Okay, so after trawling through the logs and their order, a few things jump out at me as being, for lack of a better word, odd. (Yes, I'm fully aware that this is just a game; I find it very compelling and a great deal more detailed than the plot of most games, so I look at it with a sharper eye. That's all it is. What? Stop looking at me like that. I SAID STOP!
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The Bayliss-Polito log asking about your memory restoration is, in fact, on the sixth. This would mean that Goggles has been unconscious for some time already, else Bayliss wouldn't have any idea who Polito was even talking about. The only evidence of Shodan's influence up to that point is the Polito log on the 4th, and that only because the log is abruptly cut off. It's possible, in fact quite likely, that Shodan's control at that point only extended to Polito's local terminal, as I kind of doubt an AI expert who already doesn't trust them would start playing around with an alien one without putting some pretty serious security precautions up. The first contact Shodan makes to anyone is to Marie, on the 11th, and I can't see her influence on the plot at any other point until then. I like to believe that this means Polito waited until the 10th to release Shodan, out of desperation, then realized that either she wouldn't succeed, or she would and that would be worse. Hence the suicide. It makes a bit more sense than the idea that Shodan, who is not known for her patience, would hang about quietly preparing to make her presence known for almost a week while the Many grow more and more powerful and start subverting control of Xerxes for their own. Shodan needs the sim units reprogrammed so it makes little sense to me that she'd quietly sit around waiting while Malick brings them out of her reach. Also, the Many themselves don't seem to be aware of her presence until after the 11th.
Bloom was taken on the third, Loesser noted the disappearance on the 5th and saw the first midwife working on the eggs on the 8th. Murdoch mentions the swarm eggs on the 9th and the biomass/worms are first mentioned by Delacroix on the same day. According to Rebecca, people started disappearing since deck 3 was cleared, so we can assume the first hybrids started showing up soon after. What's interesting is that according to Watts's log, Watson had been infected "since returning from the surface of Tau Ceti Five" yet dies on the 7th; meanwhile Turnbull's infection takes just a few hours. I don't understand why. It's possible the transformations all take a great deal of time as the break between the two logs from her are about 4 days, and Korenchkin kept calling for more victims at least up until the 7th. Korenchkin himself spends a lot of time doing nothing; the first log you here from him where he's acting like an annelid slave is the 28th, though he has obviously been infected since, at least, the 20th. This would go a great deal to explain why it took so long for things to erupt into violence, yet how quickly the Many aserted control once it started.
Malick hacks into the sim units on the 6th, a day after they start shooting people. However, the Yount log makes mention to it having taken "12 hours to get it online the last time." So it's possible someone else had been enslaved the day before, which would explain what caused the turrets to go nuts--but not why, really.
A lot of people seem to be very much out of the loop; almost everyone who's not involved directly in either helping or hindering the Many's takeover seem to not even be aware that it's happening. Sanger is warned of the danger she's in on the 5th and appears to take it to heart, as you don't see anything from her until the 10th. By contrast she seems to have been one of the smarter ones; though Rebecca mentions "a kind of gloom hanging over everybody" on the second, apparently a few people didn't get the memo. Watts is still acting like nothing's wrong, despite his own patient's unexplainable death, and expecting to chill out with his pals on the Rec deck after his ill-fated autopsy. Bronson's martial law notice goes out on the 8th, but apparently not everyone got the memo as Yount isn't killed until the 9th and Rosenberg is still hanging around making a nuisance of himself. Even on the 10th, when Watts, Watson, Bronson, Polito, three Malones, Cortez and his staff, Malick, the Younts, and probably Sanger and Miller and Bloom as well as several unnamed extras have all been killed, Rosenberg is still acting like nothing is particularly out of the ordinary. Either information really has a terrible time traveling on this ship, even with the modern conveniences of video email, or he's the most oblivious person ever. Then again, so is Rebecca, as she doesnt' seem to know anything more than people are shooting each other by the 11th, and doesn't kill her first spider until the 12th.