Let x=(P,T,E,H,R)x=(P,T,E,H,R). The Jacobian J=F/xJ=\partial F/\partial x inherits a characteristic sign pattern:
PTEHRP++0T0+E+00H++0R+0\begin{array}{c|ccccc} & P & T & E & H & R\\\hline \dot P & - & - & + & + & 0\\ \dot T & - & - & - & 0 & + \\ \dot E & + & 0 & - & 0 & - \\ \dot H & + & - & + & - & 0 \\ \dot R & - & + & - & 0 & - \\ \end{array}
(Entries combine direct effects and policy-mediated terms at the operating point.)
 Eigenvalues crossing the imaginary axis as rr or E\theta_E vary indicate the onset of a critical transition.
6) Digital-media coupling (exogenous--endogenous bridge)
We encode information shocks by P(t)\xi_P(t) and H(t)\xi_H(t) (viral events, disinformation bursts). Their state-dependent gain rises with PP and low TT, effectively multiplying R2 and R3. In practice, this is modeled via state-scaled noise, e.g. P(t)P1+cPP+cT(1T)(t)\xi_P(t)\sim\sigma_P\sqrt{1+ c_P P + c_T(1-T)}\,\eta(t).
7) Spatial spillovers (optional extension)
Partition xx by city/region ii with network coupling matrix WW:
Pi=...+jwij(PjPi),Ti=...+jw~ij(TjTi),\dot P_i = \ldots + \sum_j w_{ij}(P_j-P_i), \quad \dot T_i = \ldots + \sum_j \tilde w_{ij}(T_j-T_i),
capturing diffusion of tactics/perceptions. High inter-city coupling lowers the effective E\theta_E system-wide, easing cascade formation.
8) Practical loop diagnostics
Loop gain estimates: compute empirical LR1,LR2,...\mathcal{L}_{R1},\mathcal{L}_{R2},\dots from local elasticities (e.g., P/E\partial P/\partial E, T/P\partial T/\partial P) using rolling regressions or state-space filters.
Critical-slowing metrics: rising lag-1 autocorrelation and variance in P,TP,T indicate net loop gain approaching 1.
Policy lever index: rtr_t inferred from observed U,K\Delta U,\Delta \mathcal{K} responsiveness---an operational early-warning indicator.
