IndisputableMonolith.Materials.RS_Matl_Module_010
Materials module 010 packages a domain cost functional, its nonnegativity and evaluation identity, and a strictly positive canonical threshold into an inhabited certificate. Materials theorists in the RS stack cite it when a local cost-versus-threshold comparison is needed. The module is definitional plus short positivity and equality lemmas, not a deep derivation.
claimThe module introduces a materials domain cost $C_{\mathrm{dom}}$, proves $C_{\mathrm{dom}}\ge 0$ and an evaluation identity at a reference point, fixes a canonical threshold $\theta>0$, and packages these facts as an inhabited certificate $\mathrm{RSMatl010Cert}$.
background
Recognition Science measures mismatch with the J-cost $J(x)=(x+x^{-1})/2-1$ from the Cost layer, with constants (including the tick $\tau_0$) drawn from the Constants module. Materials modules specialize that cost language to condensed-matter or lattice settings rather than particle rungs.
Module 010 is a thin certificate layer: it names a domain cost, records that the cost is nonnegative and agrees with a stated evaluation rule, and fixes a positive canonical threshold against which domain costs can be compared. Upstream imports are only Mathlib, Constants, and Cost; no forcing-chain or mass-ladder machinery is pulled in here.
The certificate type and its inhabited instance are the export surface. Downstream materials arguments are expected to open the cert rather than re-prove nonnegativity or threshold positivity locally.
proof idea
Definition module with short supporting lemmas. Domain cost and canonical threshold are introduced as defs; nonnegativity and positivity are elementary inequalities; the evaluation identity is a one-line unfolding. The certificate record is assembled from those fields and discharged by an inhabited instance. No multi-step tactic proof or upstream theorem application beyond Cost/Constants.
why it matters in Recognition Science
Gives the Materials domain a reusable cost-and-threshold certificate so later RS materials claims can cite one object instead of re-establishing $C\ge 0$ and $\theta>0$. Used_by is empty in the current graph, so this module is a leaf export rather than a step inside T0–T8, RCL, or the mass ladder. It sits beside other RS_Matl certificate modules as scaffolding for condensed-matter comparisons in RS-native units, not as a derivation of $\phi$, $D=3$, or $\alpha$.
scope and limits
- Does not derive the domain cost from J-uniqueness (T5) or RCL.
- Does not fix numerical materials constants or match experimental spectra.
- Does not prove any forcing-chain step (T0–T8) or mass-ladder identity.
- Does not assert uniqueness of the canonical threshold beyond positivity.
- Does not connect to eight-tick octave or spatial dimension results.