I mean personally I see continuity as mostly being about the hardware with the process running on it only being secondary at best and disjointed, though not completely irrelevant.
So funnily, I'm "perpendicular" to the issue of if continuous consciousness is needed -- i think the only relevant question is following the original structure and processes every "tick" of the universe to see what happened.
E.g. to me if the structure is wholly or nearly-wholly preserved, "teleporting" by the mechanism of particles having an uncertain position is non destructive and retains continuity, stasis is the same, becoming part of a hivemind is the same -- though leaving one or being quickly ship-of-theseus'd would likely impair continuity, but less in a manner of "death" and merely making the question meaningless, because two new entities are born descended from the same source, akin to how who we were as children never dies, only undergoes a process of growth/change/refinement until it's something not only different but significantly more than the "original".
I think a very interesting line of thought is actually if severe brain damage essentially kills continuity / "the original", in context of how we can absolutely disable or impair parts of the brain with magnetism or drugs but then have the hardware return to original function along with apparent return of the original process. It actually feels much closer to the self-evidently wrong assertion that any mere identical copy would equally be the original, suggesting to me that damaging the original too far past a point may actively kill that thread of consciousness in a way I wouldn't consider to be the case from sleep / stasis / joining-but-not-leaving a hivemind / etc.
So funnily, I'm "perpendicular" to the issue of if continuous consciousness is needed -- i think the only relevant question is following the original structure and processes every "tick" of the universe to see what happened.
E.g. to me if the structure is wholly or nearly-wholly preserved, "teleporting" by the mechanism of particles having an uncertain position is non destructive and retains continuity, stasis is the same, becoming part of a hivemind is the same -- though leaving one or being quickly ship-of-theseus'd would likely impair continuity, but less in a manner of "death" and merely making the question meaningless, because two new entities are born descended from the same source, akin to how who we were as children never dies, only undergoes a process of growth/change/refinement until it's something not only different but significantly more than the "original".
I think a very interesting line of thought is actually if severe brain damage essentially kills continuity / "the original", in context of how we can absolutely disable or impair parts of the brain with magnetism or drugs but then have the hardware return to original function along with apparent return of the original process. It actually feels much closer to the self-evidently wrong assertion that any mere identical copy would equally be the original, suggesting to me that damaging the original too far past a point may actively kill that thread of consciousness in a way I wouldn't consider to be the case from sleep / stasis / joining-but-not-leaving a hivemind / etc.