Network coupling

What the plate operates. §8 extends the single-user apparatus to networked populations. The engagement-cascade theorem (§8.4) gives the stability condition $\rho (W)\cdot \rho (\alpha )\cdot \int \kappa <1$; the percolation theorem (§8.5) classifies the resulting cascade dynamics. The plate visualizes both: cascades live on the graph; the branching-ratio bar shows where the parameters sit relative to threshold. In the plate's reduction the temporal kernel is absorbed into a single per-edge probability $\alpha$, so the branching ratio it displays, $\rho (W)\cdot \alpha$, is the scalar stand-in for $\rho (W)\cdot \rho (\alpha )\cdot \int \kappa$.

Three regimes to explore.

• Subcritical (rho · alpha < 1). Low connection density or low per-edge propagation. Seeded cascade dissipates after a few hops; cascade size remains small; the percolation bar reads moss.
• Near threshold (rho · alpha ~ 1). Critical dynamics. Long-range temporal correlations; power-law cascade sizes; the framework's mobilization-dispersion regime.
• Supercritical (rho · alpha > 1). High connectivity or strong per-edge propagation. Cascades propagate platform-wide; the percolation bar reads oxblood. This is the framework's substantive empirical claim about contemporary platforms.

Why the threshold matters politically. The fourth-cell intervention site — regulatory imposition on the observation channel — can in principle target the graph structure itself. Lowering $\rho (W)$below threshold converts super-critical cascade dynamics into local cascades. This is the framework's formal underwriter for graph-structure regulation as a Mode A site.

What the theorem authorizes the prose to claim. The mobilization-dispersion patterns characteristic of contemporary platforms are super-critical cascade dynamics on graphs above the percolation threshold. Theorem 10 supplies the formal vocabulary; the plate shows the threshold being crossed.