Given your method for figuring out the perfidity of negative numbers, it seems natural to generalize to all the 2-adics. Indeed, in the 2-adics, we can make equivalence classes where two numbers are equivalent if their XOR has a finite, even number of 1s. This gives us classes of numbers that have the same perfidity. Also, we can say that two numbers have opposite perfidity if their XOR has a finite, odd number of 1s. At this point, it seems natural to ask whether or not there is a ‘natural’ way to assign “odious” and “evil” to these equivalence classes. However, given that we have two fine methods of assigning “odious” and “evil” to the negative numbers, I doubt that this such a ‘natural way’ exists.
Also, it may be interesting to ask how this may relate to the Thue-Morse sequence.

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