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My personal theory & thoughts about the roles of The Book.
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Atsushi = As we know now from chapter 119, he’s The Bookmark, representing the present– what's happening now in the story, marking where the story was left off, where to go back to continue reading, the function of a bookmark. He is also the guide/guidepost to The Book– this in turn led to many who were searching for The Book (The Guild and Fyodor) to come to Yokohama where it was believed to be.
But-- Here's my idea:
Natsume = Outside force, an Author (as seen in Oda’s flashback, he’s the Author of the books Oda was reading when he was a kid– it was the day they first met) or the Narrator of The Book. He was the past, before everything– before the Tripartite Framework was even made. He was the one protecting Yokohama for years before the Armed Detective Agency, Port Mafia and the Special Division for Unusual Powers. And Yokohama’s been seen as the place where The Book is in, all well as a page that was taken out of it that was being researched by the Special Division for Unusual Powers.
Oda = Reader, would have been the next Author or Narrator after Natsume–he did after all give up killing to become a writer, which was given to him by Natsume-- who book he was reading when he first met him! He was supposed to be the future (like how his Ability could see into the future, even for a few seconds), or timeline wise, be the present that the story is now. But after his death, the story couldn’t go anywhere, there's no one to keep writing the story Natsume had left– or so you’d think but he did help the story by creating a new path for the MC. Giving the MC of The Book a new drive to follow the new path, to keep living and becoming someone who would see the beauty of saving lives instead of ending them.
Dazai = The Original Book MC, the whole of the BSD story was supposed to be about him. I can’t be the only one to realize how in nearly every light novel Dazai or his presence in it, shows how important Dazai is to the world of BSD. Especially in “Bungou Stray Dogs: BEAST”, where BEAST!Dazai was able to see the original/main story of BSD and made a world where Oda– The Reader turned Author and more importantly his Best Friend who was the reason why he left the PM in the original story– was alive, The MC created another world after discovering the plot, changed said plot to ensure that his best friend would live in these one. And isn't it funny how the first person Atsushi meets in the beginning of story was Dazai, The Bookmark catching up with/following the MC, saving his life from drowning to keep the story going because there's still so many pages left, this is not where your story ends.
Sigma = The next MC after Dazai, he was or is likely supposed to be a sequel to Dazai’s story once it ended– I say this simply because he was created by the book.
Fyodor = Original Author, he created the story and world of BSD– or at least the backstory around its world. About how Ability Users came to be and other stuff. Now he wants to rewrite the story– the tipping point was losing his last reader and that's why he does what he does in the story.
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So yeah, that's my bit, please take this with a grain of salt, I just wanted to put something out there. Hope you like it!