But they are living on! Kind of. Like you are, in that body of yours, once all the cells are replaced by new cells every decade or so. Its ok; you still sound like the same person.

Another strawman. No one is claiming that identity is tied to the molecules that make up the body, even in aggregate. There's a sense in which a car remains the same car even after continuing comprehensive maintenance has replaced every single part, but that car stays different from the next car off the production line. Does that example make sense?

Come on! If its a new car, its a different car. Doesn't matter how convoluted the path to get there (replace every part or build new). Not a strawman; an example pointed right at the argument that 'a copy isn't the same thing'. Be fair.

You be fair too. The instance-of-computation model of personal identity allows for cells of your brain to come and go, but as long as the whole thing is operating continuously, you remain. It is exactly analogous to my car example.

Its also exactly analogous to my build-an-entirely-new-one and program it exactly as the previous one was programmed. It has exactly the same result. If I did it without anyone looking, they would never be able to tell the difference.

Except the original that you replaced with your copy. You're opting for an external functional description of identity but the discussion is about the individual.

It should be understand as conceded that a perfect copy of me would pass any Turing-style test applied by an external auditor that the copy is me, but that doesn't mean that I am the copy.

Its different in a sense, sure. But consider: if I replaced it so perfectly that it was atom-by-atom identical, then God himself would not be able to say if it was you or not. Unless we admit to some external agency that defines 'you' that is not present in the mechanism e.g. a soul.