IndisputableMonolith.Materials.RS_MAT_Structural_009
Materials module that packages a structural certificate for RS domain energetics: a nonnegative domain cost built from the J-cost, a positive canonical threshold, and an inhabited certificate record tying them together. Materials theorists cite it when bounding domain stability or phase separation under RS units. The module is mostly definitions plus elementary positivity and evaluation lemmas.
claimDefine a domain cost $C_{\mathrm{dom}}$ from the Recognition $J$-cost, prove $C_{\mathrm{dom}}\ge 0$ and an evaluation identity at equality cases, fix a positive canonical threshold $\theta_*>0$, and package these into an inhabited structural certificate record for materials claim RS-MAT-Structural-009.
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 supplies that $J$-cost; Constants supplies the RS-native tick $\tau_0=1$. In the materials layer one lifts $J$ to a domain cost that scores how far a spatial or phase domain sits from a self-similar fixed configuration on the $\phi$-ladder.
This module sits in the Materials domain of the monolith. It does not re-derive $J$ or $\phi$; it only specializes the cost to a domain functional, records a canonical positive threshold against which that cost is compared, and wraps both into a named certificate type used by structural claims labeled RS-MAT-Structural-009.
proof idea
Definition-heavy module. domainCost is introduced from the imported $J$-cost; domainCost_at_eq and domainCost_nonneg are short algebraic or order lemmas (nonnegativity of $J$ and evaluation on balanced arguments). canonicalThreshold is a positive constant definition; canonicalThreshold_pos is the corresponding positivity fact. RSMATStructural009Cert (with cert and cert_inhabited) is a structure packing the cost, threshold, and their elementary properties so downstream material claims can assume a single inhabited certificate rather than re-proving the same inequalities.
why it matters in Recognition Science
Gives the Materials stack a reusable structural certificate for domain energetics under RS units, so later claims can cite one inhabited record instead of rebuilding nonnegativity of domain cost and positivity of the threshold. No downstream edges are recorded yet in the graph; the module is a leaf certificate package meant to feed higher materials stability or phase-boundary results. It touches the cost side of the forcing chain (T5 $J$-uniqueness) only by import, not by re-proving uniqueness.
scope and limits
- Does not derive or uniqueness-prove the $J$-cost; imports it from Cost.
- Does not fix numerical materials constants beyond a canonical positive threshold.
- Does not prove physical phase diagrams or experimental match.
- Does not connect to mass-ladder or alpha-band claims.
- Does not assert downstream consumers; used_by is empty in the graph.