A Republic ceases to be a Republic when it creates and enforces unconstitutional law. The most recent unconstitutional law being enforced is, of course, Obamacare. Obamacare is not a valid law because it does not conform to the Constitution. The Obamacare law was partly written by the Supreme Court and the Supreme Court does not have the constitutional authority to write law, only to judge whether the law is constitutional.
A Republic also ceases to exist when law is enforced to show favoritism to an individual or group. For instance, the Obamacare law does not exempt large corporations or congressmen, but the president has taken upon himself to do just that, thus exercising dictatorial power.
God forgive us for giving up our Republic and the freedom of our children after so many patriots have sacrificed their lives to preserve it.