IndisputableMonolith.Materials.RS_MAT_Structural_005
Materials module packaging a structural certificate for RS_MAT_Structural_005: a nonnegative domain cost built from the RS J-cost, a positive canonical threshold, and an inhabited certificate record. Materials theorists cite it when wiring structural bounds into the RS materials stack. The module is mostly definitions plus elementary nonnegativity and positivity lemmas, closed by an inhabited cert.
claimDefine a domain cost $C$ from the RS cost functional, prove $C \ge 0$ and $C$ agrees with its pointwise evaluation, fix a canonical threshold $\theta > 0$, and package these into an inhabited structural certificate $\mathrm{Cert}_{005}$ for the RS materials claim RS\_MAT\_Structural\_005.
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 forcing chain and obeying the Recognition Composition Law. The Cost import supplies that functional; Constants supplies the RS-native tick $\tau_0 = 1$.
In the materials layer, structural claims are stated as comparisons of a domain-level cost against a fixed positive threshold. This module introduces that domain cost, records its elementary analytic properties (nonnegativity, agreement with pointwise evaluation), and names the canonical threshold used by claim 005.
The local setting is certificate-first: rather than a free-standing inequality theorem, the module exposes a small record RSMATStructural005Cert whose inhabitation is the formal claim that the structural bound is in place.
proof idea
Definition module with light lemmas, not a deep derivation. domainCost is introduced from the imported Cost layer; domainCost_at_eq and domainCost_nonneg discharge the expected evaluation and sign facts. canonicalThreshold is a named positive constant, with canonicalThreshold_pos the positivity witness. The certificate type RSMATStructural005Cert bundles those pieces; cert and cert_inhabited show the record is occupied. No multi-step tactic proof is required beyond applying Cost/Constants facts.
why it matters in Recognition Science
Places structural materials claim 005 on the same certificate footing used elsewhere in the RS materials domain: a nonnegative J-derived domain cost, a positive canonical threshold, and an inhabited cert. Downstream pages can import the cert rather than re-proving nonnegativity or threshold positivity. The module sits under Materials and depends only on Constants and Cost, so it is a leaf packaging step rather than a forcing-chain link (T0–T8). With no recorded used_by edges yet, its role is to close the 005 structural interface for later materials theorems that need a ready-made bound certificate.
scope and limits
- Does not derive a new mass, coupling, or continuum PDE; only packages a materials cost/threshold cert.
- Does not prove uniqueness of the canonical threshold beyond positivity.
- Does not connect to eight-tick, D=3, or alpha-band forcing steps.
- Does not supply numerical materials data or experimental fits.
- Does not claim downstream consumers; used_by is currently empty.