IndisputableMonolith.Materials.RS_Matl_Module_004
Materials module defining a nonnegative domain cost built from the RS J-cost, a positive canonical threshold, and an inhabited certificate packaging both. Materials theorists cite it when fixing the cost scale and acceptance cut for RS material models. Structure is definitional: cost and threshold lemmas plus a small cert record.
claimA materials domain cost $C$ derived from the RS cost $J$, with $C \ge 0$ and an evaluation identity at equality cases; a canonical threshold $\tau > 0$; and an inhabited certificate bundling these facts for module 004.
background
Recognition Science measures mismatch with the unique cost $J(x) = (x + x^{-1})/2 - 1$ (equivalently $\cosh(\log x) - 1$), forced by the Recognition Composition Law and the T5 uniqueness step. The Cost import supplies that $J$-infrastructure; Constants supplies the RS-native tick $\tau_0 = 1$.
This materials module lifts $J$ to a domain-level cost suitable for condensed-matter or lattice settings, then fixes a positive canonical threshold against which that cost is compared. Sibling names indicate nonnegativity of the domain cost, an equality-case evaluation identity, positivity of the threshold, and a certificate record that packages the pair for downstream discharge.
proof idea
Definition-and-lemma module, not a single deep theorem. Domain cost is introduced from the imported $J$-cost; nonnegativity and the equality evaluation are short consequences of Cost facts. The canonical threshold is a positive constant (or closed expression) with a positivity lemma. The certificate is a structure inhabited by assembling those lemmas; no multi-step tactic proof is required beyond that packaging.
why it matters in Recognition Science
Places a reusable cost-and-threshold interface inside the Materials domain so later RS material claims can cite one certified cut rather than re-deriving $J$-nonnegativity. No downstream edges are recorded yet on this page, so the module currently acts as a leaf interface: it closes local scaffolding for module 004 and stands ready for parent material theorems that need a nonnegative domain cost and a positive acceptance threshold. It sits downstream of Cost and Constants only, consistent with the forcing chain's use of $J$ as the unique mismatch measure.
scope and limits
- Does not derive microscopic Hamiltonians or band structures.
- Does not fix numerical material constants beyond the cost and threshold interface.
- Does not prove uniqueness of the domain cost outside the imported J-cost facts.
- Does not connect yet to recorded parent theorems (used_by is empty).
- Does not address experimental falsification protocols for real samples.