A massive loophole
In Mysterious Instances, finding a way to break the cycle is actually not difficult.
Take the current Instance for example; the lingering spirit causing the cycle clearly aims to annihilate or torment a specific force.
According to the information Alexander has gathered, each cycle lessens the spirit's obsession a bit until it eventually vanishes entirely.
For a cycle to reach the scale we see now, the spirit's obsession must be immense, signifying that its intent has never been fulfilled.
For instance, the person it wants to kill may have survived to the end, or their death was not sufficiently painful. As a result, repeated cycles not only fail to dissolve the lingering spirit's obsession but intensify it, turning it into a frightening Mysterious Instance.