IndisputableMonolith.Materials.RS_Matl_Module_007
Materials module 007 packages a domain cost functional, its nonnegativity and evaluation identities, a positive canonical threshold, and an inhabited certificate record. Materials and condensed-matter workers in the RS stack cite it when they need a named cost-plus-threshold bundle rather than ad-hoc constants. The module is mostly definitions and short positivity/equality lemmas over the imported Cost layer.
claimThe module introduces a materials domain cost $C_{\mathrm{dom}}$, proves $C_{\mathrm{dom}}\ge 0$ and the evaluation identity at equality cases, fixes a canonical threshold $\theta_*>0$, and packages these into an inhabited certificate record for RS materials claim 007.
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 supplies that functional and its elementary inequalities; Constants supplies the RS-native units (including the tick $\tau_0=1$).
In the materials domain one specializes the same cost to a domain cost $C_{\mathrm{dom}}$ that scores structural or energetic mismatch on the $\varphi$-ladder. A single positive canonical threshold $\theta_$ is fixed so that comparisons $C_{\mathrm{dom}}\gtrless\theta_$ are uniform across later materials lemmas. Module 007 is the thin packaging layer that names these objects and records their basic positivity and evaluation facts.
proof idea
Definition-heavy module. domainCost and canonicalThreshold are introduced as defs; domainCost_nonneg and canonicalThreshold_pos are short nonnegativity/positivity arguments that reduce to the corresponding Cost lemmas and constant positivity. domainCost_at_eq is an evaluation identity. RSMatl007Cert (with cert and cert_inhabited) is a structure bundling the above into an inhabited certificate, with no deep proof content.
why it matters in Recognition Science
Gives the materials stack a single named cost-plus-threshold certificate (claim 007) instead of scattered local constants. Downstream materials arguments that need a uniform domain cost or a fixed positive cutoff can import this module rather than re-deriving positivity from Cost. No parent theorems are recorded in the current used-by graph; the module sits as a leaf packaging layer over Constants and Cost, aligned with the RS cost infrastructure (T5 J-uniqueness, RCL) rather than with the geometric forcing steps T7–T8.
scope and limits
- Does not derive microscopic Hamiltonians or lattice models from first principles.
- Does not fix numerical material parameters beyond the named canonical threshold.
- Does not prove uniqueness of the domain cost among all possible materials scores.
- Does not connect to eight-tick, D=3, or alpha-band forcing results.
- Does not supply experimental fitting or comparison to measured spectra.