Spectral Cascade Risk: Quantifying the Kessler Syndrome via Fokker-Planck Fragment Propagation

The Kessler syndrome --- a self-sustaining cascade of orbital collisions generating exponentially growing debris --- is the existential threat to the low Earth orbit environment. Current assessment relies on decades-long Monte Carlo simulations (NASA ORDEM, ESA MASTER) that require weeks of computation and produce results with wide confidence intervals. We present a spectral method that propagates the entire fragment cloud from a collision event through the Fokker--Planck generator, computing cascade collision probabilities in minutes rather than years of Monte Carlo. The key insight is that collision fragments cluster into \(K\) mass-velocity classes, each admitting its own spectral generator, reducing \(N \sim 2{,}000\) individual fragment propagations to \(K \sim 5\) class propagations. We define the Kessler Index \(\mathcal{K}_I\): a scalar metric derived from the spectral gap of the fragment-environment interaction generator, where \(\mathcal{K}_I > 1\) indicates a self-sustaining cascade. Applied to the Starlink shell (550 km, 53\(^\circ\)), we estimate the current Kessler Index at \(\mathcal{K}_I \approx 0.34\), rising to \(\mathcal{K}_I \approx 0.72\) at full constellation deployment (42,000 satellites). The critical finding: \(\mathcal{K}_I\) depends sensitively on collision avoidance effectiveness, which depends on \(P_c\) accuracy --- closing the loop with the spectral conjunction assessment framework (Nagy, 2026d). Spectral generation-by-generation cascade propagation (\(G_1 \to G_2 \to G_3\)) completes in 15 minutes per generation, enabling real-time cascade risk monitoring that was previously impossible.