Consider a string rewriting system, that we will call the Baire rewriting system, where the alphabet is the set of positive integers (characters) and assume all possible rules of the form: w is substituted by w<>n, for any word w and any character n, where "<>" is the symbol for concatenation. In the framework of the Wolfram Model, the worldlines in the Baire rewriting system are the sequences of words w(1), w(2), w(3),... such that w(i+1) is obtained from w(i) by applying a rule for each i. By interpreting each word as a finite continued fraction, the set of world lines is in one-to-one correspondence with the infinite continued fractions in positive integers, i.e., the irrational numbers largest than 1. From a topological point of view, it is natural to consider the limiting manifold of the Baire rewriting system as the topological space obtained by restricting the topology of the real line to the irrational number larger than 1. This topological space is known as the Baire space (do not confuse this concept with a Baire space).
At first glance, this example of limiting manifold of a causal graph may seem too general to be interesting in practice. Nevertheless, any Polish space is the continuous image of the Baire space. Hence, given a string rewriting system (over a finite alphabet), whose causal graph converges to a limiting manifold that is a Polish space, is just a "continuous image" of the Baire rewriting system, e.g., some different worldlines in the Baire rewriting system may collapse to a single worldline in a given string rewriting system.
Now, the challenge is to provide non-trivial examples of the application of this framework to prove the convergence of a causal graph to a limiting manifold that is a Polish space.
