Archreality/Mathematical description

Generating Archrealities from a Function
A lower level archreality can be denoted using the notation $$R_{a}^{b}$$, where a is a set of objects (such as realities), and b is the iteration count.

We can state that $$R_{a}^{b} = R_{R_{a}^{b-1}}^{1}$$ if b > 1, allowing for the iteration piece to function.

We can also state that simply the constant $$R$$ corresponds to a unireality.

We can also define that $$R_{a}^{0} = a$$ as an identity property.

We can also state that $$R_{R_{a}^{b}}^{-b} = a$$ as the iterative property of equality.

Normally, eventually, a given equation boil down into $$R_{a}^{1}$$. Since b is not greater than 1 anymore, we need a unique ruleset to handle this. This is where the "building upon inferior realities" piece really comes into play.

Every set/reality has a defined "building property" attached as some form of "metadata". These building properties define the basic components of a reality, which defines how a reality behaves, what a reality is, etc.

The building property for a unireality is normally directly what a given thing is. It can also be something different depending on the building property.

Step of computing $$R_{a}^{1}$$

 * Make a new set/reality containing all variants of "a" (including "a" itself) with different orders of which the building properties are ordered in.
 * Give the new set/reality a building property. This building property will be about permutating sets of every single "a".
 * You are now done the process!

Downgrading Archrealities for Visualization
Although archrealities can be generated via functions, they aren't of much use if you can't visualize them as a lower-level reality. As a result, there has to be steps in order to downgrade an archreality to the previous level.

Here are the typical steps followed:

For each copy of $$R_{a}^{b}$$ (note that both arbitrary functions and the arbitrary property used in the succeeding steps must be different for each $$R_{a}^{b}$$): Once you have gotten all of the values of "H" you need, take the properties from each corresponding element in each copy of "H", then merge these into one element. Repeat for each corresponding element between the "H"s, then piece them together in the same formation that each "H" had.
 * Get a reality $$R_{a}^{b}$$.
 * For each and every property an object can have in $$R_{a}^{b-1}$$, make a new copy of $$R_{a}^{b}$$.
 * Apply any arbitrary function that can convert $$R_{a}^{b}$$ into $$R_{\mathbb{R}}^{b}$$. Call this new reality H.
 * Convert each set of only numbers within "H" into a single number using another arbitrary function.
 * Now turn each number in "H" into a property.

Hierarchy
$$r_a^b(c)$$ is a FGH-like hierarchy used for measuring archrealities. It is defined as so:
 * $$r_a^b(c)$$ if $$a$$ is not a limit ordinal = $$R^{b}_{ \{ x \mid x = r_{a-1}^{n}(c) \} }$$
 * $$r_0^b(c)$$ = $$R_{c}^{b}$$
 * $$r_a^b$$ = $$r_a^b(R)$$
 * $$r_{\alpha}^b(c)$$ if $$\alpha$$ is a limit ordinal: = $$r_{\alpha[b]}^b(c)$$