The Electoral College decides who is President and who is Vice President. Not Congress. (Article 2 Section 1.) The Constitution does not specifically make the Presidency expire after 4 years (should the Electoral College fail to elect a President in a timely fashion).  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  Until someone new is lawfully sworn in as President, the current President remains President. The 20th and 25th amendments deal mainly with death or incapacity. Congress is allowed by the Constitution to define rules of succession. The Speaker of the House and President Pro Tempore of the Senate are not explicitly granted rights of succession by the Constitution. Rather, Congress legislated succession be for their own positions (Speaker of the House, President Pro Tempore).  In Case of the Removal of the President from Office, or of his Death, Resignation, or Inability to discharge the Powers and Duties of the said Office, the Same shall devolve on the Vice President, and the Congress may by law provide for the Case of Removal, Death, Resignation or Inability, both of the President and Vice President, declaring what Officer shall then act as President, and such Officer shall act accordingly, until the Disability be removed, or a President shall be elected.  Should an election fail to conclude, none of the following criteria is met: "Death, Resignation, or Inability to discharge the Powers and Duties of the said Office"