Fish/Frog Update

So, I was just reminded again of fish/frog as it relates to Euler's Identity and the guard riddle. 

I just realized that you actually can't accept the word of the guard as a valid starting point to this riddle (one tells the truth the other lies). Unless the guard saying this is the one that always tells the truth, the whole premise is certainly a lie. The only thing that would then be certain is that they are not both compelled to always tell the truth. So at least there's that. For example, the speaker's partner could always be compelled to tell the truth, but only if the speaker himself is neither compelled to tell truth or lies. In fact, without being certain of the validity of this starting statement externally of the guards in question, there is no good way to begin this riddle, 

This idea also helps me flesh out fish/frog further. In a complex field, assuming you had a way to ensure one of the two values was "real" and "complete" from its own perspective (such as ensuring measurements of the Euler's Identity only came up fully in line with either real space or exclusively in imaginary space, you could use a wavelength measurement for this), then Euler's Identity would return a value that was "true" in one place and "false" in another at any given point. One could name |1| as true and 0 as false, or even split this into to separate realms (kind of like what we do with anti-matter) and say that in the second realm -1 is true, 0 is false (in the first 1 is true, 0 is false).

If you assume functionally infinite planes angled around each other in a kind of circle (which would be a corkscrew following the Euler value's angle as it traveled, each plane repeating only once in each "wavelength"), every result would provide a true/false answer, just in a different combination of dimensions. So each plane would be infinitesimal degrees away from each other, such that for the one halfway between real and imaginary from our perspective true must be an equal combination of real and imaginary, and false would be the resulting partnered value (that would also be "0" there in that plane, from our perspective, viewing their space. If done in this way, then at all points the fish/frog would return one truth and one lie, always within in a certain pair of dimensions that are logically perpendicular with each other (no idea on the math though). In this way a wavelength could be viewed as a perpetual string, coursing through a number of planes that are arranged in a geometrically significant and consistent way with our own, but are not what we can access as "real" space.