Phase

A Phase (位相) is a layer applied over the world, in this context referring to the entire universe or reality itself, which acts like a veil or filter, and changes how the world appears to be.

Principles
The world as is normally perceived is not pure and untouched - various layers, or phases, exist over it and act like filters, affecting how it appears. By applying new phases over it or manipulating the existing ones, the world can be altered.

Creation and Manipulation
The many religions (Christian, Buddhist, Celtic, Indian, Shinto, Incan, Aztec, Greek, Roman, Norse etc) which have arisen throughout history have had the effect of applying various phases to the world. Extremely powerful beings, such as Majins/Magic Gods, are also capable of applying new phases and manipulating existing ones.

As the Imagine Breaker is the reference and restoration point of the world, it can be used to restore the world back to a previous state before the phases were manipulated. This later proved problematic to Touma as though he knew its purpose, he didn't know how he could use it to restore the world back to his original world other than Othinus, forcing him to cast his "hesitations" in a fight with Othinus, believing that he could "reach her".

The sword Dáinsleif, when unsheathed, is capable of cutting through Phases, releasing energy stored within them and manifesting them as a figure of calamity from that mythology or religion, representing the evil imprisoned by the Gods brought forth once the domain of the Gods and their protection is removed.

Kamisato Kakeru's power, known as World Rejector, has also shown the ability to crush a new phase as it was being created, before being embedded into and changing the world.

Effects
The effects of an individual phase on the world and the effects of the phases being manipulated vary depending on the circumstances involved. The recreated world may be similar to the previous one, like the Omega World being recreated by Othinus each time Touma was near death, to radically different, like the Omega World being different from Touma's original world, where people's various traumas and tragedies have not come to pass. People and events may be completely changed; for example, in the Omega World, many who had perished in Touma's original world were alive, and Sherry Cromwell had the age of a child, despite everybody else being the same age as that of Touma's original world.

When the world is changed due to the manipulation of the phases, most of its inhabitants have no idea that the world has been changed, truly believing that the world is the same as always. However, if the Will of the Whole Misaka Network saying that Othinus "cheated" can be believed, it can be assumed that Othinus can assure that people would have memories of the previous world. Regardless, the only people that have continuous memories of all of the worlds as the phases are manipulated, are beings like Magic Gods, and to a limited extent, likely the living bearers of the Imagine Breaker, such as Touma. There are also beings such as the thought entity known as the Will of the Whole Misaka Network, referred to as beyond the living and the dead, and the collective entity St. Germain, who may not be directly or completely affected by the changing phases of the world, though they may be indirectly influenced.

The phases themselves are not usually perceived by most people, though there are beings who can detect them, including ones capable of influencing them. There is one particular layer, referred to as the Hidden World, where the true Gremlin reside, hidden but unable to leave, which was beyond being detected or destroyed by most - even Othinus, a Magic God, was unaware of its existence.

Pure World
Beyond the filters lies the 'pure world', described by Othinus as a world of science unaffected by religion - one which the man known as "The Silver Star" has been apparently been attempting to tamper with.

Background
Throughout history, the various religions which have arisen have had the effect of applying phases to the world. At some points, the phases have also been manipulated by the Magic Gods, the majority of which eventually formed the reconciliation council Gremlin and came to reside in a hidden layer, separate from the rest of the world.

After Othinus became a Magic God, she used her power to apply new phases and change the world, however as she did, she forgot how her original world was and as such couldn't go back once she wished to, leading her to recreate the world many times attempting to reach it. Her hesitations prevented her from using the Imagine Breaker to return the world back to her own original world. These hesitations and uncertainties would later build up inside of her, and she had hoped to find someone to understand her, and she believed that person could be found in her original world. She would later settle in the current world — the world where Kamijou Touma would be born as a bearer of the Imagine Breaker.

A man calling himself as the Silver Star has also attempted to tamper with the Pure World beyond all the filters and continues to aim to do so. Aleister Crowley also attempted to find and breach the hidden layer where the true Gremlin resided, leaving the world's strength to handle Othinus, despite knowing of the dangers of her running free.

Angel Fall Arc
The term 'phase' is briefly mentioned during Tsuchimikado Motoharu's expalanation to Kamijou Touma about the nature of Heaven and Hell.

World War III Arc
Aleister Crowley mentions phases, specifically the adjustment of a phase's thickness, when talking to Fiamma of the Right about the similarities between their plans.

Shifting and Fluctuating World
After acquiring Gungnir, Othinus 'ends' the world, creating a new phase over it. She then proceeds to recreate the world through multiple new phases in an attempt to break Kamijou Touma. After subjecting him to many worlds, she talks about him about phases before creating one last new world.

Failing to break Touma with the phases, Othinus attempts to break him directly, repeatedly crushing him again and again without him breaking, before becoming exhausted and trying to end him. The world created by the applied phase is then ripped apart as she throws Gungnir and when the lance is destroyed, she uses a crossbow comprised of the black world to strike Touma. However as Touma dies, he manages to remind Othinus why she was trying to return to her original world in the first place - to find someone who understood her. Realizing she had inadvertently made such a person in Touma and that there may not be anyone like that in her original world, Othinus chooses to recreate Touma's world.

VS The World
During his journey with Othinus across Denmark, Touma faces Marian Slingeneyer, who makes use of Dáinsleif to cut away phases within a barrier and summon figures of calamity to attack him.

Following Touma's attempt to save Othinus from killing herself, the Magic Gods of the true Gremlin interfere from their hidden world, a phase which Othinus didn't perceive or affect, to prevent a change occurring in Touma which would be inconvenient for their plans for his future. As they discuss the events, Aleister Crowley penetrates and compromises their hidden world, stepping into their "divine territory". On apparently being provoked by them, Aleister begins a battle with the Magic Gods, eventually escaping with a third of his body blown off but having accomplished his goal. With their hidden sanctuary gone, Gremlin proceeded to make preparations to enter the real world.

Magic God Invasion Arc
After being weakened by Aleister Crowley on entering the real world, Nephthys is left the only Magic God in the true Gremlin able to swap out the phases, due to her nature, and only able to do so once. She attempts to do so when faced with Kamisato Kakeru, who had exiled by Niang-Niang and the other Magic Gods with his World Rejector, but the phase she creates is crushed before it can be embedded and the world changed.

Created by religions

 * Heaven and Hell (天国 & 地獄) (Christianity)
 * The Underworld (冥府) (Various)
 * The Pure Land (浄土) (Buddhism)
 * Yomi (黄泉) (Shinto)
 * The Abyss (地底)
 * Olympus (オリンポス) (Greek)
 * The Fairy Island (Celtic)
 * Nirai Kanai (ニライカナイ) (Ryukyuan)
 * Asgard (アースガルド) (Norse)
 * First sun (Aztec)
 * Takama-ga-hara (Shinto)
 * Amaravati (Hindu)

Worlds created through phase manipulation

 * Black World
 * Alpha World
 * Beta World
 * Omega World

Other

 * Hidden World (Gremlin)