Oh my god thank you so much for the explanation/theory. It makes a lot of sense
oh yes, now it all makes sense. now i can leave the comment bar in peace. thank you
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I don't think Jugemu was a reincarnation. I think that the problem stems from the fact that they look alike, but according to Jugemu, he was born somewhat before Edo period or during the Edo period and it seems as the other guy's story also happened during Edo period. So if Jugemu was already alive he couldn't have been a reincarnation.
Also, usually in mangas if a reincarnation were to happen, some sort of acknowledgment from other characters is seen. Like mentions of their similarities and things like that. And in this case, where it involves a Shinigami, even more.
In the first story, the bum was told he would suffer divine punishment. I think his punishment was not dying when he was reincarnated (after some time, he thought of it as a curse, and really wanted to die). Then, when he met the shinigami in the third story, he was again advised he would face divine punishment. I think that came in the form of him aging and dying when at that point he really wanted to live to be with the shinigami