  Methods for studying zero-divisors (ZD's) in 2n-ions generated by
Cayley-Dickson process beyond the Sedenions are explored. Prior work showed a
ZD system in the Sedenions, based on 7 octahedral lattices ("Box-Kites"), whose
6 vertices collect and partition the "42 Assessors" (pairs of diagonals in
planes spanned by pure imaginaries, one a pure Octonion, hence of subscript <
8, the other a Sedenion of subscript > 8 and not the XOR with 8 of the chosen
Octonion). Potential connections to fundamental objects in physics (e.g., the
curvature tensor and pair creation) are suggested. Structures found in the
32-ions ("Pathions") are elicited next. Harmonics of Box-Kites, called here
"Kite-Chain Middens," are shown to extend indefinitely into higher forms of
2n-ions. All non-Midden-collected ZD diagonals in the Pathions, meanwhile, are
seen belonging to a set of 15 "emanation tables," dubbed "sand mandalas."
Showcasing the workings of the DMZ's (dyads making zero) among the products of
each of their 14 Assessors with each other, they house 168 fillable cells each
(the number of elements in the simple group PSL(2,7) governing Octonion
multiplication). 7 of these emanation tables, whose "inner XOR" of their
axis-pairs' indices exceed 24, indicate modes of collapsing from higher to
lower 2n-ion forms, as they can be "folded up" in a 1-to-1 manner onto the 7
Sedenion Box-Kites. These same 7 also display surprising patterns of DMZ
sparsity (with but 72 of 168 available cells filled), with the animation-like
sequencing obtaining between these 7 "still-shots" indicating an entry-point
for cellular-automata-like thinking into the foundations of number theory.
