Phi's mother died shortly after her birth, and nothing is known about her father. She was then raised by foster parents. The only memento she has of her mother is a brooch with a Latin phrase written on the back: "Elapsam semel occasionem non ipse potest Iuppiter reprehendere", which translates to: "Not even Jupiter can find a lost opportunity." This basically means that not even the Roman god Jupiter, the most powerful of gods, can turn back time to do something different.
Virtue's Last Reward
Zero Time Dilemma
Phi continues to wear the brooch during the Decision Game.
After Phi is incinerated during the Fire Decision game, only her brooch survives the flames. Diana holds onto it and later gifts it to her newborn infant daughter, named Phi in her honor, in the D-END: 2 when utilizing the Transporter device to send her daughter's atomic data through spacetime along with their son Delta. When Phi was reconstructed in 1904 and 2008, the brooch appeared with her.
However, the existence of this brooch and Phi herself is a bootstrap paradox as the brooch was given to infant Phi after the adult Phi died in the incinerator, leaving behind the brooch. In turn, Phi's brooch was given to her by her mother shortly before her death, which is revealed to have occurred in D-END: 2. This leaves the question of the brooch's origin since Phi essentially gave it to herself in a causal loop.
It is later revealed that the brooch is one of two items necessary to activate the Force Quit Box, the other being the blue bird necklace which was given to Phi's brother, Delta. Both are a "mother's mementoes" (Diana's).