Verse Mechanics Explanations
Terminology meaning to keep in mind:
The World = Everything, all of reality, dimensional space, concepts. Not limited to the Planet, the Universe, nor the Multiverse, but an all encompassing term.
Existence and Nonexistence: Everything that is and that isn't, a dualism of Yin and Yang. Everything and Nothing is part of the Taikyoku concept and is no exception to control or change.
The Universe: Hadou God's are described as walking universes, but they affect everything and transcend all dimensional and multiversal limitations, manifolds and branes, and whatever lies beyond.
Taikyoku/Taiji (referred to in Dies Irae as "Atziluth") is the beginning of all things, the root, original/first beginning, or beginning of all beginnings. Taikyoku is even the source of yin and yang. Taikyoku is the "Supreme Ultimate" state of limitless potential, the oneness before duality, all who have reached Atziluth have taikyoku. Atziluth is the emanation of the spirit/soul (Atman), which is beyond the dimensional framework, representing a cosmic law which repaints all of reality for themselves, in accordance with the desire of the god itself, bypassing any obstacles because taikyoku is primary in relation to any form of existence.
Beings who reached taikyoku are transcendent in relation to any form of existence, any form of information, any form of duality, any form of time-space, regardless of its complexity, or its size. Taikyoku is the root of all things and phenomena, therefore, regardless if an impact is tangible, intangible, spiritual, mental, logical, verbal, nothing can affect them. To oppose the law, you are required to possess an equivalent or greater Taikyoku value to compete on a similar playing field as the Atziluth user.
Everyone who has reached Atziluth is a walking universe (the universes are Taikyoku and therefore "beyond dimensions"); atoms, spirits, planets, stars, the concept of the existence and nonexistence of all this is the all-encompassing ultimate reality, which is a Hadou / Gudou god. Even a drop of Gudou God's blood weighs more than celestial bodies. The very size at the same time does not depend on the physical dimensions, from a physical point of view the universe exists "outside" them. A god with a more powerful Atman, will actually have a universe that is much "larger and deeper" than other gods.
In reality, cutting tools carry the Laws of Cutting, there is also the Law of Burning which fire carries. Lung respiration is impossible without the Law of Water, those mundane and small things are simply physical.
It is anything but difficult. If the aforementioned law of water is taikyoku, then the entire universe will be changed into water. Alternatively, if the law of the world is frozen water, then the water that cannot be frozen, but also cannot evaporate because it is an independent Law of Domain is also Taikyoku.
The former is Hadou. The latter is Gudou.
One with Taikyoku conquers all Creation.
Levels of Taikyoku
Taikyoku is also measured in units. Should a God have a higher value than another (Ex. Higher one has a value of 2 whereas the Lower one has a value of 1), the one with higher Taikyoku will always reign supreme to that of the God with the lower value. As of current, the God with the highest Taikyoku in the series is Hajun, who possess a value of Taikyoku as "Immeasurable" that is constantly rising beyond due to the Tumor in him. Outside of him, various other Gods of Hadou/Gudou nature have been ranked in normal values otherwise (albeit, the lowest known in the series has been 33, the highest going as far as 100).
The Throne (座 literally means "Seat") is a metaphysical device created as a byproduct of the First Heaven's civilization to manage all dominant Hadou Taikyoku. The Throne can not exist without taikyoku (it was likely created by First Heaven herself), the concept of taikyoku however will exist without the Throne. Taikyoku is a concept that existed long before the Throne was even created. Nothing can exist outside the Throne's Domain due to the nature of Hadou. The space Throne holds are within as well as things outside (i.e. Twilight Beach, separate infinite worlds exist as it were "under the" Throne, beyond creation) - but are still are part of the Hadou. And no matter how many of these worlds spawn from nowhere, they will be a part of it. Heaven (what Hadou Gods are called) is the alpha and omega, the beginning and the end, everything and nothing, they are what precedes duality. And even if there was something external (that is not possible, unless it's another Hadou / Gudou, nothing can be external to it.). The Dominant Hadou will swallow it. Let's take the Snake of Mercury for example. Mercurius will always be above all, the only thing that can stop him is another Hadou of a similar force. That is why when he fights, he doesn't destroy outer worlds. He takes them as part of his body and thus whatever is encompassed by the Hadou is encompassed by the Throne.
Atziluth-Tai - The law of the throne. It's the wellspring of the concepts that operate in the setting, basically the mechanics of the cosmology itself. Hadou God's define their own mechanics so Paradise lost, Dies Irae and Kajiri Kamui Kagura are governed by different set of rules/laws. Aside from the macro mechanics (Taikyoku, etc.) of setting.
すなわちその世界における絶対法であり、そうした決まり事を定めた張本人を指す。 For example would be like how life forms perceive and act, how the world looks like, how the soul is managed after death (whether or not it is erased from existence after death or reset back in time, so you practically keep reliving your life over and over again), or whether the concept of (insert here) exist or not.
Hadou Gods (Hadou meaning "Royal Path, Way, Route, etc.) are the gods that are committed to continuous, unlimited expansion, no matter how great or small the world, Hadou will flood its entirety. Essentially this means that Hadou will encompass anything and everything regardless of scale. Be it from a singular universe, a multiverse, to a dimensionless space such as the Throne (world). Because of this, until the era of the Fifth Heaven (Marie) two Hadou deities could not co-exist, there is no empty space between gods.
Hadou Gods are walking Taikyoku/Atziluth who defines their own boundaries, and they infinitely expands their desire until everything in setting has been subjugated to their law (integrated). They are the gods that encompass the throne as the next godhead.
Hadou Gods by default cannot co-exist without Marie, they would devour one another until only one them is left alive to occupy the Throne and become the Godhead.
Hadou God is the Godhead. I am everything, everything is me.
Try to reach the throne and the Hadou God's reflex kills you.
The Hadou God is the meta-scenario writer of the setting.
Next is the Singularity, the singularity is a tear in reality to the Throne where the godhead resides. Usually generated by a clash of Hadou Atziluth-Tai between the Throne and the lowest floor (Multiverse). Basically singularities are created from the collision between the "Color" or Law of Hadou Gods, the world is "unaffected" by the Law of the Hadou where God does not exist and is not affected by the Law as they block the flow of Law and became the Singularity rather than be influenced by the Supreme Law (Atziluth-Tai). In order to reach the throne, the challenger may opt to drill through the depths of the Singularity using their law.
The depth of the Singularity varies in accordance with the ruling deity’s law strength (Taikyoku/Taiji). As such, Singularities generated by a God with “higher” quality would be immeasurably “deeper” than those of lower quality, and therefore will not be “drilled” or “crossed” without adequate strength - there are few exceptions to this rule, however.
Additionally, the Singularity is described as an infinite abyss of colorless Taikyoku/Taiji that can be comprehended and interacted only by deities. It is a “place” that gives birth natural laws and new ideas/concepts, and where the afterimages or shades of Gods from the past, who must be overcome and thoroughly understood in order to reach the throne, reside.
Despite being referred to as “place”, It is utterly impossible to define the Singularity due to its abstract nature: the laws of physics simply do not exist in the Singularity, and all of creation are mixed there, side by side - be it past, present or future. Thus, the process of “drilling” can be compared to “falling” to nowhere.
Lastly, It should also be noted that despite being naturally unaffected by the laws of the Gods, the Singularity is still under the possession of the Throne God.