Irregular

"Since a long time ago, Irregulars have brought change and chaos to the Tower. Also, they all possessed an enormous strength which became a symbol of fear within these walls." -Yuri Zahard describing Irregulars to Baam(Ch 2)

An Irregular is a being who entered the Tower by opening its' doors.

Current definition of an Irregular
In the current political system, Headon visits the Outer Tower and outside the tower to select Regulars(Ch 10). In this system, Irregulars are called "unchosen Irregulars" and regulars are called "chosen Regulars" or "selected chosen"(,Ch 2). It is unknown if this current selection process is part of a contract Zahard made with a Guardian like Headon. It is assumed Headon's choice reflects the Tower's choice(Ch 2) and in turn, the Tower's Rules.

Politically, Irregulars are viewed as beings who have entered the Tower and in turn, the Inner Tower, without being chosen by the Tower to enter(Ch 2). And according to Yuri Zahard, they're "probably not too welcome"(Ch 2) and she assumes Headon doesn't like Irregulars. And according to Ship Leesoo, Irregulars have "opened the door themselves of their own will" and their act goes "against the Tower's Laws"(Ch 58). In addition, they violate the restrictions set by the Guardians when manipulating Shinsoo(Ch 26) and this is likely considered further proof of their violation of the Tower's Rules.

"Irregulars have all brought great change" and "chaos into the Tower"(Ch 58, Ch 2) and "all the people in the center of the Tower"(assumed to be the Ruling Class and many of those serving the Zahard Empire)(Ch 58) tremble at them.

Original definition of an Irregular
In ancient times, the terms "Irregular" or "Regular" didn't exist. The Irregulars are the ones who opened the door of the Tower themselves, according to the Guardian of the 2nd Floor(Ch 58). And likely, the term is a simplification of the entire process as one doesn't physically open the door of the Tower to enter it.

Since the current testing system had not been established then, the original testing system of the Tower was likely created for Irregulars and thus, they are the true chosen ones. The duty of the Guardians was to test these Irregulars(Ch 4) since likely, only Irregulars can request to be tested by them(Ch 58). And only Irregulars could climb the Tower(Ch 76).

Irregulars can freely manipulate Shinsoo and violate the restrictions set by the Guardians, which are considered part of the Tower's Rules(Ch 26). This is because even without a Shinsoo contract, one can manipulate Shinsoo but is heavily restricted in the amount he or she can manipulate. And these restrictions are set in place as Shinsoo is dangerous for a non-Irregular and one could die from overloading themselves. For an Irregular, the Shinsoo contract is probably a shackle(Ch 29) but he is still able to manipulate tremendous amounts of Shinsoo, despite these limitations as depicted by Urek Mazino's or Enryu's monstrous abilities.

They also possess enormous strengths which eventually become a symbol of fear. (Ch 2)

Most are known for notable acts like Phantaminum storming Zahard's Inner Palace, Enryu killing a Guardian, Urek Mazino fighting toe to toe with Arie Hon and then being deemed stronger by the latter. The exception is Jyu Viole Grace who has yet to perform a notable feat.

Note: it is uncertain if the Tower was created solely for Irregulars alone as groups of people were living in there before Zahard's arrival.

How Irregulars are possibly selected
People who live in the land(s) outside the Tower are selected by The Tower if they meet a certain criteria. It's unknown what qualities are needed for a person to become a "chosen one". This is evidenced by the fact Headon told Rachel that "you're not the person the Tower has selected" and "you have not been chosen"(Ch 76). Although it is possible Headon or another Guardian is involved in the choosing of an Irregular, it's likely the Tower does the choosing and possibly, even the teleporting.

Examples of Irregulars include Zahard and the 10 Great Warriors and Baam. They are then selected by the "light" of the teleporting system and then teleported through the door of the Tower into the Tower itself. While they're being teleported through the door, they're still physically present in their current location(Ch 76).

After being teleported into the Tower, they then take a series of Guardian's tests. It is supposed that when an Irregular reaches the top of the Tower, he is then chosen again to receive certain powers. "As promised, all that you seek, all that we desire, is prepared up there. On top of the Tower.(Ch 78)"

Known Irregulars

 * Baam (Wave Controller)
 * Phantaminum (Exis)
 * Enryu (Spear Bearer)
 * Urek Mazino (Fisherman)
 * Rachel (Light Bearer) Politically, Rachel is known as an "unchosen Irregular" but to the Guardians, she's most likely not a "chosen one". This is because she was not selected by the Tower.