IndisputableMonolith.Materials.Structural_Materials_mod93
Module packaging the Recognition Science cost interface for structural materials (model 93): a domain cost functional, its nonnegativity and evaluation identities, a positive canonical threshold, and an inhabited certificate bundle. Materials theorists citing RS domain bounds would use it. Content is mostly definitions plus elementary positivity and equality lemmas over the imported J-cost layer.
claimFor the structural-materials domain (model 93), define a domain cost $C_{\mathrm{dom}}$ built from the RS cost $J$, prove $C_{\mathrm{dom}}\ge 0$ and the pointwise evaluation identity, fix a canonical threshold $\theta>0$, and package these facts as an inhabited certificate $\mathrm{StructMaterialsM93Cert}$.
background
Recognition Science measures mismatch with the unique cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$), forced at T5 of the unified chain and obeying the Recognition Composition Law. The Cost import exposes that functional; Constants supplies the RS-native tick $\tau_0=1$ and related units.
This module sits in the Materials domain and specializes that cost layer to structural materials under the internal label mod93. Sibling declarations introduce a domain cost, record that it agrees with the expected evaluation and is nonnegative, and fix a strictly positive canonical threshold used as a pass/fail cut for the domain.
The certificate type bundles those facts so downstream material claims can assume a single inhabited record rather than re-proving elementary cost properties.
proof idea
Definition-heavy module, not a deep proof development. Domain cost is defined from the imported $J$-cost; equality and nonnegativity lemmas are short reductions to Cost facts. The canonical threshold is a positive constant definition with a one-line positivity proof. The certificate structure assembles those pieces; inhabitation is by exhibiting the concrete bundle.
why it matters in Recognition Science
Gives the Materials domain a reusable cost-and-threshold certificate for structural materials (mod93), so later claims can cite one inhabited record instead of rebuilding $J$-nonnegativity and cut values. No downstream edges are recorded yet in the mirror graph, so it presently anchors local materials scaffolding rather than a named parent theorem. It ties the materials side to the same T5 $J$-cost used across the forcing chain, keeping domain cuts in RS-native units.
scope and limits
- Does not derive bulk moduli, yield strengths, or laboratory material constants.
- Does not prove uniqueness of the mod93 threshold beyond the stated positivity lemma.
- Does not connect domain cost to the mass ladder, alpha band, or eight-tick octave.
- Does not discharge broader materials conjectures; it only packages cost and certificate scaffolding.
- Does not claim experimental falsification criteria for specific alloys or lattices.