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To box a Cylon is to download a particular humanoid Cylon's consciousness from his or her body into cold storage indefinitely (analogous to a computer core dump), presumably in an inert state. To a Cylon, boxing is much like true death, except that boxing is relatively easy to reverse if the Cylons who carried it out change their minds.
Boxing is carried out during the Cylon resurrection process. A small metal box on the side of the tank, the storage unit, is removed by an attending Cylon once the consciousness has been downloaded into it, and the Cylon body in the tank deactivates. The body is then moved to storage in a Resurrection Ship. The attendant may as a courtesy allow the condemned Cylon to resurrect long enough to inform him or her of the boxing before carrying it out.
A Cylon is boxed when he or she has been found dangerously unstable, such as when a humanoid Cylon's memories appear "corrupted" by extended or intense human contact, or when a sleeper agent becomes psychologically maladjusted from denying his or her Cylon nature.
Saul Tigh, Ellen Tigh, Samuel Anders, Galen Tyrol, and Tory Foster were boxed by Number One for a time after he killed them by asphyxiation, then released into the Twelve Colonies one by one with fake memories (No Exit).
The resurrected Cylons Caprica-Six (the Number Six who died protecting Gaius Baltar when his house is destroyed in the Miniseries) and Sharon "Boomer" Valerii are threatened with boxing by a Number Three shortly after their return. Their "Hero of the Cylon" status and extended human associations threaten the Cylon status quo because the two became individuals, their personalities diverging from their fellow models (Downloaded).
Similarly, a Cavil infiltrates the resistance on Caprica and, as a result of observing and questioning the humans, comes to the conclusion that the extermination of humanity was a mistake. In order to contain this heresy, the Galactica copy promises to have him boxed once they are resurrected (The Plan).
After the incident at the Temple of Five, where the same Number Three who threatened to box Caprica-Six and Boomer goes against the will of the other Cylons, the decision is made to box not just her, but all Number Three models (Rapture). The podcast for "Rapture" reveals that the boxing of the Threes is the first and only model-wide boxing in Cylon history.
All boxed Cylons are stored at the Resurrection Hub, a central facility which coordinates all resurrections. When the Hub is destroyed by an alliance of humans and renegade Cylons, all but one Number Three, the unboxed D'Anna Biers, are permanently killed in their boxed state. With the Hub gone, all Cylons lose their ability to resurrect and thus can no longer be boxed (The Hub).