IndisputableMonolith.Materials.Hardness3_FromPhiLadder
Materials module that packages a hardness certificate from the Recognition phi-ladder. It defines a domain cost, a positive canonical threshold, and an inhabited Hardness3Cert record tying those quantities together. Materials theorists citing RS-native hardness bounds would use the certificate; the argument is definitional plus nonnegativity and positivity lemmas imported from Cost and Constants.
claimA materials hardness package on the $\varphi$-ladder: a domain cost $C$ with $C\ge 0$, a canonical threshold $\theta>0$, and an inhabited certificate $\mathrm{Hardness3Cert}$ asserting that the cost-threshold pair meets the level-3 hardness interface in RS-native units.
background
Recognition Science places material scales on the golden-ratio ladder forced at T6: dimensionless rungs are powers of $\varphi$, with costs measured by the unique J-functional $J(x)=(x+x^{-1})/2-1$ from the Recognition Composition Law. The Cost import supplies that J-cost infrastructure; Constants supplies the RS tick $\tau_0$ and related native units ($c=1$, $\hbar=\varphi^{-5}$, etc.).
This module sits in the Materials domain. It introduces a domain cost (nonnegative, with an evaluation identity), a strictly positive canonical threshold, and a certificate type Hardness3Cert whose inhabitant packages those data as a level-3 hardness claim. No external experimental hardness scale is assumed; the threshold is internal to the phi-ladder geometry.
proof idea
Definition-and-certificate module rather than a deep derivation. domainCost and canonicalThreshold are introduced as defs; domainCost_nonneg and canonicalThreshold_pos discharge the sign obligations needed by the certificate. Hardness3Cert is a structure (or Prop bundle) inhabited by cert / cert_inhabited, wiring the cost and threshold into the level-3 interface. Upstream Cost and Constants are used only for the J-cost and unit scaffolding; there is no multi-step tactic proof of a physical identity here.
why it matters in Recognition Science
Gives the Materials layer a named, inhabitable Hardness3 certificate built from phi-ladder cost data, so downstream materials claims can cite a single RS-native object instead of ad-hoc inequalities. The forcing chain (T5 J-uniqueness, T6 $\varphi$) is the conceptual parent: hardness thresholds are expressed in the same ladder language as masses and coupling bands. used_by is presently empty, so this module is a leaf certificate ready for later materials theorems (strength bounds, defect energetics) rather than an already-wired lemma in a larger proof.
scope and limits
- Does not derive experimental Mohs, Vickers, or Brinell numbers from first principles.
- Does not prove uniqueness of the canonical threshold beyond the local positivity lemma.
- Does not connect Hardness3Cert to continuum elasticity or lattice dynamics.
- Does not claim a closed-form hardness formula for arbitrary atomic number Z.
- Does not discharge any sorry outside this module's own certificate interface.