California election law stipulates as long as you sign the registration affidavit, that is sufficient proof of US citizenship.
The cittizenship status is typically not investigated, registration signature becomes the signature on record and the statement "under penalty of perjury" is not verified or prosecuted.
CA Elec Code § 2111 (2017)
A person may prove he or she is a citizen by his or her certification under penalty of perjury on the affidavit of registration.
CA Elec Code § 2112 (2017)
Notwithstanding any other provision of law to the contrary, the fact that a person certifies to his or her United States citizenship by signing his or her affidavit of registration shall be deemed evidence of citizenship for voting purposes only.
https://law.onecle.com/california/elections/2111.html