Cyclic Monitoring Games and the Unique Equilibrium of the Directed 3 Cycle

The prisoner's dilemma has no cooperative equilibrium. We show that adding a single structural element — cyclic monitoring among three agents — inverts this result: cooperation becomes the unique equilibrium, not one among many. We define a three-stage game on the directed 3-cycle C₃ where monitoring is a costly strategic choice and sanctions are automatic. Monitoring is strictly dominant (constant advantage δ+γ−m, independent of opponents' play), making the cooperative profile simultaneously the unique Nash equilibrium, subgame perfect, coalition-proof, and trembling-hand perfect. The profile is the unique stochastically stable state with global basin of attraction and stochastic potential zero. C₃ is the unique graph satisfying four stability axioms and three is the minimum number of agents for which the mechanism exists. The only welfare loss is the monitoring cost m, which each agent has a private incentive to minimize: the price of anarchy R/ (R−m) → 1 as technology improves. The system bootstraps automatically: the founding act makes non-cooperation immediately irrational. All results are derived symbolically with zero fixed parameters; a companion script (271 tests, all PASS) provides a reproducible audit trail.