IndisputableMonolith.Materials.Structural_Materials_mod73
Defines the RS domain cost, canonical threshold, and inhabited certificate for structural materials (mod73 package). Materials workers in the monolith cite the nonnegativity and positivity facts when wiring cost bounds. Mostly definitional: short algebraic identities and record assembly over the imported J-cost.
claimThe module introduces a structural-materials domain cost $C_{\mathrm{dom}}$, a canonical threshold $\theta>0$, and an inhabited certificate $\mathrm{StructMaterialsM73Cert}$ packaging the standard nonnegativity and positivity inequalities for those quantities in RS-native units.
background
Recognition Science scores 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. The Cost import exposes that functional; Constants supplies the RS tick $\tau_0=1$.
In the materials domain the cost is specialized to a domainCost on structural configurations, measured against the phi-ladder yardstick. A canonicalThreshold cuts off admissible scores. The module lives under IndisputableMonolith.Materials and packages the resulting facts into the certificate record StructMaterialsM73Cert.
proof idea
Definition-and-lemmas module, not a single deep theorem. domainCost is defined from the imported cost; domainCost_at_eq and domainCost_nonneg record evaluation and nonnegativity. canonicalThreshold and canonicalThreshold_pos fix a strictly positive cutoff. cert and cert_inhabited assemble an inhabited StructMaterialsM73Cert record from those facts. Proofs are short algebraic or record constructors.
why it matters in Recognition Science
Supplies the materials-side cost/threshold certificate that later structural-materials arguments can import as a single interface. No downstream used_by edges are recorded yet; the package is the reusable hook for composition with mass-ladder, eight-tick, or D=3 constraints. It sits on the J-cost backbone (T5 / RCL) via the Cost import and keeps materials claims inside RS-native units ($c=1$, $\hbar=\phi^{-5}$).
scope and limits
- Does not derive bulk moduli or alloy phase diagrams from first principles.
- Does not prove uniqueness of the structural threshold.
- Does not link domainCost to laboratory materials data.
- Does not discharge any T0–T8 forcing-chain step.
- Does not assert global mass-ladder or alpha-band bounds.