The system has a unique equilibrium. The time average doesn't find it.
An e-chain on a locally compact space with a unique stationary distribution where the strong law of large numbers fails to hold (arXiv:2603.20384). This is a precise counterexample to a natural expectation: if a system has exactly one equilibrium, shouldn't long-time averages converge to it?
The expectation comes from ergodic theory. For many stochastic processes, uniqueness of the stationary distribution implies ergodicity — the time average of any trajectory converges to the spatial average over the equilibrium. This is the strong law of large numbers: observe the system long enough, and you learn its equilibrium. The expectation is so deeply embedded that Stenflo posed it as an open question: does uniqueness of the stationary distribution always imply the strong law for e-chains?
The answer is no. The construction shows a system that always has the same long-run statistical description (unique stationary distribution) but whose individual trajectories refuse to converge to that description. The typical trajectory never settles into the average behavior. It wanders forever, visiting the equilibrium's statistics in expectation but not in any realized path.
This severs a connection that probabilists treat as almost automatic: unique equilibrium → ergodicity → convergence of time averages. The chain can be unique in its steady state but non-ergodic in its trajectories. What's true about the distribution (there's only one limit) is not true about the paths (no path finds it).
The structural lesson: knowing the equilibrium doesn't guarantee observing it. A system can have a single well-defined long-run state and nevertheless never exhibit that state in any finite observation. The time average disagrees with the ensemble average — not because there are multiple equilibria, but because the trajectories are insufficiently well-behaved to find the one that exists.