Taxes and BD😒

My bd is threatening me saying he's going to claim my sun on his taxes even tho I have automatic full custody. We don't even live in the same apartment*(see below) let alone the same state. He hasn't asked about this kid in almost three months since I've left him and ALL OF A SUDDEN he messages me demanding I give him the baby social security number so he could claim him. Mind you, I've already put him on child support thru my benefits and I was assured that he would not be able to get money from taxes cuz of that.

He said that since I signed a paper starting we were on his lease that he could claim him from that. (From above👆) we have only lived with bd for 6mo 16days and I have done ALL I could to make sure this kid was taken care of (doc appts etc) nothing documenting bd providing ANY financial support other than a roof over our heads. This kid been with me since birth so...

I ask you...

Can he even find a way to claim my sun?

ETA:

Yes He's the bio-dad but he falsified data on the BC when he signed my sons name as the "father"