In the preceding table, magical categories are associated with integers and grouped accordingly. For example, 1,11, 21, and 31 are extensions of the Creation category. 6, 16, 26, and 36 are extensions of the Illusion category. Since there is an infinite number of integers, there is an infinite number of possible instances. The emerging system is a system of sigils because the entities in the categories can be denoted with sigils and related to each other in equivalence relations.
The sum of element #616 in the Creation category (with a value of 6151) and the element #201 of the Form category (that has a value of 2002) is 8153mod10. 8153Mod10≅3, so the sum of those elements is an instance of Alteration. A sorted list of quotient sets provides the element number from the position of the ordered list. In a list for the remainders of 1 for kmod10, the 616th element would have a value of 6151.
It is possible to model these magical entities in an atomic sense by looking at the factors or the addends of that entity. An entity of Creation in the second position (that has the value of 11) is a sum of 3+3+3+2≡11mod10. The respective properties of this entity are Alteration (3), Alteration (3), Alteration (3), and Form (2). A different formula is 1+2+3+5≡11mod1. The respective properties of that formula are Creation (1), Form (2), Alteration (3), Conjuration (5).
Magical Classes and Additive Inverses
In many magical systems, there are categories of entities that have dualistic connotations of opposites. It is possible to derive opposites in an ordered set from the direction of the iteration. Positive properties are positive values added to move from the start of the loop to the end. Negative properties are values subtracted to move from the end of the loop to the start. An intentional system of opposites creates a balanced magical system. If it is not balanced, counter-intuitive relationships that you did not intend will emerge, so it is important to include that balance in the design of the list. You can derive the opposite of a magical character from an additive inverse. For example, 1-1=0 and 2-2=0. A balanced magical system considers the opposite of every entity. An additive inverse is the number of a pair of numbers that, when added with its pair, has a sum of zero. To calculate the additive inverse for an integer in modulo n, you can flip the sign. For example, -1 is the additive inverse of 1, so the expression is -1mod10≅9. In a circle of ten sequential points that start and end at 0, moving to clockwise one position is equal to 1, and moving counterclockwise is equal to 9. The reflection of 1mod10≅1 would be -1mod10≅9. The reflection of 2mod10≡2 would be -2mod10≅8. An antonym of Creation is Destruction, and an antonym of Form is Entropy.
Higher-Dimensional Magical Properties and Networks
Higher-dimensional categories of a magical system are inductions of the lower category’s so that the structure is self-similar. That fractal relationship is evident with recursive algorithms. The quotient set of 10 encapsulates 9 and 1. 10-1=9, so 9 is in the set of 10. 9-1 is 8, so 8 is in the set of 9. Semantically this means that inside Magic, there is Destruction, and inside Destruction is Entropy. In addition to that, the reflection of Destruction (9) is Creation (1), so if you draw a path from Destruction, to every other category until you get back to Destruction, you will draw the reflections of Creation and Destruction that are lower forms of the archetype Magic. You can apply this property to create structures comprised of how entities relate to their opposites recursively. For 9-1, the reflection of Entropy (8) is Form (2), so if you draw a path from Entropy to every fourth category until you get back to Entropy, you will draw the reflections of Form and Entropy that are lower forms of the archetype Destruction. When you start somewhere and end at the place you started after some number of paths, you have a circuit and a loop. The illustrations in this section are networks for the magical classes introduced earlier. If the nodes are denoted by sigils, it is a network of sigils. Even though I arbitrarily created the network, the flow of the network is intuitive because I made sure to keep the concepts balanced.