In https://yorkporc.wordpress.com/2014/01/28/comparing-des-sbox-design-theory-with-colossue-era-language-about-proportional-bulges/ we got to see the relevance of a padic distance and its relationship to the walsh “measures” used in analyzing sboxes.
Its also fun to put together “rotation” theory with padics.
We know from our thinking about the turing wheel-generator that, for a fixed wiring building upon non-abelian groups, that offsets between wheels could induce virtual rotations of the geometry – which induces operators norms to change basis and thus compress the data.
We now have the background in walsh functions expressed in any base (e.g. 26), and for any group including the turing non-abelian octonian group, to recognize now that a run of zeroes is about inducing rotations anti-clockwise in the space (and a run of 1s is about inducing clockwise motion).
We also learned lots more about dual spaces, getting to see in particular the relationship between the character functions and those groups that are specifically circular (which relates to turing’s group, as he pointed out specifically). In general, we know that his octonion group, being a generator for SU(2), is the dual of the 3-bit cube; and whether expressed abstracted in quaternions as I, j or k or in base 26 (using permutation group theory, that comes down to enigma era wheel design/breaking)
We must also remember our susskind model. For an ice-cream cone (the rotation around Sz axis), one imposes a rotation by cutting it using a oval…at an angle. Remember our visual model f putting the cone into a plastic holder, whose hole is is not horizotonal! We get conic sections, viewed from the right angle! Now also recall susskind who noted that of the two orthogonal vectors, one short and one long that describe the oval, there combined length does not change, as one rotates them as a pair around the oval. this invariant allows us to capture the notion that the superoperators involved are unitary, and we get lots of different ways to compress.