IndisputableMonolith.StandardModel.RS_STD_Structural_009
Defines the domain-cost functional, its nonnegativity, and a positive canonical threshold packaged as RS Standard Model structural certificate 009. Standard-model auditors cite the certificate type and the threshold positivity lemma. The module is mostly definitional: equalities are by unfolding, positivity is elementary from the cost axioms.
claimThe module introduces a domain cost $C_{\mathrm{dom}}$, proves $C_{\mathrm{dom}}\ge 0$ and an evaluation identity, fixes a canonical threshold $\theta_*>0$, and packages these into an inhabited RS-STD structural certificate 009.
background
Recognition Science measures mismatch with a nonnegative cost built from the unique $J$-functional forced by the Recognition Composition Law. The Cost import supplies that cost layer; Constants supplies the RS-native tick and related units.
This module sits in the StandardModel domain and isolates one structural slice labeled 009: a domain-level cost, a comparison threshold, and a small certificate record that witnesses the expected sign and evaluation facts. Sibling names indicate the cost is evaluated at a point (domainCost_at_eq), shown nonnegative (domainCost_nonneg), and compared against a strictly positive canonical threshold.
No forcing-chain step (T5–T8) is re-proved here; the file assumes the ambient cost and constant infrastructure and only records the 009 structural package.
proof idea
Definition-first module. domainCost and canonicalThreshold are introduced as defs; domainCost_at_eq is an unfolding or evaluation identity. Nonnegativity and threshold positivity are short lemmas from the Cost axioms and constant positivity. The certificate type RSSTDStructural009Cert bundles those facts; cert and cert_inhabited supply a canonical inhabitant so downstream code can assume the package without rebuilding the inequalities.
why it matters in Recognition Science
Structural certificates are the bookkeeping layer that lets later Standard Model claims cite a single named package instead of re-opening cost inequalities. Certificate 009 freezes domain-cost nonnegativity and a positive threshold in one place. The module currently has no recorded downstream edges in the mirror graph, so it functions as a leaf certificate definition ready for SM structural theorems to import. It does not itself derive masses, couplings, or the alpha band; it only stages the cost/threshold facts those arguments need.
scope and limits
- Does not derive the J-cost uniqueness or RCL; imports Cost.
- Does not prove particle masses, mixings, or gauge quantum numbers.
- Does not fix numerical SM observables beyond the abstract threshold sign.
- Does not discharge forcing-chain steps T5–T8.
- Does not assert global completeness of all RS-STD structural certificates.