It doesn't matter.
If your index key is only N bits long, you cannot index more than 2^N files. No matter how much more you post, this will continue to be true.
Pi is an irrational number, which means there is no fixed sequence of recurring digits which continues indefinitely.
It does not mean that every sequence of Y digits is guaranteed to be unique.
In fact, even a few moments of thought will tell you that there must be repeating sequences, otherwise Pi would be representable by a finite string of digits, which it isn't.
So there is a possible way to end up with the same index key from two different files, and that is how you end up with only 2^N possible answers.
I appreciate you taking the time to explain matters, but I haven't your background, and I'm not sure what you mean by 2^N, so I can't actually formulate any response to your critique that will make sense to you.
Every unique file is, itself, made up of repeating strings of data, but what makes the data unique from other files is the arrangment of the data, how the 0s and 1s are arranged. Thats true for almost everything. Some popular music differs from other songs only by a few notes and the main singer, but we view almost all of it as unique, the small differences do a lot, even if its almost the same.
What I've created is a way for every bit (0 or 1) to affect the path taken to reach the index in Pi. Therefore, its impossible for any index in Pi to hold more than any one outcome.
Take 0000001 and 0001000 and 100000 for example. The index for each is, respectively:
BYTE EXAMPLE: 0000001: 0001000: 100000:
Pi Index: (57) (85) (103)
And that's just with one byte of data. The larger the file size, the more unique it becomes. You could have billions of 1 megabyte files, and if even one bit in their internal structure was different, the outcome would be different. This is truly the butterfly effect at work.
Lets go in stages:
a) If you have billions of unique files, they must map to billions of unique Pi indexes, otherwise two files would map to the same index, and you would not be able to to determine which of the two files was the original. Do you agree?
b) If you had hundreds of billions of unique files, they must map to hundreds of millions of unique Pi indexes, otherwise two files... Do you agree?
c) Therefore the number of possible indexes increases as the number of possible input files increases. Do you agree?
d) Therefore the number of possible indexes increases as the size of the input files increase. Do you agree?
e) Therefore there is no fixed length of index string which can possibly represent all of the possible indexes for all possible files. Do you agree?
If you have got all the way through to e) and still agree, then you should see that your scheme as presented, with a fixed index length which represents all possible files, is impossible.
To take a small example, lets look at all files of 32 bits in length.
There are 2^32 of them. (2*2*2*2... 32 with 32 numbers 2s in).
That is 4294967296 in decimal.
So there are 4294967296 possible input files.
They each need to have a unique Pi index.
Therefore there must be 4294967296 Pi indexes for them.
That is 2^32 Pi indexes.
Therefore you need 32 bits to represent the possible Pi indexes.
Therefore, as explained before, the average file size after compression will be the same (or greater, with a poor compression scheme) that the average input file size.