Chiralkine counting works like me and you, where each of us self-identifies as me, not you (1) as opposed to you, not me (0). Each of the words me (1) and you (0) has two, mutually exclusive meanings in the context of our relationship.
In chiralkine counting, a mathematical object is defined by a relationship between what it is (1) as opposed to what it is not (0), where each side of the relationship self-identifies as itself as opposed to the other side, not itself. From one side of the relationship, the difference between 1 and 1 is 1, but on the other side the difference between 0 and 0 is 0. Two mutually exclusive, mirror opposite ways of distinguishing between same and different are operating together symmetrically.
Chiralkine counting is constructed from interpenetrating truth tables for the logic gate XOR and its mirror opposite, XNOR. There are four columns of six rows in a cycle. Each row contains two 1s and two 0s. A distinction defining a relationship between is and not relationship consist of a 1 and a 0. Each row differs from the next by the exchange of a 1 and a 0. Exchange of a 1 and a 0 results in one pair of a 1 and a 0 in a row becoming two 1s or two 0s. When that happens, a relationship between an is and a not is erased. Another relationship between and is and a not is created in its place. For example, consider the two rows and their mirror counterparts below:
0 1 0 1 and 1 0 1 0
0 1 1 0 and 1 0 0 1
Comparing first and second rows, two of the 1s and 0s have been exchanged and two have remained unchanged. A distinction drawn between a 1 and a 0 in the first row has been erased in the second row.
Each step in the cycle is controlled by the logic of the interpenetrating XOR and XNOR truth tables. In the next step the second row will act on the first row to generate a third row.
0 1 0 1 and 1 0 1 0
0 1 1 0 and 1 0 0 1
0 0 1 1 and 1 1 0 0
The quantity of distinctions is conserved every step.
When a mathematical object is counted, the relationship that defined it is cleared (the different symbols 1 and 0 on the two sides become the same on each side) and a record is made in the form of another mathematical object, also defined by a relationship between what it is (1) as opposed to what it is not (0). Put another way, ordered pairs of 1s and ordered pairs of 0s exchange 1s and 0s, producing differently ordered pairs of 1s and ordered pairs of 0s. The two mutually exclusive, mirror opposite ways of distinguishing between same and different that are operating together control the counting process such that it repeats in a six-step cycle. During this cycle the original meanings are inverted after three steps and restored after six steps. For example, goods and services – what is mine and what is yours – can be exchanged under the control of the cycle.
Chiralkine counting is inconsistent with traditional mathematics, because that cannot be based on using two mutually exclusive, mirror opposite ways to distinguish between same and different together symmetrically. Traditional mathematics embodies the concepts of an equation, zero (the difference between two mathematical objects that are the same), positive and negative numbers, a number line and real and imaginary numbers. These are not defined in chiralkine counting, but the quantity of objects is conserved in chiralkine counting, so it is no less suitable for performing computation. It should prove superior to traditional accounting when applied to socioeconomic relationships, because it keeps track of the symmetry inherent in relationships, and hence provides control against one side of a relationship being privileged over the other.