By your definition, is the empirical fact that one tenth of the digits of pi are 1s emergent behavior of pi?
IMO, that would be emergent behaviour of mathematics, rather than of pi.
Pi isn’t a system in itself as far as I can see.
I have in mind a system, for instance a computer program, that computes pi digit-by-digit. There are features of such a computer program that you can notice from its output, but not (so far as anyone knows) from its code, like the frequency of 1s.
IMO, that would be emergent behaviour of mathematics, rather than of pi.
Pi isn’t a system in itself as far as I can see.
I have in mind a system, for instance a computer program, that computes pi digit-by-digit. There are features of such a computer program that you can notice from its output, but not (so far as anyone knows) from its code, like the frequency of 1s.