IndisputableMonolith.Materials.RS_Matl_Module_011
Materials module 011 packages a domain cost functional, a positive canonical threshold, and an inhabited certificate bundle for Recognition Science materials work. A materials theorist cites it when binding local cost comparisons to the global J-cost and constants stack. The file is mostly definitions plus short nonnegativity and positivity lemmas, closed by a certificate record.
claimDefine a materials domain cost $C_{\mathrm{dom}}$ built from the Recognition cost $J$, a canonical threshold $\theta>0$, and a certificate record asserting the basic positivity and evaluation identities needed by downstream materials arguments in RS-native units.
background
Recognition Science measures mismatch with the cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$), forced uniquely in the T5 step of the unified forcing chain. The Cost import supplies that functional; Constants supplies the RS-native scale (including the tick $\tau_0$).
This materials module sits in the Materials domain and introduces a domain-level cost $C_{\mathrm{dom}}$, its pointwise evaluation identity, and nonnegativity. It also fixes a canonical threshold $\theta$ with a positivity lemma, then wraps those facts in a certificate type RSMatl011Cert with an inhabited instance so later materials developments can assume a single bundle rather than re-prove the local arithmetic.
proof idea
Definition-first module. Domain cost is introduced as a def tied to $J$; domainCost_at_eq is an evaluation identity; domainCost_nonneg is a short nonnegativity argument from Cost. Canonical threshold is a positive constant def with a one-line positivity proof. The certificate record packages those facts; cert_inhabited supplies a canonical inhabitant. No deep tactic scripts; structure is defs plus thin lemmas plus a cert bundle.
why it matters in Recognition Science
Gives Materials a named cost/threshold/certificate surface so later RS materials claims share one local interface instead of ad hoc $J$-rewrites. It does not yet feed recorded downstream theorems in the graph (used_by is empty), so it is infrastructure for materials certificates rather than a forcing-chain step. Landmarks it touches only indirectly: T5 $J$-uniqueness and the Constants stack ($c=1$, $\hbar=\varphi^{-5}$, etc.). Parent consumption is expected in later materials modules that import this cert.
scope and limits
- Does not derive bulk material properties or band structures from first principles.
- Does not prove uniqueness of the domain cost beyond the imported $J$ facts.
- Does not connect the canonical threshold to measured critical temperatures or energies.
- Does not discharge any forcing-chain step (T0–T8) or fix $\alpha$, masses, or $D=3$.
- Does not yet appear as a dependency of recorded downstream theorems.