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

Abstract

This note studies a concrete bitwise and triangular coordinate model for the central product of n copies of the quaternion group Q8. The positive basis elements are words of length n in the alphabet {1, 2, 4, 7}, identified with i, j, k, and the identity element e. The signed basis group Fn is the corresponding central product of n copies of Q8, and the real algebra generated by the basis words is H^{\otimes n}. The contribution is the coordinate model: in this basis, Boolean multiplication, recursive triangular tilings, digitwise S3-actions, reflection anti-automorphisms, parity cancellation, centralizer tile sets, and axis-landing phenomena can be expressed in a single language. A local XNOR/AND rule recovers quaternionic basis multiplication and gives a table-free digitwise multiplication rule in every order. The associated centroid map to a recursive triangular tiling is equivariant for the digitwise S3-action and the dihedral action on the triangle. Odd digit permutations reverse multiplication order, yielding ordinary or twisted commutation criteria for products of elements symmetric about triangular axes. Synchronized cyclic changes of selected noncentral digits give equilateral triangles of centroids. For every non-unit basis word, the centralizer in the signed group has cardinality 4^n, and its associated positive tile set occupies exactly one half of the order-n tiling. This positive tile set also has a natural decomposition according to the sign of the common product cb=bc. A parity-dependent family built from this decomposition shows that products of elements symmetric with respect to one triangular axis can land on another axis, lose all triangular reflection symmetry in odd orders, or, in the order-two case, acquire enhanced symmetry.