From Wiki
In the legal system of the United States, the Supreme Court is the final interpreter of federal constitutional law, although it may only act within the context of a case in which it has jurisdiction.
NBC clause in our Constitution.
No person except a natural born Citizen, or a Citizen of the United States, at the time of the Adoption of this Constitution, shall be eligible to the Office of President; neither shall any Person be eligible to that Office who shall not have attained to the Age of thirty-five Years, and been fourteen Years a Resident within the United States.
The House is not allowed to reinterpret the meaning of the constitution. That is the pejorative of the Supreme Court
Again from Wiki
Under Marshall, the Court established the power of judicial review over acts of Congress,[10] including specifying itself as the supreme expositor of the Constitution (Marbury v. Madison)
The House can do whatever it wants but if the Supreme Court rules at candidate for POTUSA is not a NBC then that candidate is not eligible for the office and will not be placed on the ballot.
See same sex marriage for a recent example of the Supreme Court forcing it views on an unwilling Congress, an unwilling several states and an unwilling nation.
https://www.law.cornell.edu/constitution/articleii#section1
US Constitution, Article 2, Section 1:
The executive power shall be vested in a President of the United States of America. He shall hold his office during the term of four years, and, together with the Vice President, chosen for the same term, be elected, as follows:
Each state shall appoint, in such manner as the Legislature thereof may direct, a number of electors, equal to the whole number of Senators and Representatives to which the State may be entitled in the Congress: but no Senator or Representative, or person holding an office of trust or profit under the United States, shall be appointed an elector.
The electors shall meet in their respective states, and vote by ballot for two persons, of whom one at least shall not be an inhabitant of the same state with themselves. And they shall make a list of all the persons voted for, and of the number of votes for each; which list they shall sign and certify, and transmit sealed to the seat of the government of the United States, directed to the President of the Senate. The President of the Senate shall, in the presence of the Senate and House of Representatives, open all the certificates, and the votes shall then be counted. The person having the greatest number of votes shall be the President, if such number be a majority of the whole number of electors appointed; and if there be more than one who have such majority, and have an equal number of votes, then the House of Representatives shall immediately choose by ballot one of them for President . . .
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My interpretation is that this gives the states control over ballot access and Congress jurisdiction over presidential eligibility. Neither is subject to the Supreme Court on this issue. When the states and then Congress allowed the communist to take power on January 8, 2009, that settled the question of whether an anti-American traitor of questionable background could hold that office. The same determination will be made on or before January 6, 2017 - if the President of the Senate, in front of Congress, counts the ballots and recognizes the votes of the electors, then the question is settled. [Under the generous assumption that the Supremes stay within their proper bounds.]