Paper Dress

yah, i think this song is about how being in a relationship of some kind and screwing up something really important, and no matter what, that problem will always be there, and not forgotten??

a paperdress is what a female wears at the doctor's.

This song makes me think of incest when I hear it. Like when it says it would redefine the perfect function it makes me think that he meant that it would make the family bond more dysfunctional and now it will always be there haunting them.

The most obvious thing is science fiction based, similar to the story line of "V for Vendetta" or "Project Alice" in the "Resident Evil" series of games; it would be a recount from a scientist about creating super beings. As mentioned by another commenter, a paper dress is a garment worn by patients, which could allude to experiments (as well as that she looks the same, yet is not inside). "Underneath" then specifically refers to her body, and "we're different" means that something has both affected the narrator and girl physically. "...redefine the perfect function [of the body] to realise some kind of potential" then quite clearly marks that any such operations would be to improve the subjects physically. "Shoulders and neck shrug away this distance" could then refer to how they are able to handle what would be great burdens to other people with ease. "Now in my home and in my car I can't help but hold my thoughts eventually drink back to this day" then would definitely refer to the narrator holding fear or regret for the operation or it's effects, and having a drinking problem on account of that. "Wherever we go this ghost will follow" then also refers to how the narrator cannot escape the memories of the operation. Also, the girl might have amnesia and be a threat to the narrator, as otherwise "she knows we're different" would be insignificant; why would the narrator mention it if not?