Why the Born probabilities?
I see no reason why a non-conscious machine, say a bayesian superintelligence, would not encounter the Born probabilities. As such, consciousness seems unlikely to be related to them - it's too high-level to be related to quantum effects.
Continuity of consciousness. Part of the hard problem of consciousness - why do I wake up tomorrow as myself and not, oh, EY?
How do you define "I" that you can credibly imagine waking up as Eliezer? What difference do you expect in the experience of that Eliezer? I think it's a bug in the human brain that you can even ask that question; I think it's incoherent. You tomorrow is the entity that carries on all your memories, that is most affected by all of your decisions today; it is instrumentally useful to consider yourself tomorrow as continuous with yourself today.
Their brains are spread out across their whole body, so the splitting process necessarily requires the slow split of their brains while they are functioning. The question is: at what point are there "two" Ebborians?
This only is a problem if you insist on being able to count Ebborian individuals. I see no reason why the number of Ebborians shouldn't start out as 1 at the point of split and quickly approach 2 via the real numbers as the experiences diverge. As humans we have no need to count individuals via the reals because in our case, individuals have always been cleanly and unambiguously differentiable; as such we are ill-equipped to consider this situation. I would be highly surprised if, when we actually encountered Ebborians, this question was in any way confusing to them. I suspect it would just be as intuitively obvious to them as counting individuals is to us now.
Why aren't I a Boltzmann Brain?
That one seems hard but, again, would equally confound a non-conscious reasoning system. It sounds like you're taking consciousness as the big mystery of the human experience and thus pin on it everything marginally related that seems too mysterious to answer otherwise.
Highlighting for you a section you missed, as I think it important:
"Does falling asleep constitute "death" - is there any difference at all we can point to?"
Although Elizier has dealt with personal identity questions (in terms of ruling out the body theory), he has not actually, as far as I know, "solved" the problem of Personal Identity as it is understood in philosophy. Nor, as far as I know, has any thinker (Robin Hanson, Yvain, etc) broadly in the same school of thought.
Why do I think it worth solving? One- Lesswrong has a tradition of trying to solve all of philosophy through thinking better than philosophers do. Even when I don't agree with it, the result is often enlightening. Two- What counts as 'same person' could easily have significant implications for large numbers of ethical dilemnas, and thus for Lesswrongian ethics.
Three- most importantly of all, the correct theory has practical implications for cryonics. I don't know enough to assert any theory as actually true, but if, say, Identity as Continuity of Form rather than of Matter were the true theory it would mean that preserving only the mental data would not be enough. What kind of preservation is necessary also varies somewhat- the difference in requirement based on a Continuity of Consciousness v.s a Continuity of Psyche theory, for example should be obvious.
I'm curious what people here think. What is the correct answer? No-self theory? Psyche theory? Derek Parfit's theory in some manner? Or if there is a correct way to dissolve the question, what is that correct way?