We construct an explicit deterministic system in which non-injective projection produces stochastic effective behavior and a sharp knee-like transition in first-passage statistics. Using rigorous deterministic homogenization, we derive an Ornstein–Uhlenbeck limit for the projected dynamics and show that the knee arises when an effective stability margin crosses zero. The phenomenon requires no intrinsic noise, thermodynamic limit, or discontinuous dynamics and provides a stress test for projection-based descriptions of collapse and threshold behavior.