IndisputableMonolith.Materials.RS_Matl_Module_009
Materials module 009 packages a domain cost built from the Recognition J-cost, a positive canonical threshold, and an inhabited certificate record for that pair. Materials workers in the RS stack cite it when fixing a cost scale and pass/fail threshold for a materials domain. The file is mostly definitional: nonnegativity and positivity follow from the imported Cost and Constants facts.
claimModule introducing a materials domain cost $C_{\mathrm{dom}}$ derived from the Recognition cost $J$, a canonical threshold $\theta>0$, and an inhabited certificate asserting $C_{\mathrm{dom}}\ge 0$ and $\theta>0$ for materials unit 009.
background
Recognition Science measures mismatch with the J-cost from the Cost module (the unique cost forced by the Recognition Composition Law and T5). Constants supplies the RS-native tick $\tau_0=1$. This materials unit sits in that setting: it specializes the global cost to a domain-level functional and names a positive cutoff against which materials configurations are scored.
Sibling declarations define domainCost and prove it agrees with evaluation at a point, is nonnegative, and that the canonical threshold is strictly positive. A small certificate record bundles those facts so downstream materials lemmas can assume a single inhabited witness rather than re-proving the inequalities.
proof idea
Definition module with thin lemmas. Domain cost is introduced as a Cost-derived functional; equality-at-a-point and nonnegativity are one-line transfers of Cost facts. Canonical threshold is a positive Constants-scale quantity; positivity is immediate. The certificate is a structure packing those propositions, with an inhabited instance assembling the proofs.
why it matters in Recognition Science
Gives the Materials domain a named cost-and-threshold pair and a reusable certificate so later RS materials results need not rebuild nonnegativity or positivity. No downstream used-by edges are recorded yet; the module is a leaf scaffold for materials scoring against the J-cost ladder. It does not itself touch T0-T8, the eight-tick octave, or the mass ladder, but inherits J from Cost and the tick from Constants.
scope and limits
- Does not derive a materials mass or coupling formula.
- Does not prove uniqueness of the domain cost among all materials functionals.
- Does not connect the threshold to alpha, G, or the phi-ladder rungs.
- Does not supply numerical materials data or experimental bounds.
- Does not discharge any forcing-chain step T0-T8.