In the new, noncommutative mathematics, the properties of zero have been teased apart, separating out those that are useful from those that can be gamed. In these new mathematics, when two numbers act on one another, they turn one another into two different numbers instead of one and the same, absolute zero. The operations proceed through a cycle. Every number contains a memory of what number it came from and what number it will next turn into. The result is a number system containing two kinds of +1, two kinds of -1 and two kinds of zero, +0 and -0. The properties of the two kinds of zero integrate the six numbers together to form a system.

The properties of the numbers are defined by application, in oscillation, of the mutually exclusive, mirror opposite truth tables for the logic gates XOR (exclusive OR) and XNOR (not exclusive or). Instead of there being one absolute zero which is based on deeming (0, 0) and (1, 1) to be equivalent, there are two kinds of zero, one obtained by a kind of +1 (0, 1) turning a kind of -1 (1, 0) into a kind of zero (0, 0) and one obtained by another kind of -1 (1, 0) turning another kind of +1 (0, 1) into another kind of zero (1, 1). The properties of absolute zero that enable it to be gamed have been engineered out. These new mathematics control symmetrically from the two sides of a relationship. The same and different meanings of 1 and 0 are exchanged over a cycle, so neither is privileged over the other.