Box-Kites III: Quizzical Quaternions, Mock Octonions, and Other Zero-Divisor-Suppressing "Sleeper Cell" Structures in the Sedenions and 2^n-ions

Abstract

Building on two prior studies of zero-divisors (ZD's) generated by the Cayley-Dickson process, algebras we call "lariats" (Line Algebras of Real and Imaginary Axis Transforms), linkable to quantum measurement, are discovered in the Sedenions, complementing the 7 isomorphic "box-kites" (pathway systems spanning octahedral lattices) interconnecting all primitive ZD's. By switching "edge-signs," products among the diagonal line-pairs associated with each of a box-kite's 4 triangular, vertex-joined, "sails" generate not 6-cyclic ZD couplings when circuited, but 28 pairs of structures with Quaternionic multiplication tables -- provided their symbols represent the oriented diagonals as such, not point-specifiable "units" residing on them. If a box-kite's 3 "struts" (pairs of opposite vertices, the only vertex pairings which do not contain mutual ZD's) each be combined with the ZD-free Quaternion copy uniquely associated with said box-kite, 21 lariats with Octonionic multiplication, one per each box-kite strut pair, are generated. Extending this approach to "emanation tables" (box-kite analogs in higher 2^n-ions) indicates further ZD-masking "sleeper cell" structures, with renormalization's basis possibly amenable to rethinking, thanks partly to the ZDs' newfound "Trip Sync" property, inhering throughout the 2^n-ion hierarchy.