IndisputableMonolith.Mathematics.RS_MTH_Structural_006
Structural mathematics module that defines a domain cost functional, records its nonnegativity, and introduces a strictly positive canonical threshold. It packages both facts as an inhabited certificate object for downstream RS math. The argument is definitional plus elementary positivity checks against the imported cost layer.
claimThe module introduces a domain cost $C$, proves $C \ge 0$, defines a canonical threshold $\tau_*>0$, and supplies an inhabited structural certificate bundling these facts for Recognition Science mathematics.
background
Recognition Science measures mismatch with a nonnegative cost built from the unique $J$-functional $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$), forced by the Recognition Composition Law. The Cost import supplies that layer; Constants supplies the RS-native tick $\tau_0=1$.
This module sits in the mathematics structural series. It names a domain-level cost (evaluation of the cost on a designated domain), records the equality that pins its value at a reference point, and isolates a canonical positive threshold used as a comparison scale.
No forcing-chain step (T0–T8) is proved here. The local job is bookkeeping: make the cost bound and threshold available as a single certificate type rather than scattered lemmas.
proof idea
Mostly a definition module. domainCost and canonicalThreshold are introduced by definition; domainCost_at_eq is an evaluation identity; domainCost_nonneg and canonicalThreshold_pos are short nonnegativity/positivity arguments over the imported Cost facts. The certificate record RSMTHStructural006Cert bundles those properties, and cert_inhabited exhibits a witness.
why it matters in Recognition Science
Gives the mathematics layer a reusable structural certificate for domain cost nonnegativity and a positive threshold, so later lemmas can depend on one inhabited object instead of re-proving elementary cost facts. Downstream use edges are not yet wired in this snapshot (used_by empty), so the module is infrastructure rather than a cited parent theorem. It supports any argument that needs a clean cost lower bound or threshold comparison inside the RS $J$-cost framework, without touching mass ladders, $\alpha$, or the T5–T8 forcing steps.
scope and limits
- Does not derive uniqueness of $J$ or invoke the Recognition Composition Law.
- Does not prove any T0–T8 forcing-chain step or dimensional claim.
- Does not compute physical constants, masses, or the fine-structure band.
- Does not assert a numerical value for the threshold beyond positivity.
- Does not yet show downstream consumers; used_by is empty in this snapshot.