IndisputableMonolith.Materials.RS_Matl_Module_004
Materials module 004 packages a domain cost functional on the RS cost landscape together with a canonical positive threshold and an inhabitation certificate. Condensed-matter and materials workers in the RS stack cite it when they need a nonnegative cost score and a fixed comparison level for a materials domain. The module is mostly definitional: cost is specialized from the global J-cost, nonnegativity and positivity are recorded, and a cert record is inhabited.
claimOn the Recognition cost landscape, fix a materials domain cost $C_{\mathrm{dom}}$ derived from the J-cost, prove $C_{\mathrm{dom}}\ge 0$, fix a canonical threshold $\theta>0$, and inhabit a certificate record packing these facts for downstream materials arguments.
background
Recognition Science measures mismatch with the J-cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$), forced unique by the T5 step of the unified forcing chain and obeying the Recognition Composition Law. The Cost import exposes that functional and its elementary inequalities; Constants supplies the RS-native tick $\tau_0=1$ and related units.
This materials module specializes that global cost to a domain-level score $C_{\mathrm{dom}}$ appropriate for condensed-matter or materials configurations. It also names a canonical threshold $\theta$ used as a fixed comparison level (for example, a creation or acceptance barrier) and records positivity of $\theta$. The certificate bundle simply packages the definitions and the elementary inequalities so later materials lemmas can depend on one inhabited object rather than a scatter of facts.
proof idea
Definition-and-certificate module, not a deep proof development. Domain cost is introduced as a specialization of the imported J-cost; an equality lemma records the specialization at a point; nonnegativity is inherited from the known nonnegativity of $J$. The canonical threshold is defined as a positive real, with a one-line positivity fact. A certificate structure packs these pieces and is inhabited by a trivial constructor, giving a single named witness for downstream use.
why it matters in Recognition Science
In the Materials domain of the RS monolith, later arguments need a uniform, nonnegative domain score and a fixed positive threshold rather than ad-hoc constants. Module 004 supplies that local interface: domain cost, threshold, and an inhabited cert. No downstream edges are recorded yet on this page, so the module currently sits as a leaf packaging layer between Cost/Constants and future materials theorems (gap formulas, rung placements, or acceptance criteria on the phi-ladder). It does not itself force dimensions, the eight-tick octave, or coupling constants; it only localizes cost bookkeeping for materials work.
scope and limits
- Does not derive a microscopic Hamiltonian or lattice model from first principles.
- Does not prove uniqueness of the domain cost beyond the imported J-cost facts.
- Does not fix numerical materials constants (masses, gaps, conductivities).
- Does not connect the threshold to Berry creation $\phi^{-1}$ or $Z_{cf}=\phi^5$ unless a later lemma does.
- Does not feed a recorded parent theorem on this page (used_by is empty).