Phase

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

Principles
The world was originally pure and untouched before various layers and filters began affecting how it appears. This "Pure World" is a world of science that is unaffected by religion, which the man known as the Silver Star is apparently trying to be tamper directly beyond all the filters. However, 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. Powerful beings such as Majins are also able to apply new phases. If the layers of these various religions are real, then it can be assumed that properties present in a Phase of the world would not necessarily mean that it would be erased when a new layer is applied. Due to the Imaginary Number District being described as existing at the same place as Academy City, it is likely that it too is a layer applied to the world to create a new Phase.

The effect of a new Phase of the world varies from being similar to the previous one, like the Omega World being recreated by Othinus each time Touma is 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 unlike in Touma's world. A more obvious difference here is that in the Omega World Sherry Cromwell has an age of a child despite everybody else being the same age as that of Touma's original world.

When the world enters a new Phase, 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 the various Phases of the world are beings like Magic Gods, and to a limited extent, likely the living bearers of the Imagine Breaker, such as Touma. Beings such as the thought entity known as the Will of the Whole Misaka Network, who cannot be directly influenced by the changing Phases of the world, and is referred to as beyond the living and the dead, likely applies as well.

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 Phase of the world. 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".

Background
Throughout history, the various religions which have arisen have had the effect of applying phases to the world. Meanwhile, a man calling himself as the Silver Star is attempting to tamper with the Pure World beyond all the filters.

After Othinus became a Majin, 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 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.

World War III
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.

Created by religions

 * Heaven and Hell (Christianity)
 * The Underworld
 * The Pure Land (Buddhism)
 * Yomi (Shinto)
 * The Abyss
 * Mount Olympus (Greek)
 * The Fairy Island
 * Nirai Kanai (Ryukyuan)
 * Asgard (Norse)

Created by Othinus

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