It doesn’t (it could do a discrete leap forward in time). Still, why do I wake up as myself every morning? One of the papers linked in this thread discusses (lack of) object permanence from a QM perspective, I’d like to understand how subject permanence works as well, and functionalism does’t handle that well.

Why do waves split and gather again ?

These Socratic games are fun, and I guess I get where you’re coming from. .. but that doesn’t really answer my question regarding subject persistence and the subjective continuity of identity. What emerges tomorrow when I wake up could be another me, like one can argue that a new wave forms after the split.

Most of the atoms in our bodies are replaced on a regular basis (5 years on average) - are we another person then ?

The ship of Theseus doesn’t help your cause here. I’m not the same person I was 5 seconds ago. The self is obviously mutable, which kind of makes sense as long as there is a continuity of consciousness.

What puzzles me is the persistence of identity across interruptions of consciousness, even though your brain can be mutated while you are not there.

The only mostly permanent thing in the brain is neuronal DNA since most neurons don’t undergo mitosis.

IIRC consciousness is about a regime of brainwaves, while memories (and therefore "continuity" of consciousness) are about the actual neuronal connections, and so "survive" sleep ? (Though sleep is also known to be used to integrate new memories, so in that sense a (slightly) different person wakes up each time.)