IndisputableMonolith.Foundation.RS_FDN_Structural_008
Foundation ledger entry 008: defines a domain-level cost functional, records its nonnegativity and pointwise evaluation identity, and fixes a strictly positive canonical threshold. Packages the package as an inhabitable structural certificate. Anyone auditing the RS foundation checklist for cost-threshold consistency would cite it. Argument is mostly definitional, with short nonnegativity and positivity lemmas from the Cost and Constants layers.
claimIntroduce a domain cost $C_{\mathrm{dom}}$ with $C_{\mathrm{dom}}\ge 0$ and a pointwise evaluation identity; fix a canonical threshold $\theta>0$; package these facts as an inhabitable structural certificate for foundation item 008.
background
Recognition Science measures mismatch by a nonnegative cost built from the unique J-functional $J(x)=(x+x^{-1})/2-1$ forced at T5. The Cost import supplies that layer; Constants supplies the RS-native tick $\tau_0=1$ and related units. Structural foundation modules pin discrete ledger facts that later forcing and phenomenology steps may assume without re-deriving them.
This module sits in that ledger as item 008. Its siblings name a domain cost, an evaluation identity at a point, nonnegativity of that cost, a canonical threshold with a positivity proof, and a certificate type with an inhabitation witness. The setting is therefore definitional bookkeeping plus the minimal inequalities that make the threshold usable as a cutoff.
proof idea
Definition-and-certificate module rather than a deep derivation. Domain cost is introduced as a named functional; a short lemma records its value at a distinguished argument; nonnegativity is inherited from the Cost layer. The canonical threshold is a named positive real; positivity is a one-line or short inequality proof. The certificate structure bundles these facts, and inhabitation is witnessed by assembling the definitions and lemmas into a single term.
why it matters in Recognition Science
Keeps foundation item 008 explicit in the RS structural ledger: domain cost is nonnegative and a positive canonical threshold exists. Downstream used-by edges are empty in the current graph, so the module presently acts as a leaf certificate rather than a direct lemma feed. It still matters for auditability: later forcing-chain or phenomenology developments that need a domain-level cost cutoff can import a single named cert instead of re-proving nonnegativity and positivity. Ties to the Cost layer that implements the T5 J-cost, and to Constants for RS-native units.
scope and limits
- Does not derive J-uniqueness or the Recognition Composition Law.
- Does not fix numerical values of c, hbar, G, or alpha.
- Does not prove mass-ladder or eight-tick octave statements.
- Does not claim downstream consumers; used-by is currently empty.
- Does not replace a full Cost-module development; only packages domain-level facts.