IndisputableMonolith.Mathematics.RS_MTH_Structural_010
Structural mathematics module packaging a domain cost functional, its nonnegativity and evaluation identities, and a strictly positive canonical threshold. A certificate record bundles the proved facts for downstream RS structural claims. The argument is definitional plus short positivity and evaluation lemmas over the imported cost layer.
claimDefine a domain cost $C$ with $C\ge 0$ and an evaluation identity at equality cases; define a canonical threshold $\theta>0$; package these as an inhabited structural certificate for RS-MTH-010.
background
Recognition Science measures mismatch with a nonnegative cost built from the unique J-functional forced by the Recognition Composition Law, $J(x)=(x+x^{-1})/2-1$. The Cost import supplies that layer; Constants supplies the RS-native tick $\tau_0=1$.
This module sits in the Mathematics domain and introduces a domain-level cost (evaluation and nonnegativity) together with a canonical positive threshold. Those objects are the local vocabulary for a numbered structural claim (RS-MTH-Structural-010), not a full forcing-chain step (T5–T8).
Sibling declarations name the cost, its equality evaluation, nonnegativity, the threshold and its positivity, then a certificate type with an inhabited instance.
proof idea
Definition module with thin lemmas. Domain cost and canonical threshold are introduced as defs; nonnegativity and positivity are short proofs over the Cost/Constants imports; an equality-evaluation lemma records the on-shell identity. A certificate structure aggregates the facts, and inhabitation is by direct construction of that record. No deep tactic script or multi-hop forcing argument.
why it matters in Recognition Science
Gives a reusable certificate surface for structural mathematics claim 010: domain cost well-behaved and a positive canonical threshold available. Downstream graph edges are empty here, so the module is a leaf packaging layer rather than a parent of named theorems in the supplied slice. It anchors cost-threshold language that later RS mathematics or physics bridges can import without re-proving nonnegativity or positivity. It does not itself force $\varphi$, the eight-tick octave, or $D=3$.
scope and limits
- Does not prove J-uniqueness (T5) or force $\varphi$ (T6).
- Does not derive the eight-tick octave or $D=3$.
- Does not state mass-ladder or $\alpha$ numerical claims.
- Does not expose used_by parents in the supplied graph slice.
- Does not replace the full Cost theory; only packages domain-level facts.