IndisputableMonolith.Materials.RS_Matl_Module_006
Materials module 006 packages a domain cost functional, its nonnegativity, and a positive canonical threshold into a single inhabited certificate. Materials theorists working in the RS ledger cite it when they need a certified cost-threshold pair rather than ad-hoc cutoffs. The module is mostly definitional: it wires the Cost import to a materials-facing certificate type and proves elementary positivity facts.
claimDefine a materials domain cost $C_{\mathrm{dom}}$ (from the RS $J$-cost), prove $C_{\mathrm{dom}}\ge 0$ and an evaluation identity, fix a canonical threshold $\theta>0$, and package these into an inhabited certificate $\mathrm{RSMatl006Cert}$.
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. The Cost import supplies that functional and its elementary calculus; Constants supplies the RS-native tick $\tau_0=1$.
In the materials layer one needs a domain-restricted cost and a fixed positive threshold against which to compare it, rather than a free parameter. Module 006 introduces those objects under the names domain cost and canonical threshold, then wraps them in a certificate record so downstream materials lemmas can assume a single inhabited bundle instead of re-proving positivity each time.
proof idea
Definition module with light supporting lemmas. Domain cost is introduced as a materials-facing wrapper on the imported Cost layer; an evaluation identity and a nonnegativity lemma discharge the basic algebraic obligations. Canonical threshold is a positive constant (positivity proved directly). The certificate type bundles these facts; inhabitation is a one-line constructor application. No deep tactic scripts or external paper propositions are required.
why it matters in Recognition Science
Gives the Materials domain a certified cost-threshold pair so later RS materials results can cite one inhabited certificate rather than re-open Cost and Constants. Feeds any downstream materials argument that needs $C_{\mathrm{dom}}\ge 0$ or a fixed positive cutoff; the present graph lists no direct used-by edges yet, so the module is a leaf package awaiting consumers. Anchors materials work to the same $J$-cost forced at T5 and to RS-native units from Constants, keeping the materials layer on the same ledger as the forcing chain.
scope and limits
- Does not derive a numerical materials constant or mass rung.
- Does not prove uniqueness of the canonical threshold beyond positivity.
- Does not connect domain cost to experimental spectra or band gaps.
- Does not discharge any forcing-chain step T0–T8.
- Does not assert a used-by consumer; the module is presently a leaf.