IndisputableMonolith.QFT.RS_QFT_Structural_003
Structural certificate module for RS-QFT item 003: a nonnegative domain cost, a strictly positive canonical threshold, and an inhabited certificate packaging both. QFT auditors cite it when checking that the structural layer has a well-defined cost floor. Content is definitional plus elementary nonnegativity and positivity lemmas from the Cost import.
claimDefine a domain cost $C$ on the RS-QFT structural layer and a canonical threshold $\theta$. Prove $C \ge 0$ and $\theta > 0$, and exhibit an inhabited certificate $\mathrm{RSQFTStructural003}$ bundling these facts.
background
Recognition Science builds QFT structure on the same cost calculus used for the forcing chain. The Cost module supplies the J-cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$) and related nonnegativity facts; Constants supplies the RS-native tick $\tau_0=1$.
This module sits in the QFT domain and introduces a domain-level cost functional together with a canonical numerical threshold against which structural inequalities are checked. Sibling declarations name the cost, its evaluation identity, its nonnegativity, the threshold, and the threshold's positivity.
The certificate object packages those lemmas so downstream QFT structural claims can assume a single inhabited record rather than re-proving elementary bounds.
proof idea
Definition module with short positivity wrappers. domainCost and canonicalThreshold are introduced as defs; domainCost_nonneg and canonicalThreshold_pos discharge nonnegativity and strict positivity, almost certainly by reduction to Cost-library lemmas on $J$ (or a monotone transform of $J$). domainCost_at_eq records an evaluation identity. RSQFTStructural003Cert, cert, and cert_inhabited assemble the record and prove it is inhabited. No deep tactic proof; the argument is packaging plus Cost nonnegativity.
why it matters in Recognition Science
Gives the QFT structural layer a named, inhabited certificate (item 003) so later RS-QFT results can cite one object for "domain cost is nonnegative and the canonical threshold is positive." That pattern matches other RS certificate modules: close elementary analytic gaps once, then reuse. It rests on the Cost import (J-cost nonnegativity from the T5 uniqueness lineage) and Constants ($\tau_0$). No downstream used_by edges are recorded yet; the module is a leaf certificate for structural audits rather than a step in the T0-T8 forcing chain itself.
scope and limits
- Does not derive the Standard Model Lagrangian or interaction vertices.
- Does not prove uniqueness of the domain cost beyond the stated defs.
- Does not connect the threshold to measured coupling constants or alpha.
- Does not address renormalization, anomalies, or continuum limits.
- Does not discharge any T0-T8 forcing-chain obligation.