Should have been, “You can’t just say the claim ISN’T true and expect any reasonable person to believe you unless you’ve got sources that trump the 15+ that I’ve given to document the claim.
All the 15+ documents you posted say is that the DOH is supposed to say no record exists if no record exists. They do not say that a denial of access is tanemount to an confirmation that a record exists.
A DOH denial of access to amendment records could mean such records exist, but it also could mean it doesn't exist and the DOH bureaucrat screwed up and didn't do what he was supposed to do.
But of course, in the birther fantasy world, bureaucrats never screw up. They always follow protocal, and any violation of must be attributed to some nefarious conspiracy theory, never mere incompetence.