Let's see... neither the Constitution nor the Declaration of Independence explicitly use the word "republic" for the United States as a whole, so you should perhaps reconsider that argument.
Constitution, Article 1, Section 2, and Section 3, in their first paragraphs clearly define the legislative branch as a representative democracy.
Constitution, Article 1, Section 2, and Section 3, in their first paragraphs clearly define the legislative branch as a representative democracy.