My (37M) GF's (34F) daughter (13f) falsely accused me of sexually assaulting her. Now I'm proved innocent, my gf wants to reconcile but I'm not sure I want to?

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We've been going out for the best part of 2 years. It was going amazingly well and because of lockdown, we moved in together. She has just the one daughter and I always thought we got on well.

Last week, I got home from work and I noticed her and her daughter we're not in. I didn't really think it was strange, but I then was contacted by the police asking me to go in and see them which I did.

Basically, her daughter accused me if sexually assaulting her. I didn't if course, absolutely denied it and the police asked me permission to search my house which I did. They examined the daughter, didn't find anything and she basically was interviewed and admitted she made it up. She admitted her dad, who's a real wrong un, was telling her she needed to say that so it could get me out the picture and he might be able to be back in her life - he's not been in it for a while and hasn't long started to make contact and spend time with her and we've noticed she's changed a bit lately. Acting out, being disrespectful that sort of thing. If I ask her to do something, "you're not my fucking dad" etc.

Naturally I'm relieved but I'm angry, upset, and just really confused. My girlfriend made contact, saying she wants to move back in and we look into picking up where we left off. I told her no way, because I can't trust her daughter yet and don't really want to be around her and I'm not sure I want her living under my roof. She said she's only a kid, and I shouldn't hold it against her and was being manipulated - I know that's true, but I can't look past how horrible it was. She also says as a mum she did what she has to do and as horrible as it was, she'd do it again in a heartbeat so can't say she apologises. I mean I get that, but it doesn't make anything easier and I can't apologise for how I feel too.

So here I am, I don't know what to do. She's staying with her mum until we can talk but I genuinely don't know what way it'll go. Do I take her back and move on? Or shall I cut my losses?

Edited to add it's bad enough what happened. What's making it feels l worse is that I feel like she should be a bit more empathetic and apologise at least for what happened and what the kid did. But she's not doing that.

Tldr: GF's kid accused me of sexual assault and it later was admitted she made it up. GF wants to get back together and I'm not sure I want to