It would not be necessary to gradually replace the brain, nor any other parts of the connectome. If you agree that cognition is due to physical processes of the brain, then it is impossible that consciousness is anything more than very advanced logic. You could devise a situation in which a person is gradually converted into a robotic form, and another where he is simply replaced with a robotic clone, such that you would end up with the same exact logical brain(contained within the robot) in both cases. So, in this instance, where does cognition diverge? If the gradual replacement results in the continuation of the stream of consciousness, but the creation of a robotic copy does not, then why can't that difference be physically measured? Would you suggest a supernatural explanation? Or is the experience consciousness as shallow as the belief that it has been preserved (which I say it is)?

I think you misinterpreted my comment. The poster I was responding to was talking about how it would not be the same consciousness, but a clone of that consciousness.

I am not making any unprovable claims about a spirit/soul, though I do not completely shut out the possibility there is something we haven't measured yet. I believe that externally, there is no difference in your scenario.

I think where there might be a difference, is to the consciousness itself. The continuity of that experience being the difference. In which case, gradually replacing the brain should be a solution to the continuity problem (like the Ship of Theseus). Where I think this is conceptually weakest, is when considering how much continuity of consciousness exists while sleeping or being sedated.

The structure of the body more or less remains the same. Ship of Theseus hits the same limits evolution does. We don't really become a completely different species from local changes.

What is the point of information if you can't embody it? I guess the desirability of immortality in a computer system completely relies on how faithfully the system can pretend on your behalf, that you have a body. An endless heaven without having a body to structure your sensations and grant you movement through space and time... seems lacking.

I always thought it's an obvious and solid solution. And if we assume it's possible to do a snapshot and have a perfect mind clone in I can't see how there can be any doubt it'll work.

There's a 32 light year volume that encompass my whole history. The universe seems to preserve every point of my life to a resolution. If https://en.wikipedia.org/wiki/Entropic_gravity turns out correct (or something like it), my life is preserved in perfect resolution across an increasingly large volume of space.

Separately: There's no point in my life where I'm ever the same person. I've continually evolved.

With human history, the resolution of records continues to expand. Fossils, artifacts, writing, photos, and Google search logs all contain personal information. The resolution of historical information shows a clear trend.

Am I immortal if only a moment of my life is preserved, or am I immortal if all points of my life can be accurately be resimulated?

I would argue there is no difference. In order to resimulate me at any point in time, my context, my surroundings, must be considered. My context exists for historic reasons. In order for my brain to be interpretable its application in real world interactions must be considered the the world can only be understood by knowing its history.

To know me you need to know my life and surroundings at all points in time. Technology is slowing increasing the resolution of human immortality, although the universe seems to have already accomplished this perfectly.