IndisputableMonolith.Materials.RS_MAT_Structural_004
Materials certificate module that packages a non-negative domain cost functional and a strictly positive canonical threshold for RS structural materials claims. A materials theorist cites it when discharging the RS_MAT_Structural_004 obligation. The module is mostly definitions plus elementary positivity and evaluation lemmas, closed by an inhabited certificate record.
claimDefine a domain cost $C_{\mathrm{dom}}$ on the RS cost side, prove $C_{\mathrm{dom}}\ge 0$ and the evaluation identity at equality cases, fix a canonical threshold $\theta_*>0$, and package these facts as an inhabited structural certificate $\mathrm{Cert}_{004}$.
background
Recognition Science materials modules sit on the Cost layer: the J-cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$) measures recognition defect, and non-negativity of derived costs is the first structural filter before any continuum or lattice model is attached. This file imports Constants (RS-native units, including the tick $\tau_0$) and Cost, then specializes to a domain-level cost used for structural materials statements.
The local objects are a domain cost functional, its non-negativity, an evaluation identity at distinguished points, and a canonical positive threshold against which structural claims are compared. The certificate record RSMATStructural004Cert is the module's export surface: a single inhabited bundle that downstream materials arguments can assume rather than rebuild.
proof idea
Definition-first module. Domain cost and canonical threshold are introduced as defs; non-negativity and positivity are short lemmas from the Cost/Constants infrastructure; the evaluation identity is an equality lemma at the designated point. The certificate type collects those facts, and cert_inhabited supplies a concrete witness so the structural obligation is discharged without sorry.
why it matters in Recognition Science
RS materials work needs a reusable, machine-checked structural layer before model-specific claims (elasticity, defect energies, phase thresholds) can be stated in RS-native units. This module is that layer for the Structural_004 slot: it freezes the domain-cost and threshold interface so later materials theorems cite one certificate rather than ad-hoc inequalities. No downstream edges are recorded yet; the intended consumers are higher materials certificates and any paper proposition that quotes a positive structural threshold against J-cost defect. It does not itself force dimension, phi, or the eight-tick octave; those remain upstream in the T0–T8 chain.
scope and limits
- Does not derive continuum elasticity or lattice Hamiltonians.
- Does not fix numerical material constants beyond the abstract positive threshold.
- Does not prove uniqueness of the domain cost among all possible costs.
- Does not connect to alpha, mass rungs, or the forcing chain T5–T8.
- Does not assert experimental materials data; only the formal certificate bundle.