@David Petry: I think that your project in theoretical music and its connection with the Wolfram Physics Project is interesting enough to be part of its own thread. I recommend you to write a computational essay explaining your ideas from the beginning so that people without the knowledge of theoretical music (like me) could understand your project in detail and eventually contribute.

Does this help? I’m creating a frequency of 11 from 8. The foundation is based on a wave down and up. Using 2’s and 3’s (6) The rule is 2 becomes 3, 3 becomes 2, 4 becomes 3, 3 becomes 4

Yes -if I well understood- basically you are working with a string rewriting system motivated by applications in theoretical music. Maybe there is a function such that when it is applied to any vertex of the Baire string rewriting system (topological space of irrational numbers), the result is your string rewriting system. In order to explore deeper this idea, it is important to see the code of your method expressed as a string rewriting system. Here are some functions that may be useful.

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.

Not sure how your example can be used to solve the challenge. There are also several mistakes in the second paragraph. First, not every Polish space can be mapped continuously onto the Baire space (example: real line, which is a Polish space, cannot be mapped continuously onto the Baire space, which is isomorphic to the space of irrational numbers). Second, the causal graph for the Baire substitution system is just a chain, and thus, it is isomorphic to the natural numbers, and not reals or irrationals. What is isomorphic to irrational numbers, is the resulting string of the Baire substitution system; the multiway system of the substitution system is a tree (with infinite branching at each node); And the causal graph of such system is just a chain with a limiting manifold being a single point (which is not very useful for proofs in general case).

Just to clarify that my quotation "Nevertheless, any Polish space is the continuous image of the Baire space." is not contradicted by your observation that "not every Polish space can be mapped continuously onto the Baire space".

You are right. I initially thought it is not possible to map irrationals continuously onto reals in such a way so the image is the whole real line, but it turns out it is possible: https://mathoverflow.net/questions/112127/the-reals-as-continuous-image-of-the-irrationals/112130

I do not get it. It seems to me that it is a forest since there are several possible substitutions at every string.

I am not sure I understood your substitution system correctly then. Could we say, that on each step you are adding a number to the end of a string? Or, rather, you are inserting a number into any place of the string? If the first case is true, then the causal graph for any evolution history will be a single chain, since at each step you are using the output of the previous step (and thus each update is connected to the previous update). If the second interpretation is true, then the causal graph can be pretty complex, but does not have to have a tree structure (can still be a chain like in the first interpretation though).