They have done neither. They can't by law. One official said the Certification of Live Birth(NOT Certificate) posted on line *appeared* authentic, and two others (or one may have been the same as the first) said they had seen that there is Certificate of Live Birth for Obama. However there are ways, both legal and illegal, that could be true, and he would still not have been born there. The certificate *should* have his place of birth, including the hospital if in Hawaii, date and time, parents names and places of birth. Then current residence address of the mother, name and signatures of the attending doctor, nurse, midwife or other person attending the birth, as well the "local registrar" (Hospital administrator for example) It should look pretty much like this:
That one is from '63, only two later than Obama's should be from. But if not born in a Hawaiian hospital, the date of birth and the dates of registration could be over a year apart. The place of birth might not even be in Hawaii. If either of the last two is true, then block 23 would have supporting information, as would a certificate issued upon adoption, where the parents names and information would that of the adoptive parents. Place and date of birth is supposed to remain correct in those cases though.