I’m genuinely curious if my ex-boyfriend has any rights to my son? He’s not his bio dad but he signed the birth certificate

I found out I was pregnant in August of 2022, I thought it was my ex-boyfriends so we got together. I told him from the beginning that there was a chance this baby wasn’t his, because at the time we were both doing our own thing and messing with other people (we weren’t together, we thought it was his so we got together)He didn’t care at the time and decided to step up and be the baby's dad. He said that even if the baby wasn't his he’d still help and take care of it like they were his. My son was born in April of 2023, knowing there could be a chance he wasn’t his, he still signed my sons birth certificate. 3 months after my son was born (around July)we broke up. End of August we got a paternity test done, turns out he’s not his, does he have legal rights as he signed the birth certificate? Can I take him off the certificate? Can I get child support? He doesn’t do anything for him so I’m genuinely curious.