IndisputableMonolith.Mathematics.RS_MTH_Structural_003
Structural mathematics module defining a domain cost functional, its non-negativity and evaluation identities, and a positive canonical threshold. Ships a certificate bundle (RSMTHStructural003Cert) packaging those facts for downstream RS audits. Proofs are elementary real-analysis checks against the imported J-cost and Constants layers.
claimThe module introduces a domain cost $C$ on a real domain, proves $C \ge 0$ and the pointwise evaluation identity for $C$, and defines a canonical threshold $\theta > 0$. These are packaged as an inhabited certificate record for structural claim RS-MTH-003.
background
Recognition Science measures mismatch by a nonnegative cost derived from the unique J-functional forced at T5: $J(x) = (x + x^{-1})/2 - 1$. The Cost import supplies that J-cost and its elementary identities; Constants supplies the RS-native tick and related units.
This module sits in the Mathematics layer and isolates a structural claim labeled RS-MTH-003: a domain-level cost (domainCost) together with a fixed positive threshold (canonicalThreshold). The certificate pattern (RSMTHStructural003Cert, cert, cert_inhabited) is the standard RS packaging so later forcing or audit steps can depend on a single inhabited record rather than scattered lemmas.
proof idea
Definition-heavy module with short positivity and evaluation lemmas. domainCost is introduced from the Cost layer; domainCost_nonneg and domainCost_at_eq discharge non-negativity and the evaluation identity by direct appeal to J-cost properties. canonicalThreshold is a concrete positive real; canonicalThreshold_pos is a one-line positivity check. The certificate record assembles these fields and is shown inhabited.
why it matters in Recognition Science
Gives a reusable structural certificate for domain cost and a canonical threshold inside the RS mathematics spine. Downstream forcing, audit, or mass-ladder arguments that need a packaged nonnegative cost-plus-threshold fact can import the inhabited cert rather than re-proving elementary Cost lemmas. No used_by edges are recorded yet; the module is a leaf certificate source for later structural claims in the monolith.
scope and limits
- Does not derive J-uniqueness or the Recognition Composition Law; those live upstream in Cost and the forcing chain.
- Does not fix a numerical value for the threshold beyond positivity.
- Does not connect domainCost to physical mass, alpha, or dimension claims.
- Does not assert uniqueness of the certificate beyond inhabitation.