Perturb a cellular automaton's initial state and watch the damage spread. If the dynamics are strongly irreversible, different initial states converge — the perturbation heals. If weakly irreversible, the damage persists, spreading through the system like an infection. The transition between these phases has been understood as belonging to the directed percolation universality class — one of the cleanest critical phenomena in nonequilibrium physics.
Nahum and Roy show this is only the ground floor. The full critical theory contains an infinite hierarchy of operator sectors, each with its own universal critical exponents and its own renormalization group fixed point. Directed percolation is a subsector of every level, but each level adds new structure that directed percolation alone cannot describe.
The hierarchy emerges from overlaps between multiple trajectories. Tracking two trajectories from different initial conditions gives you the damage-spreading transition — directed percolation. But tracking three, four, or n trajectories simultaneously defines higher-order overlap observables labeled by set partitions. Each partition class decays with its own critical exponent at the transition point. The universality class isn't one class — it's a tower of classes, nested inside each other, with directed percolation at the base.
The physical consequence is that entropy decay — how fast the system forgets its initial state — encodes information invisible to single-overlap measurements. The rate at which trajectory bundles distinguish themselves carries richer critical structure than pairwise comparison. The standard approach of comparing two trajectories captures the most relevant exponent but misses the entire upper architecture.
One phase transition. Infinitely many critical exponents. The same boundary, observed with different instruments, reveals different universality.