IndisputableMonolith.Mathematics.RS_MTH_Structural_004
Structural mathematics module that defines a domain-level cost functional, proves its nonnegativity and evaluation identity, and fixes a strictly positive canonical threshold. It packages these facts into an inhabited certificate type for the RS mathematics layer. Anyone citing domain cost bounds or threshold comparisons against the RS cost stack would land here. The argument is definitional plus short positivity and equality lemmas, then certificate assembly.
claimThe module introduces a domain cost $C_D$ satisfying $C_D \ge 0$ and an evaluation identity on equality cases, together with a canonical threshold $\theta > 0$, and packages these into a structural certificate for RS mathematics item 004.
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 stack; Constants supplies the RS-native units and fixed scales against which thresholds are compared.
This module lifts cost from pointwise values to a domain-level functional domainCost, with an evaluation identity when the domain data sit at equality and a nonnegativity lemma. Alongside it sits canonicalThreshold, required to be strictly positive, as the comparison scale for structural claims in the mathematics layer.
The local setting is a certificate module: named facts are collected into RSMTHStructural004Cert so downstream developments can depend on a single inhabited record rather than scattered lemmas.
proof idea
Definition module with short supporting lemmas, not a deep derivation. domainCost and canonicalThreshold are introduced by definition. Nonnegativity of the domain cost and positivity of the threshold are discharged by direct appeal to the imported Cost and Constants facts (nonnegative $J$-cost and positive RS scales). An evaluation identity lemma records the on-equality case. The certificate type is then inhabited by assembling those lemmas into one record.
why it matters in Recognition Science
Gives the mathematics layer a reusable structural certificate (domain cost bounds plus a positive canonical threshold) under the RS_MTH_Structural_004 label. No downstream consumers are wired in the current graph, so the module stands as a self-contained packaging point for later structural theorems that need a named, inhabited cost-and-threshold bundle rather than ad hoc appeals to Cost and Constants. It sits downstream of the $J$-cost uniqueness chain (T5) only through the Cost import, and does not itself advance the forcing chain T0–T8.
scope and limits
- Does not derive the $J$-cost or Recognition Composition Law; imports them.
- Does not fix numerical values of physical constants beyond the threshold positivity claim.
- Does not prove uniqueness of the domain cost among all possible domain functionals.
- Does not connect the certificate to mass, alpha, or forcing-chain theorems.
- Does not assert any downstream consumer; the use graph is currently empty.