Relic

The "Relic" refers to an object, or more actually a pair of objects, connected to a past Level 5 and Academy City's "previous generation".

Etymology
In a religious sense, a (derived from the Latin reliquiae, meaning "remains") is an object or article of religious significance from the past, such as the physical remains or personal effects of a saint or other similarly venerated person, preserved for veneration as a tangible memorial. Relics are present in Christianity, Buddhism and various other religions. A shrine housing one or more religious relics is known as a reliquary.

The term can also refer to something which has survived the passage of time, such as an object or custom whose original culture has disappeared, as well as an object cherished for historical or memorial value (such as a keepsake or heirloom).

Principles
There are numerous rumors surrounding the Relic, said to have leaked from a now-defunct laboratory. It has been said to be a malignant tumor which consumes esper powers, a precious stone which gathers lifeforce, or a secret technique to rearrange the "braid of time".

Tokiwadai Middle School also has a rumor concerning a "Relic", specifically rumored secret win condition for its student council presidential election, outside the election itself - a treasure hidden in Tokiwadai which is said to give each president a hidden power, with people relying on the rumor called Diggers.

Relics
The actual Relic (or rather Relics, given that there are at least a pair of them) is a cube which is connected to one of Academy City's past Level 5s and the city's mysterious "past generation". Among the past Level 5s, there was once one capable of spatial control, which "enveloped" a target space to create a special territory, with those in the affected area said to feel an indescribable "chill".

The Relics are capable of manipulating those who hold them, apparently putting them under the influence of the former Level 5,   and allow them to draw upon powers associated with them. Whether the powers of the two Relics are identical or differ in some ways is not currently clear, but they both involve the creation of Shadow Metal, and heralded by a drop in temperature within the area of effect to the extent of ice forming.

In the case of the Relic which Kouzaku Mitori encountered in a facility, through which the Level 5 initially communicated with her through vibrations, the power displayed was referred to as Absolute Leaf and invoked with the command Throttle. Aside from the creation of Shadow Metal and the temperature drop, it was able to neutralize an effect caused by the other Relic (namely the manifestation of large metallic spikes to skewer Misaka Mikoto when she strongly invoked her own power), with the effect of the scene shattering. It was also capable of manifesting three vaguely-insectoid limbs from the right side of its host's back.

A clash between the powers of the two Relics produced a spatial warp at the point of the clash, with a nearby observer witnessing glimpses of other places.

Previous Generation
Not much is known about the unnamed Level 5 behind the Relics. When communicating with Kouzaku, they claimed to be her "Senior" academically and the previous #1 Level 5.

There appear to be some differences between this Level 5's presence and the will in the two Relics. With Kokuchou's Relic, on taking over when Misaka Mikoto approached, they said a current Level 5 should stopped by the previous. They also expressed a wish for revenge and intent to bring down the hammer and destruction upon the city's law and its ruler. They were initially unaware of the other Relic and its similar power, later describing it as its copied baby. The Relic in Kouzaku's hands intend to stop its counterpart's plots, describing itself as the same as the other and it itself as its own greatest enemy, as well as accepting the 'humiliation' of returning to a world it considered hellish to do so. In response to its counterpart's 'arrogance' towards its coped baby, it claimed that people evolved by copying and there was nothing to say a successor was inferior to the original.

Aside from the Level 5, there is also another unnamed group associated with the Relics who are also said to be part of the mysterious and rarely mentioned "previous generation" before the current generation. This group used two squares (visual representations of the cubes) as their symbol. Aside from the Relic, the group also sought Shadow Metal, said to exist as cubic crystals, with an increase in activity after a large amount was generated in an event during the Daihaseisai. The group had a number of facilities and personnel at their disposal.

Other
Aside from the two with the Relics in hand, one of the other election candidates who was involved and emphasized in the events surrounding the election and the cubes, possesses traits notably similar to the Relics and their associated aspects. Kamino had square-shaped eyes (somewhat similar to the square representation used by the group connected to the cubes) and used a spatial control ability affecting a cube-shaped area, the usage of which provoked a feeling that reminded Shokuhou of the past Level 5. Aside from sealing off space (with the effect represented visually with a dark miasma), the ability was also capable of disintegrating objects within a designated area. Whether there is a connection and the exact nature if there is one is currently unknown.

Background
The exact time frame which the past Level 5 and the "previous generation" were active, and the circumstances which led to their obscurity and the creation of Relics, are currently unknown.

In a conversation with Kamijou Touma one year prior to the main story, Shokuhou had mentioned that there was once a Level 5 capable of spatial control.

Tokiwadai Election Arc
The Relic(s) played a key role in the events surrounding Shokuhou Misaki's involvement in Tokiwadai's student council presidential election in October. Prior to her involvement in the election, Shokuhou began investigating the Relic, learning of a supposed close connection to someone at Tokiwadai. Although it had started as a simple investigation, Shokuhou ended up taking out a related institute, with individuals connected to the group involved attacking her in the next day.

Three days after this attack, Kouzaku Mitori had a phone conversation with Shokuhou about these circumstances, with Kouzaku subsequently infiltrating a related institute and coming into contact with the Relic housed there. Around the same time, the rival election candidates conducted an ambush on Shokuhou and Gaouin Tsukasa, with the similar-traited Kamino and Relic-influenced Kokuchou Mayui playing a role in these events. These events eventually led to the two Relics clashing with one other.

Trivia

 * When the Relics communicate, through a person or otherwise, the dialogue box is square shaped.
 * The ability and situation concerning the Level 5 behind the Relics bear a number of similarities to the case of Kakine Teitoku.  The two espers manipulate a substance which does not naturally exist in the world (Shadow Metal for the Relic Level 5, Dark Matter for Kakine) and use it to produce various phenomena.  Both of these substances carry their thoughts in some manner,   both seemingly left behind their will and power in some form separate from their original human body (the unnamed Level 5 in the Relics, Kakine through his Dark Matter with his mangled remains preserved),   and both have had iterations of themselves produced in this state act against them.