IndisputableMonolith.Materials.RS_MAT_Structural_002
Certificate module for Recognition Science materials structural claim 002. It defines a domain cost built from the RS J-cost, a positive canonical threshold, and an inhabited certificate record packing nonnegativity of the cost with positivity of the threshold. Materials workers citing the structural layer would reference this as a typed witness. Content is definitional plus elementary positivity facts.
claimThe module defines a domain cost $C_{\mathrm{dom}}$ from the Recognition $J$-cost, a canonical threshold $\theta>0$, and a certificate record asserting $C_{\mathrm{dom}}\ge 0$ together with $\theta>0$.
background
Recognition Science materials modules attach structural claims to the same cost calculus used in the forcing chain. The Cost import supplies the $J$-functional $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$), which measures multiplicative defect away from the identity. Constants supplies the RS-native tick $\tau_0=1$.
In this setting a domain cost is a nonnegative real functional on a materials parameter domain, typically assembled by evaluating $J$ (or a simple transform of it) at a dimensionless ratio. A canonical threshold is a fixed positive cutoff against which that cost is compared when stating a structural inequality.
The module sits in the Materials domain and exposes named objects for the cost, its value at equality cases, nonnegativity, the threshold, threshold positivity, and a certificate record with an inhabited instance.
proof idea
Definition module, not a deep proof development. It introduces domainCost and canonicalThreshold as defs, then records elementary facts: evaluation identities, nonnegativity of the cost (inherited from $J\ge 0$), and positivity of the threshold. The certificate type bundles those propositions; cert_inhabited supplies a canonical inhabitant by assembling the positivity lemmas. No multi-step tactic argument beyond what nonnegativity of $J$ already gives.
why it matters in Recognition Science
Gives the Materials layer a typed structural-002 witness: cost, threshold, and certificate in one place, aligned with the RS cost calculus rather than an ad-hoc inequality. Downstream used-by edges are empty in the current graph, so the module presently stands as a leaf certificate package rather than a lemma feeding a named parent theorem. It anchors the materials side of the framework next to the Cost and Constants imports, so later structural claims can cite a uniform nonnegativity-plus-threshold pattern instead of re-proving $J\ge 0$ locally. No T0–T8 forcing step is discharged here; the link is infrastructural.
scope and limits
- Does not derive material-specific numerical mass or gap predictions.
- Does not prove uniqueness or optimality of the canonical threshold.
- Does not connect the certificate to experimental materials data.
- Does not discharge any T0–T8 forcing-chain obligation.
- Does not define dynamics or time evolution of the domain cost.