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No, you are absolutely wrong.
She needed only his BC to demonstrate his date of birth; nothing else.
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You know that you are incorrect. Everyone here who researched it even a little bit knows that you are incorrect.
Ted was in Canada until the age of four. The parents had a business in Canada, and his father certainly did apply for citizenship in Canada. Seems that the family initially planned on remaining in Canada as citizens.
To establish US citizenship rights for a child born in Canada the US parent had to make an appointment, fill out forms, present certain documents and pay fees. It was a process. You don’t show up with a kid and your passport, and a Canadian birth certificate for said kid, and they just had over a passport for the kid. He didn’t get a passport until he was a teen - per Cruz’ own bio. Ted C had to go through a process to renounce his Canadian citizenship also - forms, documents and fees. Fact; his campaign detailed that process he had gone through.
I read the State Dept website also - it seems to be circular and actually gives conflicting information, if you are reading beyond the snippet that supports what you claim. The Canadian housed US Consulate website gives clearer information. Forms must be completed, submitted, fees paid.