I think the idea you’re looking for is Martin-Löf random sequences.
possibly an easier entry point to the topic is herehttps://en.wikipedia.org/wiki/Chaitin%27s_constant
which is a specific construction that has some relation to the ideas OP has for a construction
I think the idea you’re looking for is Martin-Löf random sequences.
possibly an easier entry point to the topic is here
https://en.wikipedia.org/wiki/Chaitin%27s_constant
which is a specific construction that has some relation to the ideas OP has for a construction