IndisputableMonolith.Materials.RS_Matl_Module_001
Materials module 001 packages an RS domain cost for materials, its nonnegativity, a positive canonical threshold, and an inhabited certificate type. Materials-side RS claims cite the cost lemmas and the certificate bundle. Structure is mostly definitions plus short positivity and equality facts drawn from the Cost and Constants layers.
claimModule objects: a materials domain cost $C_{\mathrm{matl}}$ (from the RS cost $J$), equality and nonnegativity facts for $C_{\mathrm{matl}}$, a canonical threshold $\theta>0$, and an inhabited certificate asserting those cost and threshold properties.
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 cost layer; Constants supplies RS-native units, including the time quantum $\tau_0=1$ tick.
This materials module sits in the Materials domain of the monolith. It specializes the global cost to a domain cost $C_{\mathrm{matl}}$ and fixes a canonical threshold used as a materials decision or stability cut. Sibling names indicate equality-at-a-point, nonnegativity of the domain cost, positivity of the threshold, and a certificate record bundling those facts for later checks.
proof idea
Definition-heavy module, not a single deep theorem. Domain cost is introduced as a thin specialization of the imported Cost API; equality and nonnegativity are short transfers of Cost lemmas. Canonical threshold is a positive constant (positivity proved directly or from Constants). The certificate type packages those propositions; inhabitation is a constructor application assembling the proved facts.
why it matters in Recognition Science
Gives the Materials domain a named cost, threshold, and certificate so later materials claims can cite one bundle instead of re-deriving nonnegativity and positivity. No downstream edges are recorded yet; the natural parents are future materials theorems that need $C_{\mathrm{matl}}\ge 0$, $\theta>0$, or an RSMatl001-style cert. Ties to the framework via J-cost (T5) and RS-native constants rather than via T6–T8 geometry. Closes local scaffolding for module 001 certification only.
scope and limits
- Does not derive microscopic materials Hamiltonians or band structures.
- Does not prove numerical match to experimental material constants.
- Does not force phi, eight-tick period, or D=3 (those live upstream).
- Does not define mass-ladder rungs or the alpha band.
- Does not assert uniqueness of the canonical threshold beyond this module.