3. Significance
This variable set forms the foundation of a predictive framework capable of:
Identifying early warning signals of bifurcation.
Estimating the likelihood of emergent "Black Horse" actors reshaping governance.
Quantifying resilience thresholds for strategic intervention.
III. Mathematical Model Formulation
A. System of nonlinear differential equations
Notation clarification (to avoid ambiguity).
 Earlier discussion used the letter R both for a "repressive response" and for "resilience." in this section we keep the earlier variable names for the state variables but introduce a distinct symbol for the coercive policy. Concretely:
T(t)T(t) --- institutional trust, 0T10\le T\le 1.
E(t)E(t) --- economic stress (index, 0\ge 0).
P(t)P(t) --- protest / unrest intensity (non-negative).
H(t)H(t) --- black-horse potential (latent attractiveness of outsider actors).
U(t)U(t) --- accommodative policy (social relief, transfers, concessions; control input).
K(t)\mathcal{K}(t) --- coercive policy (law-and-order/repression; control input).
R(t)R(t) --- resilience coefficient of the socio-political system (ability to absorb shocks).
All variables are time dependent. We also include stochastic forcing terms i(t)\xi_i(t) to model exogenous shocks (viral events, sudden price spikes, scandals).
Below is a compact but operational system of coupled nonlinear ODEs that embodies the interaction structure discussed earlier.
1) Functional forms (building blocks)
We use saturating / switch-like nonlinearities to model thresholds and limited capacities.
