IndisputableMonolith.Materials.RS_MAT_Structural_007
Materials module that packages a domain cost functional, its nonnegativity and evaluation identities, and a positive canonical threshold into a single inhabited certificate. Materials theorists citing RS structural claims use it as the local ledger for Structural_007. The argument is definitional: cost is built from the RS J-cost, threshold positivity is elementary, and the certificate is a pure data bundle.
claimDefine a domain cost $C$ on material configurations from the RS cost $J$, prove $C\ge 0$ and the pointwise evaluation identity, fix a canonical threshold $\theta>0$, and package $(C,\theta)$ into an inhabited Structural\_007 certificate.
background
Recognition Science measures mismatch by the unique 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 inequalities. Constants supplies the RS-native tick $\tau_0=1$ used to normalize material time scales.
In the materials layer, a domain cost lifts $J$ from dimensionless ratios to configuration-level defects (grain boundaries, phase interfaces, lattice strain). The canonical threshold is the positive cutoff against which that cost is compared when deciding whether a structural motif is admissible.
This module sits in the Materials domain and only imports Mathlib plus those two RS cores; it does not yet depend on mass-ladder or eight-tick machinery.
proof idea
Definition module with thin lemmas. domainCost is introduced from the imported $J$-cost; domainCost_at_eq is an evaluation identity; domainCost_nonneg follows from nonnegativity of $J$. canonicalThreshold is a positive constant; canonicalThreshold_pos is the corresponding positivity proof. RSMATStructural007Cert bundles these facts; cert and cert_inhabited discharge inhabitation. No deep tactic scripts: algebraic reduction plus structure packing.
why it matters in Recognition Science
Structural_007 is the local certificate that a materials claim has a well-defined nonnegative domain cost and a positive comparison threshold inside RS units. Downstream used_by edges are empty in the present graph, so the module is a leaf ledger rather than an intermediate lemma. It anchors later materials arguments that need a named, inhabited certificate before comparing defect costs to the Berry or rung thresholds of the broader framework. It does not itself force $D=3$, the eight-tick octave, or the fine-structure band; those remain upstream forcing-chain results.
scope and limits
- Does not derive the explicit numerical value of the canonical threshold from first principles.
- Does not prove uniqueness of the domain cost among all possible material defect measures.
- Does not connect domain cost to the phi-ladder mass formula or atomic rungs.
- Does not establish experimental falsifiers for any concrete alloy or crystal.
- Does not feed any recorded downstream theorem in the current dependency graph.