IndisputableMonolith.Materials.RS_Matl_Module_003
Materials module 003 packages a domain cost functional and a positive canonical threshold for RS materials analysis, together with a small certificate bundle. A materials theorist cites it when fixing the cost scale against which phase or defect thresholds are compared. The module is mostly definitional: nonnegativity and positivity lemmas discharge the certificate inhabitance obligations.
claimDefine a materials domain cost $C_{\mathrm{dom}}$ (built from the RS $J$-cost), prove $C_{\mathrm{dom}}\ge 0$ and an evaluation identity, fix a canonical threshold $\theta>0$, and package these facts as an inhabited certificate $\mathrm{RSMatl003Cert}$.
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 Recognition Composition Law. The Cost import supplies that functional; Constants supplies the RS-native tick $\tau_0=1$.
In the materials layer one needs a domain-level cost $C_{\mathrm{dom}}$ that inherits nonnegativity from $J$, plus a fixed positive threshold $\theta$ against which costs are compared (activation, defect, or phase boundaries on the $\varphi$-ladder). Module 003 is the third materials packaging unit: it names those objects and records the elementary positivity facts required by later certificates.
proof idea
Definition module with short supporting lemmas. domainCost is introduced from Cost; domainCost_nonneg and domainCost_at_eq are direct consequences of $J\ge 0$ and evaluation. canonicalThreshold is a positive constant; canonicalThreshold_pos is immediate. RSMatl003Cert bundles the above; cert and cert_inhabited witness that the bundle is inhabited. No deep forcing or uniqueness argument lives here.
why it matters in Recognition Science
Gives the materials stack a reusable cost-and-threshold certificate so later modules can compare domain costs to a fixed positive scale without re-proving nonnegativity. Downstream used_by edges are empty in the current graph, so this is a leaf packaging unit rather than a step in T0–T8. It sits under the Materials domain and depends only on Constants and Cost, keeping the materials cost scale aligned with the global $J$-cost and the RS tick.
scope and limits
- Does not derive the numerical value of the canonical threshold from T5–T8 forcing.
- Does not prove uniqueness of domainCost among possible materials costs.
- Does not connect thresholds to mass rungs, alpha, or eight-tick dynamics.
- Does not supply experimental materials data or fitting procedures.
- Does not discharge any sorry outside its own certificate inhabitance.