The Dynamical Laws of NSD

The core of the theory is a three-equation slow-fast dynamical system governing continuous accumulation, non-linear activation, and structural degradation.

Law 1: Stress Evolution (The Slow Drive)

Stress accumulates via exogenous load, dissipates naturally over time, and redistributes rapidly during network cascades: \(\\frac{dS_i}{dt} = L_i(t) - \gamma S_i(t) + \sum_{k} \sum_{j} W_{ji}^{(k)} F_j^{(k)}(t) - \sum_{k} F_i^{(k)}(t)\) Where \(\gamma\) is the natural localized dissipation rate.

Law 2: Differentiable Cascade Activation (The SOC Trigger)

Nodes become highly contagious sources of stress only when \(S_i(t) \gg C_i(t)\). We define a smooth, differentiable activation function to permit Lyapunov stability analysis: \(F_i^{(k)}(t) = \kappa^{(k)} (S_i(t) - C_i(t)) \sigma(a(S_i(t) - C_i(t)))\) Where \(\sigma(x) = \\frac{1}{1+e^{-x}}\), \(\kappa^{(k)}\) is the propagation speed in layer \(k\), and \(a\) dictates the threshold sharpness. As \(a \to \infty\), this recovers the hard boundary of classical sandpile models.

Law 3: Dynamic Capacity Degradation (Structural Fatigue)

Capacity is not static. It naturally recovers toward a healthy baseline \(C_i^*\) but suffers immediate, permanent degradation when forced to cascade stress: \(\\frac{dC_i}{dt} = \mu(C_i^* - C_i(t)) - \eta \sum_{k} F_i^{(k)}(t)\) Where \(\mu\) is the institutional recovery rate and \(\eta\) is the burnout penalty.

Law 4: Global Leaky Conservation

By summing Law 1 over all \(N\) nodes and applying the topological normalization condition, the internal propagation terms strictly cancel out: \(\sum_{i=1}^{N} \\frac{dS_i}{dt} = \sum_{i=1}^{N} L_i(t) - \gamma \sum_{i=1}^{N} S_i(t)\) This mathematically proves the system is globally conserved with localized dissipation.