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Why does Hei have such a large appetite? He's eating for two!
Amber is more of a Magnificent Bastard than she initially appears. Amber pretty much knows the future, and thus the reason she had no problem having Maki and Wei in her group, is that she knew both of them would act against her wishes and die in the process. It's also interesting now that the Gaiden has shown Amagiri to have survived. In the first season, it looked like while Amber had tried to do something to save both Amagiri and Brita, it didn't work, and both were burned to death. Now that it's clear Amagiri survived (but ended up badly scarred), maybe Amber was only trying to save him—because he was loyal to her and the nicest of her main supporters—and she succeeded.
This show does not like to explain things, so a few things listed as fact on its page are the product of Fridge Brilliance. Namely, that Mundane Utility moment in the first episode: we just see that Li smacks a broken TV, some sparks fly out of it and he jumps, and all of a sudden the picture clears up. Since it takes a while before they explain much about who and what he is, you may have to watch it again before you realize that he had his eyes closed when he laid his hand on it; Hei used his Shock and Awe abilities to fix the TV, and the Eyes Always Shut smile hid the telltaleredglow.
Anything launched high enough into the atmosphere disappears without a trace. NASA and the space agencies of other countries would find themselves defunct, except for the fact that in the new sky every star is connected to a contractor. If a contractor uses their powers, their star shines brightly. When a contractor dies, their star falls from the sky. Analyzing the sky is the main way to determine what is happening with Contractors, so NASA and other governmental space agencies, (which should have fallen apart when the stars vanished and space essentially stopped existing) have become intelligence agencies whose sole purpose is to analyze the sky and the stars to tell what is going on with the Contractors.
The tape recorder:
Hell's Gate is Mind Screw incarnate, to the point where the characters know and comment on it. But a bit of sideways What Do You Mean, It's Not Symbolic? logic and remembering that its main rule is, "It'll give you back what you lost, but for a price" can make sense of some of the oddities that happen around it. Misaki's reappearing tape recorder, for example. She lost it, needed it, and wanted it back badly, so when everything went really weird, it showed up back in her pocket. If you want to get overanalytical, she finally got the opportunity to reveal the crimes of people who thought they were above the law, and in exchange she lost her illusions about a person she liked and even loved.
Quite a few bits of the (first) Gainax Ending actually make a surprising amount of sense if you think about it way too much. For instance, special attention is drawn to the fact that Kirihara gets her tape recorder taken when she goes through Pandora security, but it appears in her pocket a little later, when she challenges Director Horai. The reason? Amber put the recorder back in her pocket while time was stopped. Why Amber? Because in Gemini of the Meteor, Amber's message concerning the Mitaka Document is on that very same recorder.
In episode 9, Hei (undercover in a hotel) is talking to Saitou (much more clumsily undercover in the same hotel) about Kirihara. There's an exchange along the lines of: "She's beautiful." "Yeah, but once you get to know her, she's actually pretty scary." Now think about who one of theparticipants inthis conversationis.
It's also Remembrance Day, which honors those who died in the first world war.
Until Evening Primrose shows up, November 11 is the only Contractor badass enough to survive more than a two-parter fighting Hei, making him essentially the main villain for a while. Based on that, his white suit and light-colored hair and eyes could be a deliberate contrast with our hero, the Black Reaper. The same thing goes for Hei's Morally Ambiguous Counterpart Nick.
At one point during Heaven's war, Hei was apparently about to strangle Pai as she paid her price when Amber walked up and started taking about how cute/romantic they looked. Later, when Amber and Hei were alone in Hell's Gate, she said that she must have fought Heaven's War a hundred times trying to get it right. That means in several timelines Pai wasn't alive to seal the area and win the war.
Remember Hei's coat and how it's supposed to be bulletproof only when he wears it? Maybe it's made of a fabric or something that becomes rigid when electricity is passed through. (Hei's powers are not 'making electricity' it's quantum manipulation. This is stated in his fight with his morally ambiguous counterpart. It's easy to make electricity when you can manupulate atomic structures.)
In the second season, The original Suou died early on and the copy was born two years before the start of the second series. Since, by his own admittance, Shion has no normal human emotions, he did a lot of guess work in creating her. What results is what Shion envisions his sister would be like had she survived, i.e. a Mary Sue. It's not great for her but it works for the series.
Hei uses his telescope to look for his sister. He knows her star is BK 201, and knows she is still alive because it hasn't fallen. But he is never using his powers when looking through the telescope, so it always appears inactive. And when he is using his powers, he's fighting and is too busy to peek at the sky. To him, it must appear that she is having a quiet life somewhere.
Hei's coat is most likely filled with armor plating, which he no doubt powers himself. and since this series is very Roadside Picnic, scientific research would surely be able to produce a small scale version.