How about a record that he was ever adopted?
I agree that we do not have that record as well nor either a mention of whether he was or not by Obama himself or anyone in the media.
Being that it has been claimed that he was born with the name Barrack Hussein Obama but was registered in school in Indonesia as Barry Soetoro (as an Indonesian citizen) then there is very strong evidence to suggest that he was adopted.
There are also accoutns that he didn’t start using the Obama until much older as well. Did he just choose to start using this name but never have it legally changed?