Back when Q&A was in production, there was a casting call for school kids. Led to this theory that post-MIA, The Machine'd have Root be volunteering as a substitute there. She'd be initially against it of course, but would keep the facade going as long as possible (The Stepford Smiler). It would've also been a way for Amy to play something true to her own real life demeanor.
Figured Finch'd try to get her back in the game which she'd decline. But, she'd keep a close eye on him, and would end up saving him from the threat of the episode (which I thought was Greer, going by a BTS shot), without the Machine's guidance, to illustrate character growth. Even considered Uncle John biting it in that episode, as a message to Samaritan that the team's done hiding. Felt it would kick off the second half of the season in a really effective way, and would also introduce either Greer's Decima superior(s) (the dude who he was talking to in the phone in Trojan Horse, unless the old man wiped out the entire company's higher-ups after the S2 finale and spearheaded the Samaritan mission himself), or the whole '001 Control' & '002 Control' entities (as well as 'Admin_Control <LOCKED>, which maybe the superior of them or their collective name'), that are always seen on the SPOV screens. I think they might refer to the 2 remaining people of the six within the Government (Control, Special Counsel, Denton Weeks, Alicia Corwin) who knew about the original Machine. But I maybe wrong.
On another note, if 13 episodes is all we get this season (not the last), I'd go with The Machine re-awakening in the finale. And, also playing on the aspect of evolution, I'm also game for the whole physical embodiment aspect to serve as a cliffhanger to close the season, where one of the leads die, and their body is given a cybernetic makeover to serve as the Machine's host. The visual of it opening its eyes, the MPOV text displaying the date & 'Day 1', and the first thing it sees is Harold would make for a wonderful callback to both The Contingency & Prophets, too!
For a hard science approach on this, I'd tie it with recent developments in cybernetic technology. Hell, it could even come from the Thornhill Corporation. The Machine's had it running for atleast 2 years, and given how it funded the establishment with just micro-transactions, it is plausible enough to assume that it had a bunch of side projects on this front, much like the Utilities one for spreading itself in the power grid.
On a disturbingly note, Episode 13 would be Number 103.... the same as Angel's 'A Hole in the World'. Could history be repeating itself here?
On a creepier note: What if the Machine were to physically embody itself through Root (either after she sacrifices herself for it, or as the next step of its evolution), and Samaritan were to do the same through Shaw (either as failsafe to ensure its survival after its destroyed like how The Joker did to Robin in the Batman Beyond flick Return of The Joker, or when it's still active)? Artificial Super Intelligences waging war in the avatars of two women who love each other? How is this not a subject of slash fics already?!
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http://anthony258.deviantart.com/. Just found out that my profile pic is her work too.