Bitwise Triangular Coordinates for Central Products of Quaternion Groups: Floretion Base Vectors, Digitwise S3-Actions, and Centralizer Tiles

Abstract

We give a self-contained treatment of the floretion coordinate model for the central product of $n$ copies of the quaternion group $Q_8$. Positive basis elements are words of length $n$ in the alphabet $\{1,2,4,7\}$, identified with $i,j,k,e$, and the resulting real algebra is $\mathbb{H}^{\otimes n}$. In these coordinates, Boolean multiplication, recursive triangular tilings, digitwise $S_3$-actions, reflection anti-automorphisms, centralizer tiles, and axis-landing phenomena admit a common description. A local XNOR/AND rule recovers quaternionic basis multiplication in every order. The centroid map for the recursive triangular tiling intertwines the digitwise $S_3$-action with the dihedral action on the triangle, while odd digit permutations reverse multiplication order. Synchronized cyclic changes in selected coordinates produce equilateral centroid triangles; their oriented vertex product gives an explicit symbolic center, and a parity criterion characterizes when this agrees with the Euclidean center. For every non-unit basis word, the centralizer in the signed group has cardinality $4^n$ and its positive tile set occupies one half of the order-$n$ tiling. Its positive-product component $C_+(b)$ is invariant under the global cyclic digit action and, apart from the identity tile, decomposes into centered equilateral $C_3$-orbits; the negative-product component $C_-(b)$ is a transversal of the complementary orbit family. A parity-dependent family from the signed centralizer decomposition further shows that products symmetric about one triangular axis may land on another axis, lose all triangular reflection symmetry in odd orders, or acquire enhanced symmetry in order two.