In my rudimentary understanding of the many-worlds interpretation of quantum mechanics, one posits the existence of a universal wave function - the state of the entire universe - wherein the various superposition of states never actually collapse, yet appears to collapse to an observer (or macroscopic system) through the mechanism of quantum entanglement.
One says for example that the observer is in any instance measuring precisely one classical state out of a superposition of many by following a certain 'branch'. Or one says that the universe keeps splitting or 'branching' off, which makes intuitive sense in a localized laboratory setting. However, the more I think about it, in particular when I abstract away from the duality between the observer and the observed, the more confused I become.
What is the precise, mathematical definition of a 'branch'? Can it be defined independently of a (classical) observer? If the ontological base is the universal wave function, how are its branches defined, or made sense of?