How we cite our quotes: (Article.Sentence)
Quote #1
It having pleased the Divine Providence to dispose the hearts of the most serene and most potent Prince George the Third, by the grace of God, king of Great Britain, France, and Ireland, defender of the faith, duke of Brunswick and Lunebourg, arch-treasurer and prince elector of the Holy Roman Empire etc., and of the United States of America, to forget all past misunderstandings and differences that have unhappily interrupted the good correspondence and friendship which they mutually wish to restore, and to establish such a beneficial and satisfactory intercourse, between the two countries upon the ground of reciprocal advantages and mutual convenience as may promote and secure to both perpetual peace and harmony. (Intro.2)
This is subtle, but the first admittance of legitimacy is right there in the first sentence, not once but three times. The first is implicit: you don't make a treaty with yourself, so any treaty is by definition going to be with another country. Secondly, the United States of America are named as an equal power to King George III. And lastly, the treaty's stated purpose is to restore friendship and so on between two countries.
Quote #2
His Brittanic Majesty acknowledges the said United States, viz., New Hampshire, Massachusetts Bay, Rhode Island and Providence Plantations, Connecticut, New York, New Jersey, Pennsylvania, Maryland, Virginia, North Carolina, South Carolina and Georgia, to be free sovereign and independent states, that he treats with them as such, and for himself, his heirs, and successors, relinquishes all claims to the government, propriety, and territorial rights of the same and every part thereof. (1.1)
Granting independence is a sure indication of legitimacy on the part of the treaty. The King, under no uncertain terms, is calling the former colonies a country in their own right.
Quote #3
And that all disputes which might arise in future on the subject of the boundaries of the said United States may be prevented, it is hereby agreed and declared, that the following are and shall be their boundaries. (2.1)
The second Article goes through the boundaries of this new state. Weirdly, by placing boundaries on the United States, it makes the entire thing more legitimate. After all, an infinite border would be meaningless, impossible to enforce, and really just kind of silly. Making concrete edges defines the United States as a real place.