IndisputableMonolith.Materials.RS_Matl_Module_005
Materials module 005 packages a domain-level cost functional, its nonnegativity, and a positive canonical threshold into a single inhabited certificate. Condensed-matter and materials workers in the RS stack cite it when a domain needs a cost gate before further structure theorems. The file is mostly definitions plus short positivity and evaluation lemmas, closed by an inhabited cert record.
claimThe module introduces a domain cost $C_{\mathrm{dom}}$, proves $C_{\mathrm{dom}}\ge 0$ and an evaluation identity at equality cases, fixes a canonical threshold $\theta>0$, and packages these facts as an inhabited materials certificate $\mathrm{RSMatl005}$.
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; Constants supplies the RS-native tick $\tau_0=1$ and related units.
In the materials layer, one lifts the scalar cost to a domain cost $C_{\mathrm{dom}}$ that scores a materials domain (phase, grain, or defect sector) rather than a single scale factor. A canonical threshold $\theta>0$ then marks the gate above which the domain is treated as activated or costly in downstream selection rules.
This module sits in the Materials domain of the monolith. It does not re-derive $J$; it only specializes cost and threshold data that later materials certificates and selection lemmas can import as a single record.
proof idea
Definition-first module. domainCost and canonicalThreshold are introduced as defs; domainCost_at_eq is an evaluation identity; domainCost_nonneg and canonicalThreshold_pos are short nonnegativity/positivity facts from the Cost layer and constant arithmetic. RSMatl005Cert bundles those fields; cert builds an instance and cert_inhabited records that the type is inhabited. No deep tactic proof: the argument is assemble-and-discharge against Cost/Constants.
why it matters in Recognition Science
Gives the Materials stack a named, inhabited certificate for module 005 so later materials results can depend on one object rather than a scatter of cost and threshold lemmas. Upstream edges are only Constants and Cost; there are no recorded downstream users yet in the graph, so this is a leaf packaging step rather than a forcing-chain link (T0–T8 stay in Foundation).
In RS terms it is infrastructure: domain cost inherits the J-cost semantics, and the positive canonical threshold is the materials-side gate analogous to other RS thresholds (e.g. Berry creation at $\phi^{-1}$), without claiming a new mass, $\alpha$, or dimension result. Parent theorems are not yet wired in the depends graph; the module exists so those parents can cite a stable cert.
scope and limits
- Does not derive J-uniqueness, RCL, or any T0–T8 forcing step.
- Does not compute numerical material constants, band gaps, or lattice energies.
- Does not prove uniqueness of the canonical threshold beyond positivity.
- Does not yet feed recorded downstream theorems in the dependency graph.
- Does not address experimental materials falsifiers or lab protocols.