That “something written down” in the archives just might be a form filled out by somebody who did one of those “home birth” delayed cert.
That could have been done anytime, given the corruption and collusion in that state and dept.
Or, just one person, attending physician David Sinclair, MD. did the Dunhams or Obama Senior a favor and filled out a birth certificate and had Stanley Ann Dunham sign it and he then submitted it to the state Health Bureau with all the other August 4, 1961 birth certificates from Kapi’olani Hospital and it was reported in the local newspapers and it remained in the vital records files for 47 years.
Sometimes the simplest explanation is the best...no grand conspiracy, just one person with access and authority helping out a friend.