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

Abstract

We give a self-contained and systematic 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 and gives a table-free digitwise rule in every order. The centroid map associated with the recursive triangular tiling intertwines the digitwise (S_3)-action with the dihedral action on the triangle. Odd digit permutations reverse multiplication order, yielding ordinary and twisted commutation criteria for products symmetric about triangular axes. Synchronized cyclic changes in selected coordinates produce equilateral centroid triangles. Their oriented vertex product equals an explicit symbolic center with a determined sign, while their Euclidean center is obtained directly from the unchanged coordinates; a parity criterion characterizes exactly when the symbolic and Euclidean centers coincide. For every non-unit basis word, the centralizer in the signed group has cardinality (4^n), and its positive tile set occupies exactly one half of the order-(n) tiling. A parity-dependent family derived from the signed centralizer decomposition 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.