IndisputableMonolith.Mathematics.RS_MTH_Structural_008
Certificate module packaging a nonnegative domain cost functional and a strictly positive canonical threshold used in RS structural mathematics. A working mathematician cites it when a proof needs a packaged witness that the cost is well-behaved at equality and the threshold is positive. The module is mostly definitions plus elementary nonnegativity and positivity lemmas, closed by an inhabited certificate record.
claimDefine a domain cost $C$ on the RS cost side, record the identity $C$ at the equality locus, prove $C \ge 0$, and introduce a canonical threshold $\tau_*>0$. Bundle these facts into an inhabited structural certificate $\mathsf{RSMTHStructural008Cert}$.
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 infrastructure; Constants supplies the RS-native tick scale used when thresholds are stated in discrete time.
This module sits in the Mathematics structural series. It isolates a domain-level cost (a restriction or evaluation of the global cost on a designated domain) together with a canonical positive threshold against which that cost is compared. The equality lemma records that the cost collapses correctly on the diagonal or fixed locus, which is the usual sanity check before any forcing or uniqueness argument.
Sibling names indicate the exported surface: domainCost, its equality and nonnegativity facts, canonicalThreshold with positivity, and a certificate record with an inhabitation proof.
proof idea
Definition-heavy module. The cost and threshold are introduced as defs; nonnegativity and positivity are short lemmas (likely direct from the Cost library or elementary arithmetic). The certificate is a structure packing those fields, discharged by cert_inhabited constructing a concrete witness. No deep tactic proof; the work is packaging and interface.
why it matters in Recognition Science
Gives a reusable structural certificate for later RS mathematics that needs a nonnegative domain cost and a positive comparison threshold in one place. Downstream edges are not yet wired in the graph (used_by empty), so the module presently serves as a self-contained Math certificate rather than a direct lemma in the T0–T8 forcing chain. It supports any argument that quotes a domain cost bound or a canonical cutoff before invoking mass-ladder, eight-tick, or dimension results. Closes a named structural slot (008) in the Mathematics layer without introducing new physics constants.
scope and limits
- Does not derive the global J-uniqueness theorem (T5) or the RCL identity.
- Does not fix numerical values of c, hbar, G, or alpha.
- Does not prove mass-ladder or eight-tick octave statements.
- Does not assert which downstream theorems consume the certificate.
- Does not claim the threshold equals any named RS constant without further lemmas.