Somatic Hawkes

What the plate operates. Definition 3.1 gives the multi-mark Hawkes intensity:

lambda^k(t) = lambda_0^k + sum_l alpha_kl · integral_0^t kappa(t-s) dN^l_s

The plate uses an exponential kernel $\kappa (s)=e^{-s/{\tau }_k}$ and a symmetric $\alpha$ matrix (diagonal = self-excitation; off-diagonal = cross-coupling). Events sampled per-tick at rate ${\lambda }^k(t)\cdot dt$.

The stability condition. Per §3.5, stability requires the branching ratio $\rho (\alpha )\cdot \int \kappa <1$. For the symmetric $\alpha$ matrix used here, the dominant eigenvalue is ${\alpha }_{kk}+3{\alpha }_{kl}$. The readout flips between “stable” (cascades dissipate) and “supercritical” (cascades blow up) as the parameters cross.

Three regimes to explore.

• Independent registers. Set cross- coupling = 0. Each track behaves as its own self- exciting Hawkes process; tracks are dynamically decoupled.
• Cross-register cascades. Raise the cross-coupling. An event on one track now triggers events across the others. The intensity curves visibly couple — bursts on S produce bursts on C, P, K.
• Supercritical. Push the parameters past $\rho (\alpha )\int \kappa =1$. Cascades grow without bound; events fill every track.

What the proposition authorizes the prose to claim. The dynamical complement to Prop 7''s static explaining-away argument. The off-diagonal ${\alpha }_{kl}$is the dynamical machinery behind libidinal routing. The platform's optimization (Theorem 9) selects policies that increase the off-diagonals wherever cascades raise engagement.