Traditionally, computers define logical operators on real values:

AND

0 0 | 0

1 0 | 0

0 1 | 0

1 1 | 1

OR

0 0 | 0

1 0 | 1

0 1 | 1

1 1 | 1

XOR

0 0 | 0

1 0 | 1

0 1 | 1

1 1 | 0

I believe a similar construct can be made for complex values except that logical equivalency could maybe have two interpretations: either the magnitude is the core comparison value or the frequency (radian representation of the complex value in the unit circle) is the core comparison value.

From a magnitude perspective, we could say that X AND Y for two complex values X and Y is itself if and only if both X and Y lie on the same circle. In a sense, this is establishing a primitive group structure because the value of the operator depends on whether or not an item belongs in a set. Likewise, we could define an XOR operator the same way, just in the inverse, that both X and Y do not belong to the same circle.

From a frequency perspective, we could say that X AND Y for two complex values X and Y is itself if and only if both X and Y lie on the same group structure generated by the radian. Is there an isomorphism between the spirals generated by different magnitudes, but for the same radian? Fundamentally, it feels like there should be a common structure. When the magnitude is iteratively scaled to one, it feels like these structures should grow from the same point of the unit circle. Maybe it is even a fractal?

If the magnitudes of X and Y are both 1, then the frequency check is a simple group check whereby the radian operator is applied iteratively until a group is defined. If the magnitudes of X and Y are both less than 1, then there is some infinite structure that converges to zero? If the magnitudes of X and Y are both greater than 1, then there is some infinite structure that converges to infinity?

Maybe we can define X AND Y to be itself if both X and Y lie on the same spiral that begins on the same point of the unit circle. And similarly, maybe we can define X XOR Y to be itself if X and Y do not lie on the same spiral.

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