SatchelSF, I bolded the parts of the statute I think relevant.
quote A person who is a national of the United States whether by birth or naturalization, shall lose his nationality by voluntarily performing any of the following acts with the intention of relinquishing United States nationality—
[(1)-(6) don’t apply here]
(7) committing any act of treason against, or attempting by force to overthrow, or bearing arms against, the United States, violating or conspiring to violate any of the provisions of section 2383 of title 18, or willfully performing any act in violation of section 2385 of title 18, or violating section 2384 of title 18 by engaging in a conspiracy to overthrow, put down, or to destroy by force the Government of the United States, or to levy war against them, if and when he is convicted thereof by a court martial or by a court of competent jurisdiction.
(b) Whenever the loss of United States nationality is put in issue in any action or proceeding commenced on or after September 26, 1961 under, or by virtue of, the provisions of this chapter or any other Act, the burden shall be upon the person or party claiming that such loss occurred, to establish such claim by a preponderance of the evidence.**
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First, the person has to commit the treason, etc. with the intent of relinquishing her citizenship; it’s not enough to just do it. Second, she has to be convicted of the treason etc. by a court martial (irrelevant here) or a court. In other words, she gets due process. A cabinet secretary making an assertion is not a conviction in a court.
I’m not sure why you are so certain she wasn’t a citizen at birth. It seems to me she’s entitled to a hearing about that. It’s undisputed she was born here, so the issue is the citizenship status of her father at the time of her birth.