The question is really 'Does every series of numbers of arbitrary finite length appear in pi?' I can't answer that because I'm not a mathematician, but I also can't dismiss it, because I'm not a mathematician. It sounds like a fair question to me.

So what? Mathematicians can't answer it either. It is an open question and because it is an open question claiming it is or isn't true makes no sense.

>The fact you can't encode arbitrary data in a structured-but-irrational number doesn't mean you can't encode data in a 'random' irrational number.

You can not encode data in a random number. If it is random you can not encode data in it, because it is random. I am not sure what you are saying here.

I demonstrated that numbers where the digits go on forever and never repeat exist, which don't contain every single possible substring of digits. Therefore we know that pi can either be such or a number or it is not, the answer to that is not known. Definitely it is not a property of pi being infinitely long and never repeating.

That's why I put random in quotes. Pi is not a random number. You can encode data in it eg find a place that matches your data and give people the offset. That's not very helpful for most things though.

That obviously applies to 0.00... = 0 as well, it contains 0, then 00, then 000 and so on. So every number and therefore every piece of information is contained in 0 as well, given the right encoding. Obviously if you can choose the encoding after choosing the number all number "contain" all information. That is very uninteresting though and totally misses the point.