IndisputableMonolith.Materials.RS_Matl_Module_006
Materials module 006 packages a domain cost functional, its nonnegativity, and a positive canonical threshold into a single certificate type. Condensed-matter and materials workers in the RS stack cite it when they need a uniform cost gate on material configurations. The file is mostly definitions plus short positivity lemmas, closed by an inhabited certificate record.
claimOn the materials side one introduces a domain cost $C_{\mathrm{dom}}$ (built from the RS cost $J$), proves $C_{\mathrm{dom}}\ge 0$ and an evaluation identity at equality cases, fixes a canonical threshold $\theta>0$, and packages these facts as an inhabited certificate $\mathrm{RSMatl006Cert}$.
background
Recognition Science measures mismatch by the unique cost $J(x)=(x+x^{-1})/2-1$ forced at T5 of the unified forcing chain; the Cost import supplies that functional and its elementary identities. Constants supplies the RS-native tick $\tau_0=1$, the time quantum against which material relaxation scales are later compared.
This module sits in the Materials domain. It does not re-derive $J$; it specializes cost to a domain-level functional suitable for bulk or lattice configurations, then isolates a single positive threshold that later gates will treat as the acceptance cutoff. Sibling names indicate the usual pattern: a cost def, an evaluation lemma, nonnegativity, a threshold def with positivity, and a certificate record that bundles them.
proof idea
Definition-heavy module. domainCost is introduced from the imported Cost layer; domainCost_at_eq and domainCost_nonneg are short algebraic or order lemmas. canonicalThreshold is a positive constant (canonicalThreshold_pos). RSMatl006Cert (with cert / cert_inhabited) is a structure packing those facts so downstream material gates can assume one inhabited certificate rather than re-proving positivity each time.
why it matters in Recognition Science
Gives the Materials stack a reusable cost-and-threshold certificate rather than ad-hoc inequalities at each call site. No downstream edges are recorded yet (used_by is empty), so the module is a leaf package: later material selection, phase-stability, or defect-cost theorems are expected to import RSMatl006Cert instead of rebuilding nonnegativity of domain cost. It sits downstream of T5 J-uniqueness and the Cost library, and upstream of any materials claim that needs a uniform positive gate on configuration cost. It does not itself force dimension, the eight-tick octave, or the mass ladder.
scope and limits
- Does not derive the form of J; that is imported from Cost / T5.
- Does not prove any concrete material spectrum, band gap, or phase diagram.
- Does not fix numerical values of c, hbar, G, or alpha.
- Does not assert D=3, eight-tick structure, or mass-ladder rungs.
- Does not yet feed a recorded parent theorem (used_by is empty).