> Are you the same person after you take a nap?

Yes.

> Doesn't your consciousness basically dissolve and then reconstitute itself when "you" wake?

Yes.

> What about from second to second, between moments of inattention?

Yes.

> Are we relying on the "stability" of your constituent atoms?

Yes.

> On some "arrangement" of them?

Yes.

> On a particular locus of sensation (the goo in your brain pan)?

Yes.

> What if we "clone" you by copying every other atom into a "left" clone, leaving the original in the right, and doing the reverse on the alternating atoms? Which resulting "clone" is you?

My father replaced the handle, and I replaced the blade, but it's still my grandfather's axe?

Neither. You ripped my brain in half, I am dead.