Registration papers mean NOTHING as far as ownership.
Incorrect. The registry goes by who is listed as the owner of the animal. In order for the animal to be registered to the new owner, the previous owner has to sign the transfer line. Otherwise, the animal will stay registered to the previous owner, and the new owner may not attend breed shows with the animal.
Have a new coggins pulled on the gelding IMMEDIATELY IN YOUR NAME!
Coggins do not denote ownership. The merely indicate who authorized the coggins pulled. A coggins cannot be used as a legal document proving ownership.
A signed contract or bill of sale denotes ownership, and will trump registration papers. No contract or bill of sale, and the registration is considered to be the fallback document proving ownership.
You can claim abandonment if the animal has been left in your care for a certain amount of time, but you'll need written, verifiable proof. Such as receipts for vet care, feed, farrier, etc.
If all those are in order, you can get a legal ruling in your favor and obtain a hardship registration transfer.