There is a requirement to take the Oath in order to get the Certificate of Citizenship. It is a statutory requirement, clarified in the regulations.
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INA: Sec. 341. [8 U.S.C. 1452] CERTIFICATES OF CITIZENSHIP OR U.S. NON-CITIZEN NATIONAL STATUS; PROCEDURE
(a) A person who claims to have derived United States citizenship through the naturalization of a parent or through the naturalization or citizenship of a husband, or who is a citizen of the United States by virtue of the provisions of section 1993 of the United States Revised Statutes, or of section 1993 of the United States Revised Statutes, as amended by section 1 of the Act of May 24, 1934 (48 Stat. 797), or who is a citizen of the United States by virtue of the provisions of subsection (c), (d), (e), (g), or (i) of section 201 of the Nationality Act of 1940, as amended (54 Stat. 1138; 8 U.S.C. 601), or of the Act of May 7, 1934 (48 Stat. 667), or of paragraph (c), (d), (e), or (g) of section 301 of this title, or under the provisions of the Act of August 4, 1937 (50 Stat. 558), or under the provisions of section 203 or 205 of the Nationality Act of 1940 (54 Stat. 1139; 8 U.S.C. 603, 605), or under the provisions of section 303 of this title, may apply to the Attorney General for a certificate of citizenship. Upon proof to the satisfaction of the [Secretary of Homeland Security/USCIS] that the applicant is a citizen, and that the applicant's alleged citizenship was derived as claimed, or acquired, as the case may be, and upon taking and subscribing before a member of [USCIS] within the United States to the oath of allegiance required by this Act of an applicant for naturalization, such individual shall be furnished by the [Secretary of Homeland Security/USCIS] with a certificate of citizenship, but only if such individual is at the time within the United States.
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8 CFR § 320.5 Decision.
(a) Approval of application. If the application for the certificate of citizenship is approved, after the applicant takes the oath of allegiance prescribed in 8 CFR 337.1 (unless the oath is waived), USCIS will issue a certificate of citizenship.
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8 CFR § 322.5 Decision.
(a) Approval of application. If the application for certificate of citizenship is approved, after the applicant takes the oath of allegiance prescribed in 8 CFR 337.1 (unless the oath is waived), USCIS will issue a certificate of citizenship. The child is a citizen as of the date of approval and administration of the oath of allegiance.
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8 CFR § 341.5 Decision.
(a) Adjudication. USCIS may adjudicate the application only after the appropriate approving official has reviewed the report, findings, recommendation, and endorsement of the USCIS officer assigned to adjudicate the application.
(b) Approval. If the application is granted, USCIS will prepare a certificate of citizenship and, unless the claimant is unable by reason of mental incapacity or young age to understand the meaning of the oath, he or she must take and subscribe to the oath of renunciation and allegiance prescribed by 8 CFR 337 before USCIS within the United States. Except as provided in paragraph (c), delivery of the certificate in accordance with 8 CFR 103.2(b)(19) and 8 CFR 103.8 must be made in the United States to the claimant or the acting parent or guardian.