"some left their poor die" -> "let" or "poor to die"
My main input is that I don't like the name "Svartalfheim", too much fantasy with the "svartalf"-part.
The cities here are built according to the "Sub-Terran" architecture : vast underground volumes, usually former quarries and mines, provide a public space surrounded by subterranean buildings.
This makes me think of Asimov, who referred to this sort of habitat as "steel caves" or "caves of steel", in the novel with that title. Maybe something you want to throw in? "... quarries and mines forming steel caves of human haibtats"? You decide if you like it, I'm just throwing ideas, feel free to ignore.
Hmm, OK having read your full description to end, I guess "steel cave" gives a more claustrophobic impression, where as this world is large and high on volume, so my suggestion might be something for another future system.
Speaking of future system, I thought "Nimis" might be a name one could use for some future system, it's a large structure built of drift wood in Sweden, making up the fake nation of "Ladonia" [1].
I have a list of names to use for systems from mythology, or sci-fi, literature, that I've collected over the years, but I've never had the motivation to hand craft systems, plus I don't write as good back stories as you.
More name-inspiration from my reading today: "The Panic of 1825 was a stock market crash that started in the Bank of England, arising in part out of speculative investments in Latin America, including the imaginary country of Poyais." [2]
[1]
https://en.wikipedia.org/wiki/Ladonia_(micronation)
[2]
https://en.wikipedia.org/wiki/Panic_of_1825