We do not yet know “what matters by our constitution” as SCOTUS has not yet ruled on an on-point case other to say that “there is doubt” as to whether persons who do not have two citizen parents and born on US are NBC.
The 1948 BNA explicitly excludes Barry from being a UK subject if he was the illegitimate child of a bigamous marriage.
See 1948 BNA:
(2) Subject to the provisions of section twenty-three of this Act, any reference in this Act to a child shall be construed as a reference to a legitimate child; and the expressions father, ancestor and descended shall be construed accordingly.
Obama’s people would love to have concerned citizens following every possible thread. But we have Barry's assertion about who his father was, and his claim in “Dreams...” that he has seen his birth certificate, and thus knows that he is legitimate. So unless you represent a group of people wishing to challenge the legitimacy of the marriage of Barry's parents, it makes no sense to question the status of Obama Sr’s marriage to Stanley Ann. It makes as much sense as attempting to discover if Stanley Ann forsook her U.S. citizenship as did so many 60s radicals. To claim that Obama’s legitimacy to the presidency is only valid if his father can be shown to be a bigamist is not consistent with the intent of the founders that the allegiance of the commander in chief be insured, and that there be no legal or emotional ties to any other country.