Telomeres (テロメア, teromea?), also known as Soul Tickets, are the parts of the Reyvateils that rule over their life span. While the Telomeres in human DNA have the effect of determining the life span of the cells, the Reyvateils' Telomere are completely different things, and besides both having the same name due to their functions, they don't have anything else in common. In the case of the Reyvateils, the Telomere are located in the hexagonal section of the hinges in the Triangular Nuclear Loop, which serves as the Heart of the Reyvateils, and they determine how long the Reyvateil will live. In other words, they are like a countdown timer that determines when the Reyvateil will die. However, they don't serve as a power source, or anything similar.
Reason for Existing
The reason why the Reyvateils have them is because, once they have exceeded their functional time limit, it will be impossible to guarantee that they will continue correctly functioning. In the case of the Pureblooded β-type Reyvateils, their Triangular Nuclear Loops have a functional time limit of approximately 150 years. Once that time limit has expired, the Triangular Nuclear Loop automatically shuts down to prevent any weird actions on part of the Reyvateil, and avoid a sudden failing from manifesting on the Triangular Nuclear Loop itself. This is also possible to happen if the Loop receives unexpected damage, which would cause the Reyvateil's mind to go astray.
The Origin Reyvateils Shurelia and Frelia don't actually have Telomeres. When they were developed, the Triangular Nuclear Loop system was still in mid-development, and while the mechanism that acts as their hearts is also called a Triangular Nuclear Loop, it's completely different from the version used in the β-types. On the other hand, while Tyria is also an Origin, she uses a newer kind of Triangular Nuclear Loop, so she does have Telomere, which gives her a limited life span, and makes her mortal. However, and unlike the life span of the β-types, her remaining life time isn't set under a specific time interval.
Whenever a Reyvateil overexerts herself, she depletes her Telomere, which also make possible for Pureblooded β-type Reyvateils to live much less than the 150 years she is supposed to live. Not even Tyria is exempt from this, since actions like restoring the Tower and operating the XP Shell placed a large burden on her, continually depleting her Telomere, causing her to end with very few of them remaining at the end of the game. Also, the use of EXEC_FLIP also brings a heavy burden, as also does the persona exchange performed by both Saki and Finnel. Finnel has four Soulspace Addresses occupying a single body, which has the side-effect of consuming her Telomere at four times the normal rate. As for Saki, since her maximum Soulspace range it's so large, having many personae inside her body doesn't bring her much burden, but using her Miracles (a form of EXEC_FLIP) depletes her Telomere at an alarming speed. Having said this, it's pretty much inevitable that both of them ended having such short life spans.
Ways of Replenishing Telomere
The only way for replenishing Telomere is through a Telomere Transfer. This means that when Tyria replenished Saki's and Finnel's Telomere during the game, she actually consumed her own to do it.
On the other hand, replenishing Tyria's Telomere is completely impossible. While it's possible to do this process at the Rinkernator, it's something that can only be done to β-types. This is because the Triangular Nuclear Loop of the β-types is located inside the Pod, or in other words, inside the body of the Reyvateil that entered the Pod. On the other hand, Tyria's Triangular Nuclear Loop is located in a completely different place within the Tower, due to her being currently an Holography. Also, the rate at which Tyria consumes her Telomere is completely abnormal. If it were possible to find the place where Tyria's Triangular Nuclear Loop is located and discover the appropriate mechanisms for treating it, it would be possible to provide her with Telomere from a donating β-type, but this is a very unlikely scenario.