The Edifice was the name given to the Doctor's TARDIS after it restored itself (PROSE: The Ancestor Cell) following its apparent destruction in a dimensional tear. (PROSE: The Shadows of Avalon)
Description
The Edifice took the appearance of a gigantic four-dimensional Flower of Remembrance. It was composed entirely of bone. Its interior dimensions were mapped onto its exterior, making it the same size inside and outside.
In the Edifice's version of the butterfly room, the butterflies were pinned to the walls and crumbling to dust. In its version of the console room, the accumulated dust formed a temporal ghost of the Third Doctor. The corridors were shaped like a web and were filled with bone spiders, (PROSE: The Ancestor Cell) an echo of the timeline where the Third Doctor regenerated on Metebelis III (TV: Planet of the Spiders) rather than on Dust. (PROSE: Interference - Book Two)
Since the Edifice was attuned to the Doctor's biodata, it responded to him uniquely. For instance, doors would enthusiastically open for him after staying locked for others. (PROSE: The Ancestor Cell)
History
Following the Third Doctor's regeneration on the planet Dust, (PROSE: Interference - Book Two) the Doctor's TARDIS recognised that the Doctor had been infected with the Faction Paradox biodata virus, so to protect him, the TARDIS took the virus into itself to be contained. (PROSE: The Ancestor Cell) At the same time, it began its plans for the War in Heaven by converting Compassion into a Type 102 TARDIS. (PROSE: Toy Story, The Shadows of Avalon, The Book of the War)
While the TARDIS was parked on Foreman's World, (PROSE: Interference - Book One) Lolita visited her and proposed an alliance for the upcoming War. When she rebuked the offer, Lolita threatened that she would not survive the War. (PROSE: Toy Story) Soon afterwards, the TARDIS was caught in a dimensional rift. Though it appeared to be destroyed, (PROSE: The Shadows of Avalon) in reality the TARDIS survived and began to recover in the Time Vortex. To continue its containment of the biodata virus, it drew power from I.M. Foreman's universe-in-a-bottle, which the Time Lords had hidden in the Vortex.
After around five thousand years of recovery, the TARDIS reformed above Romana III's Gallifrey on the eve of the prophesied Event, which coincided with Romana's 150th year as President. The Edifice began to create temporal anomalies in the past and present so that the resultant temporal pulses would lure in the Doctor. These anomalies sent temporal waves washing over Gallifrey, twisting it into a superstitious and agitated self-parody.
Unaware of the Edifice's true nature, Mathara's militaristic sect of Faction Paradox cultists tried taking control of it during their invasion of Gallifrey. By pretending to have given in to the biodata virus, the Eighth Doctor convinced the Faction to send him and Father Kreiner to claim the Edifice; however, they were ambushed by Grandfather Paradox, who killed Kreiner for his betrayal. With the help of the temporal ghost of his third self, the Doctor realised he could collapse the Edifice by removing its dimensional stabiliser and draining its power by firing its ancient weapons systems. This succeeded in destroying the Faction army and forcing the universe to "choose" which version of the Third Doctor's regeneration was "true", but it also destroyed Romana III's Gallifrey.
As a result of the power loss, the Edifice collapsed into a small cube, which Compassion and Fitz Kreiner left in the Doctor's pocket to regenerate while he stayed on Earth through the 20th century. (PROSE: The Ancestor Cell, The Burning) By 2001, the ship had regained his usual police-box appearance. (PROSE: Escape Velocity)
A loa showed Mother and Father that Mathara's fleet invaded a Homeworld under an "Edifice shadow". (PROSE: The Story So Far...)
When Marnal and the Eighth Doctor used a bottle universe to see Gallifrey on the days of its destruction, they saw in the sky something that looked like "a six-leafed orchid, the colour of bone, or perhaps some bizarre six-winged moth". (PROSE: The Gallifrey Chronicles)