I'm not sure if lives down means forgotten or that she lives in a floor down the protagonist:
"When your teardrops go sour and no longer fall
the splash cross the virgin that lives down your hall"
Learn English – What does “live down” mean in this sentence
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Related Solutions
Look at the definition of "down" as a preposition here. "He is walking down the street" will mean "He is walking along the street". "He lives down the hall" will mean "He lives on the same floor, along the hallway, a few doors away."
Given that your source says that it was published in The New Yorker, I imagine the author is American.
In Europe, floors are described as follows: ground floor (street level), first floor (first floor above street level), second floor, third floor, etc.
However, in America, they call the street-level floor the first floor. So if the publication is American, then "first floor" will be the ground floor.
As for "down the hall" it means "along the hall" in this context. So it's further along the hall, not necessarily at the end. It could be at the end, but all it says is that it's closer to the end than the current position is.
Here's a diagram for example:
Doors 1-3 could be described as "down the hall" from a person standing at X. Door 5 is right next to him so it wouldn't be described as "down the hall". Door 4 probably wouldn't be described as down the hall since it's a bit too close, but it could be.
Best Answer
In this case, 'lives down' means a neighbor (in this case the virgin), probably within an appartment complex, whose door shares an internal hallway with your own. You would say that said neighbor 'lives down the hall' from you. You run into this phrase at work or schools as well when telling someone to go further 'down the hall' to find the door to their office or classroom. A hotel also would be a place where this phrase could apply as well.
You may also see 'a few doors down' or even possibly 'up the hall' to mean the same thing, though 'up the hall' feels to me like the direction specified is going the opposite direction than what is expected.