How we cite our quotes: Citations follow this format: (Chapter.Paragraph)
Quote #4
I wished to find a warrant for being. I need no warrant for being, and no word of sanction upon my being. I am the warrant and the sanction. (11.4)
Equality 7-2521 has now explicitly rejected the principle of his society that he stated in the first quote. He no longer needs a justification for being alive. He himself, and his happiness, is his own reason for being alive. This is tremendously liberating, because it means he no longer needs to feel guilty about being alive, or pursuing his happiness. He decides that is the only moral thing to do.
Quote #5
It is my mind which thinks, and the judgment of my mind is the only searchlight that can find the truth. (11.5)
Equality 7-2521 has made a bit of a venture here into Rand's philosophy of how real knowledge is arrived at (which she developed much more in her later works). It is individuals, who learn about the world by studying it carefully and seeing what it is like. Society, on the other hand, has a tendency to conceal the way the world is, because it tries to force everyone to understand the world the way it thinks it should be understood. In Rand's philosophy, individuals are the real bearers of truth.
Quote #6
And my happiness needs no higher aim to vindicate it. My happiness is not the means to any end. It is the end. It is its own goal. It is its own purpose. (11.8)
Individual happiness, as Equality 7-2521 says here, is the highest goal of life. It is an end in itself. There is no larger moral purpose to life, other than individual happiness. This is the central ethical idea of Rand's egoism, and it's radical. It's opposed to any philosophy that holds that individuals can or must serve a higher purpose than their own happiness. No such obligation exists.