IndisputableMonolith.Materials.RS_MAT_Structural_008
Materials certificate module that packages a non-negative domain cost against a positive canonical threshold for structural claim 008. Materials theorists cite it when wiring RS cost geometry into solid-state or lattice stability checks. The module is mostly definitions plus elementary positivity and evaluation lemmas, closed by an inhabited certificate record.
claimDefine a domain cost $C$ (built from the RS $J$-cost), prove $C \ge 0$ and an evaluation identity at equality cases, fix a canonical threshold $\theta > 0$, and package the pair into an inhabited structural certificate $\mathrm{RSMATStructural008Cert}$.
background
Recognition Science measures mismatch with the unique cost $J(x) = (x + x^{-1})/2 - 1$ (equivalently $\cosh(\log x) - 1$), forced by the Recognition Composition Law and T5 uniqueness. The Cost import supplies that functional; Constants supplies the RS-native tick and related units.
This materials module lifts $J$ to a domain-level cost suitable for structural comparisons (grain boundaries, defect clusters, or lattice mismatch scores). A single positive canonical threshold then separates admissible from inadmissible configurations. The local setting is certificate-style: expose the cost, its non-negativity, the threshold, and a bundled cert record rather than a full continuum materials PDE.
proof idea
Definition-heavy module. domainCost is introduced from the imported $J$-cost; domainCost_at_eq and domainCost_nonneg are short algebraic or order facts. canonicalThreshold is a positive constant (canonicalThreshold_pos). RSMATStructural008Cert / cert / cert_inhabited assemble those pieces into an inhabited certificate structure. No deep tactic development; the argument is packaging plus elementary positivity.
why it matters in Recognition Science
Places structural materials claim 008 on the same cost geometry that drives the forcing chain (T5 $J$-uniqueness, RCL). Downstream pages can import the inhabited cert instead of re-proving non-negativity or threshold positivity. With no recorded used_by edges yet, the module is a leaf certificate in the Materials domain, ready for lattice-stability or defect-energy arguments that need a named RS cost bound.
scope and limits
- Does not derive continuum elasticity or phonon spectra.
- Does not fix numerical materials constants beyond the canonical threshold schema.
- Does not prove uniqueness of the domain cost among all possible materials functionals.
- Does not connect to mass-ladder or alpha-band results.
- Does not discharge experimental materials datasets.