IndisputableMonolith.Materials.RS_Matl_Module_005
Materials module 005 packages a domain cost functional, its nonnegativity and evaluation identities, and a positive canonical threshold into an inhabitable certificate. Condensed-matter or materials theorists working in the RS stack would cite it when they need a certified cost-threshold pair for a materials domain. The module is mostly definitional: cost is tied to the RS J-cost import, with short positivity and equality lemmas and a certificate record.
claimDefine a materials domain cost $C_{\mathrm{dom}}$ (built from the RS cost $J$), prove $C_{\mathrm{dom}}\ge 0$ and the pointwise evaluation identity, introduce a canonical threshold $\theta>0$, and package these into an inhabited certificate $\mathrm{RSMatl005Cert}$.
background
Recognition Science measures mismatch with the J-cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$), imported here via the Cost module. Constants supplies the RS-native tick $\tau_0=1$. Materials modules specialize that cost language to condensed-matter domains: a domain cost scores configurations or defects relative to a recognition baseline, and a canonical threshold marks the scale at which a materials process is treated as on or off.
This file is the fifth materials packaging unit. It does not re-derive $J$ or the forcing chain; it binds a named domain cost and threshold to the shared Cost and Constants infrastructure so downstream materials claims can quote a single certificate rather than ad-hoc inequalities.
proof idea
Definition-heavy module, not a deep proof development. domainCost is introduced from the Cost import; domainCost_at_eq and domainCost_nonneg are short algebraic or order lemmas. canonicalThreshold is a positive constant (canonicalThreshold_pos). RSMatl005Cert (with cert / cert_inhabited) is a structure bundling those facts so the module exports one inhabitable certificate object.
why it matters in Recognition Science
Gives the Materials domain a certified cost-threshold pair aligned with RS Cost and Constants, so later materials results can depend on one certificate rather than scattered lemmas. No downstream used_by edges are recorded yet; the module is an upstream packaging node for materials claims that need nonnegativity of a domain cost and a strictly positive canonical threshold. It sits beside the RS landmarks (J-uniqueness, phi ladder) without extending T0-T8 itself.
scope and limits
- Does not derive J-uniqueness, phi, or any T0-T8 forcing step.
- Does not identify the physical materials system beyond a generic domain cost.
- Does not prove numerical values for alpha, masses, or continuum PDEs.
- Does not assert experimental materials data; only formal cost and threshold facts.
- Does not list downstream consumers; used_by is currently empty.