According to the Indonesian Law in effect at the time, even had Obama been adopted he could not have been a citizen.
According to Article 2, Section 1 an adopted child becomes a citizen automatically only if he is under age 5. After that he or she has to wait until they are 18 to apply for naturalization. Obama moved to Indonesia in 1967 when he was 5 or 6. Link
He appears to have registered under the name Barry Soetoro.
How do they know that?
If 'b' then, according to INA § 349, 8 USC § 1481, Obama isn't even an American citizen, let-alone "natural-born."
Except that he would have to inform the U.S. authorities of that. Which would leave a paper trail.
If Obama was adopted in Hawaii by by Lolo Soeroto, it could explain why he won't release his original birth record.
As an Indonesian citizen, Soetoro could not adopt Obama in a U.S. court. The adoption, if it happened at all, would have had to be done in an Indonesian court.
Or, his passport information