The City of San Jose Charter calls for filling City Council vacancies in the following way, SECTION 410. Filling of Vacancies. If, for any reason, a vacancy, as defined by Charter Section 409, occurs in the office of Mayor or Council member, the Council shall either fill the vacancy by appointment by a majority of its remaining members, or call an election for the purpose of filling such vacancy. 

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(a) APPOINTMENT. If the vacancy is filled by appointment, the appointment shall be effective until the end of the unexpired term of office or January 1st following the next Regular Municipal Election after the appointment, which ever first occurs. 

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(b) ELECTION. If the vacancy is to be filled by election, the election will be for the entire unexpired term of the office. The election will either be conducted at a Regular Municipal Election, a General Election or at a Special Municipal Election, as determined by the Council. The election will be conducted in accordance with Section 1600. 

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(c) INTERIM APPOINTMENT. If a vacant office is to be filled by election, the Council may make an interim appointment to fill the office until a candidate has been duly elected and the results of the election have been officially certified. A person who is appointed during the interim period shall meet the eligibility requirements to hold office under Section 404 of this Charter.

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City Charter link - https://www.sanjoseca.gov/home/showpublisheddocument/13907/637532449706900000  Pages 9 and 76-77 are the most relevant.