> believes that as long as continuity is maintained between the totally biological brain and the synthetic rebuild, there will be the same person looking out from the inside.

So when you fall asleep at night, you die, and are replaced by an imposter the next day with the same memories?

If not, how about people who undergo deep hypothermic circulatory arrest, in which heart and brain activity completely cease?

I don't believe continuity of consciousness is required for continuity of personhood.