A presidential candidate's party affiliation

[SillyAmerican]

What are the requirements for running as a Democrat or Republican?

The reason for the question? Something Dr. Carson says on page 158 of his book America the Beautiful: "Over the years, I found that no political party really represented my views of fairness, decency, and adherence to the principles set forth by the United States Constitution of 1787. So I became a registered Independent and have remained so until this day."

As a registered independent myself, I find Dr. Carson's point refreshing: the two political parties have both let the electorate down, and change is needed. What's sad is the apparent admission that this change can't come from outside the framework of the two-party system.

Perhaps John Adams knew what he was talking about when he said “There is nothing which I dread so much as a division of the republic into two great parties, each arranged under its leader, and concerting measures in opposition to each other. This, in my humble apprehension, is to be dreaded as the greatest political evil under our Constitution.”

I believe that the current party structure places unnecessary stress on candidates to adhere to positions that don't make a whole lot of sense, and to bash those of the other party for no other reason than that they're "the opposition". Not a good recipe for governance. So the question in this case boils down to the following: is Dr. Carson an independent candidate running as a Republican (and how does that work exactly), or is he a "true" Republican (and if so, what does that mean exactly?)