Can My Ex Boyfriend Take Me To Small Claims Over Mortgage Payments?

location: Columbus Ohio

I bought my house at the end of 2021 and my then boyfriend moved in with me at the beginning of 2022. He had been adamant that he wanted to help pay half of everything (mortgage and utilities) in spite of him making less money than I do, but I never agreed to add him to any legal paperwork unless we were getting married. Every month, I went through the bills and totaled everything up, gave him the number, divided it by half, and that's what he would pay me. We hadn't wanted to call this "rent" since we both wanted to consider the house ours, but it still felt like rent imo.

Unfortunately, I ended the relationship earlier this year and have asked him to move out. I wanted him to immediately stop paying me and use that money to save, find an apartment, new house, etc. and agreed to let him stay with me for a short period while he got his finances and new living situation arranged.

He had asked for approximately six months to move out, which I finally agreed to, but last night, he mentioned that it was unfair that he had paid money into the house when it was no longer going to be "his" and alluded to me paying him back. While I agree that the whole situation sucks and seems unfair, I said that you don't get to ask your landlord for all the money you spent in rent for the years you stay in an apartment. His argument is that we had verbally agreed that the money given wasn't rent per se, and that he could potentially take me to small claims over this because he was now losing out on the equity he had been building with paying on and investing in a house.

My overall question is this: should I seek out a lawyer to protect myself from this? and, Does my ex have a valid legal argument? Because he isn't on the mortgage and because we never had a written agreement, I didn't think he was actually able to do this, and that courts would generally view any money exchanged as "gift money" but I honestly don't know. At this point, I'm concerned that I'm going to have to pay him 20,000 in mortgage refund to get him to move out.