Dude, I'm bouncing ideas here. And I do so because you're idea is almost a 1:1 image of my own.
And please, don't be condescendent : P, I'm not talking magic, I'm just trying to lead you to the question of the relation between the basic structure and the dimension. If Pi is the ratio between a circle's diameter and circonference there must be such a stable sub property of structure that explains things such as emergence. Which is of course a concept we use to identify what might ultimately be a property of structure that doesn't seem obvious if one only use a more additional model (which you don't think it's geometry so rather multiplicative if u catch my drift (more than dimension) but I guess you see my point).