IndisputableMonolith.Materials.RS_MAT_Structural_010
Materials module that packages a domain-wall cost functional, its nonnegativity, and a positive canonical threshold into a single inhabited certificate for RS structural claim 010. Condensed-matter and materials theorists working in the Recognition ladder would cite it when bounding domain energetics. The file is mostly definitional: cost is specialized from the global J-cost, then wrapped as a cert record.
claimDefine a domain cost $C_{\mathrm{dom}}$ on materials configurations by specializing the Recognition $J$-cost, prove $C_{\mathrm{dom}}\ge 0$ and an evaluation identity, introduce a positive canonical threshold $\theta_{\mathrm{can}}>0$, and package these facts as an inhabited structural certificate $\mathrm{RS\_MAT\_Structural\_010}$.
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 functional; Constants supplies the RS-native tick $\tau_0=1$ used to normalize material time scales.
In the materials layer, macroscopic domain structure is scored by a domain cost obtained by evaluating or integrating $J$ on local order-parameter ratios (or lattice-scale stretch factors). Nonnegativity of that cost is inherited from $J\ge 0$ with equality only at balance. A canonical threshold then marks the minimal positive cost at which a domain wall or structural defect is counted as present rather than a fluctuation.
This module sits in the Materials domain of the monolith and does not itself derive $J$ or the forcing chain T5–T8; it only specializes those objects to structural claim 010.
proof idea
Definition-first module. domainCost is introduced as a specialization of the imported $J$-cost; domainCost_at_eq and domainCost_nonneg are short algebraic or inheritance lemmas. canonicalThreshold is a positive constant (positivity discharged by canonicalThreshold_pos). The record RSMATStructural010Cert bundles these fields; cert and cert_inhabited exhibit a concrete inhabitant so downstream code can assume the structural package without rebuilding the inequalities.
why it matters in Recognition Science
Gives the Materials tree a named, machine-checkable handle for structural claim 010: domain energetics bounded below by a positive canonical threshold in RS units. Downstream consumers (none linked in the current graph) would import the inhabited cert rather than re-proving nonnegativity of domain cost. The construction stays inside the RS cost calculus (T5 $J$-uniqueness) and the native constants, so it is ready to connect to phi-ladder mass or defect counting once parent materials theorems are wired. No open scaffold is exposed at module level; the cert is already inhabited.
scope and limits
- Does not derive uniqueness of $J$ or the Recognition Composition Law.
- Does not compute numerical material constants or experimental critical temperatures.
- Does not prove existence of physical domain walls in a specific lattice model.
- Does not link the threshold to the eight-tick octave or $D=3$ forcing steps.
- Does not supply used-by parents; the export graph is currently empty.