Let’s suppose a time travel event occurs in which an agent with free will travels to their own causal past, and let’s suppose this creates a parallel timeline which can differ from the first (leading to a new version of the agent which creates a third timeline, and so on).

We can consider this time-travel event as a function in which one timeline maps to a successor timeline — or in general, the event is an iterative map from the space of possible timelines to itself. If this map meets a few general criteria, we can apply the fixed point theorem and conclude that, after enough iterations, the process will converge to some fixed point that maps to itself (that is, the agent causes the past of their own timeline, even though they have free will). This timeline maps to itself—but it is also mapped to by an infinite succession of timelines in which the agent is free to alter their successor timeline, converging on one in which their choices cause no further alteration.

At that point, we can dispense with the assumption that time travel creates parallel timelines, and assume instead that the fixed-point, self-causing timeline is the only real one.