The sworn statement directions in the U.S. code show that U.S. or United States is a federal corporation or the federal gov’t “in congress assembled”. It’s an entity that manages certain things for the 50 states, that are collectively called The United of America, not United States.
Here are the sworn statement directions. Notice only the first mentions U.S. of A.
Wherever, under any law of the United States or under any rule, regulation, order, or requirement made pursuant to law, any matter is required or permitted to be supported, evidenced, established, or proved by the sworn declaration, verification, certificate, statement, oath, or affidavit, in writing of the person making the same (other than a deposition, or an oath of office, or an oath required to be taken before a specified official other than a notary public), such matter may, with like force and effect, be supported, evidenced, established, or proved by the unsworn declaration, certificate, verification, or statement, in writing of such person which is subscribed by him, as true under penalty of perjury, and dated, in substantially the following form:
If executed without the United States: “I declare (or certify, verify, or state) under penalty of perjury under the laws of the United States of America that the foregoing is true and correct. Executed on (date).(Signature)”.
If executed within the United States, its territories, possessions, or commonwealths: “I declare (or certify, verify, or state) under penalty of perjury that the foregoing is true and correct. Executed on (date).(Signature)”.
Without the United States means one is not working for/in the federal government, therefore one must reference U.S.of A. to give context.
Side point is that supposedly no one should ‘swear’. Never ‘swear’ on the bible as that is blasphemy against their rule book and immediately may place thee in contempt of court.