Total Stability Condition

Canonical definition

The Total Stability Condition is the joint stability requirement of the three-layer formalism, summing the within-Individual loop gains and the network loop gains and comparing the result to the total critical threshold: $$G{total} = G{eff} + G{slow} + G{cross} + G{net}^{\Lambda} + G{net}^{K} < G{crit}^{total}.$$ The terms are: $G{eff} = G\cdot(1 - p(V,S))$ the Patch-modulated fast-layer loop gain; $G{slow}$ the slow-layer loop gain incorporating $\rho(\mathcal{H})$, $\omega(T{int})$, and $\kappa{self}$; $G{cross}$ the higher-order cross-block remainder, the leading fast-slow cross-coupling product being carried by $G{slow}$; $G{net}^{\Lambda}$ and $G{net}^{K}$ the network-channel gains defined in the Network Loop Gain entry. The additive decomposition $G{total} = G{eff} + G{slow} + G{cross} + G{net}^{\Lambda} + G{net}^{K}$ is a leading-order engineering heuristic for weakly coupled loops, with $G{slow}$ carrying the leading cross-coupling product and $G{cross}$ the corrections of higher order in the cross-coupling norms; the exact stability criterion is the largest real eigenvalue of the full block-Jacobian (Appendix A Sections A.5.6, A.5.7.4, and A.9.5). The five channels of the Knowledge Patch footprint act on four of the five additive components of Gtotal: Channel 1 on Geff, Channels 2 and 3 on Gslow (through ρ(H) and ω(Tint) respectively), Channel 4 on Gnet^K, and Channel 5 on Gnet^Λ, with Gcross carrying no channel of its own (Appendix A, Section A.5.7.7). This correspondence means the Total Stability Condition is simultaneously the mathematical expression of crisis risk and the architectural map of prevention targets: reducing any component reduces Gtotal; reducing all five to a sum below Gcrit^total achieves subcriticality. Class 1 interventions address the Ignition Input mobs(t) (Appendix A.5.1). At the network scale, Observability Governance and related reductions of ρ(Λ) reshape that ignition environment and thereby the network Observability gain (Gnet^Λ, Channel 5 of the Patch footprint); Class 1 is not identical with Channel 5 alone. Class 2 interventions address the remaining four channels (Channels 1 through 4, acting on Geff, Gslow and G_net^K). The two-class typology and the five-channel framework describe the same architecture at Individual and network scale respectively. See Appendix A.5.1 for the two-class typology; Appendix A.5.7.4 and A.5.7.7 for the five-channel derivation.