Auggie is told "we sponsor your ILR. You can't stay in this country on a work visa if you're not working" but Indefinite Leave to Remain is not a work visa and it does not require an ongoing sponsor. Once granted it is indefinite as the name says.
Jin demonstrates a system with three stars and one planet to the pope and calls it a three-body system but that would be four bodies and then referred to as an "n-body system".
The basic premise of the novel, and thereby of the series, that there is no solution to 3 body problem, is wrong. There are many 3 body systems existing and surviving, with their planet(s), in the known observed universe. Star systems have been found containing up to 7 stars, which are stable, and some of them might have planets also, which also is stable. If there are more than two stars in a system, the nearest two stars each will act like a single gravitational body each and will rotate around the largest star of that system which itself could be a two star acting like a single gravitational body.
It is claimed the three-star system is four light years from Earth. The only system that close to the Earth is Alpha Centauri, which is indeed a triple system (Proxima Centauri is part of it) but the third body is 12,000 AU from the other two stars, so three-body effects would be minimal.
At 33:20, in the game, the three Suns were shown speeding up to come in a straight line from the point of view of the viewer, in this case, Great Khan. Celestial bodies don't move at such speeds, but that could have been attributed to Copernicus (Jin) having placed her hand on the bottom of the vessel of boiling water, that makes time run faster. But, she has already removed her hand from the bottom and was holding the top edge of the pot with one hand, and pointing her other hand to the approaching Suns, when she says "Look, Great Khan". So there is no explanation why the three suns moved at such super speeds. And then there is no reason the three Suns could have stopped moving altogether when they came in a straight line. They should have continued moving with the same super speeds and could have gone on their respective destined paths, giving just a momentary alignment of 3 body. Of course, it is in the game, but the game is supposed to follow real condition of their mother planet who is actually moving in a 3 sun system.