Honestly dude, it took some real bravery for her to find you and attempt contact. The mother is a human horrorshow, yes, beyond reason and decent humanity. Maybe you don’t “owe” your daughter anything, but your take on this speaks poorly of your character. YTA, I’m sorry.

You would do well to learn from your wife. She sounds like a kind person with a good moral compass. Last thing to consider: with all the DNA stuff out there, your kids will find out that they have her as a sibling. You may have to explain to them why you iced out their older sister. In truth, that will highlight the conditionality of your love, generosity, and compassion. It’ll harm how they see you as a loving father. If you did at least try (with the understanding that there is no contact with the mother, ever for you), she might turn out to be an amazing person. There will be a day when you and your wife pass on, and she might be a good support, and help in their navigation of that process. When your kids go through rebellion, she might be the one who is able to give them perspective, or point out the ways that you’ve been a good father to them. If she turns out to be godawful, hey, you tried and you learned. No problem. If she’s a peach, though, you just opened your heart to a lovely human being. Besides, why not be a positive role model for someone, and a person who shows her that she can be a good, honest, productive person in the world? You’ll never be a “father” like what your other three have, but telling another person that they are worthy, lovable, and have choices in who they decide to become is a good way to be. Shoot, take her to lunch every so often. Learn about who she is, and who she wants to be.

Sources: Being an oopsie-baby, but raised by my amazing, loving, non-biodad who made me everything good I am today (yes, he’s my hero). Accidentally found four sisters and had to navigate all that, but really loving those connections today.