module, not a deep proof development. The domain cost is introduced as a thin wrapper around the imported $J$-cost (or a simple composition of it). Nonnegativity and the on-diagonal evaluation identity are one-line applications of the corresponding Cost lemmas. The canonical threshold is a positive constant (or a positive expression in $\varphi$); positivity is immediate from the Constants/Cost arithmetic. The certificate structure is a record of these facts, and inhabitation is by assembling the already-proved fields.
why it matters in Recognition Science
In the Materials domain of the RS monolith, certified cost-threshold pairs replace phenomenological cutoffs. This module supplies one such pair (domain cost plus canonical threshold) as an inhabited certificate, so later materials theorems can cite a single object rather than reopen Cost and Constants. No downstream edges are recorded yet, so the module currently sits as a leaf export of the materials layer; it is the natural dependency for nucleation, defect-recognition, or phase-boundary lemmas that need $C_{\mathrm{dom}}\ge 0$ and $\theta>0$ in one hypothesis. It does not itself touch the T0–T8 forcing chain, the mass ladder, or the $\alpha$ band; those remain upstream in Foundation and Constants.
scope and limits
- Does not derive the domain cost from first principles beyond wrapping the global J-cost.
- Does not prove uniqueness of the canonical threshold among all possible cutoffs.
- Does not connect the threshold to experimental materials data or lattice models.
- Does not feed recorded downstream theorems yet (used_by is empty).
- Does not address particle masses, alpha, or the T0–T8 forcing chain.