Indices and sign arrays for floretion multiplication up to order 8.

Floretion Multiplication Data, up to Order 8 Overview This dataset contains precomputed multiplication data for floretions up to order 8. The same datasets can also be generated directly from the source code by running the script create_data_dirs.py in the GitHub repository linked below. The archived files are included mainly for historical reference, comparison, and cross-checking against independently generated data. A floretion of order n is a real linear combination of basis words of length n in the alphabet {1, 2, 4, 7}. The four digits are identified with the quaternionic basis symbols 1 ↔ i, 2 ↔ j, 4 ↔ k, 7 ↔ e. Multiplication is performed digit by digit, and the local quaternionic signs are collected into one global sign. In order one this recovers the quaternion group Q8. In general, the signed basis group Fn = {±b : b in {1,2,4,7}^n} is the central product of n copies of Q8, and the real algebra generated by the basis words is the tensor product H^{⊗n}. Thus floretions do not define a new abstract family of finite groups or algebras. The value of the floretion representation is its explicit digitwise, computational, and geometric coordinate system. Coordinate and geometric viewpoint The alphabet {1, 2, 4, 7} also has a triangular interpretation. The digits 1, 2, and 4 label the three corner subtriangles in a recursive equilateral-triangle subdivision, while the digit 7 labels the central subtriangle. This gives a direct connection between algebraic basis words and triangular tilings. In this coordinate system, several structures can be studied in one language: Boolean multiplication rules using XNOR and AND; digitwise S3-actions on the symbols 1, 2, and 4; reflection anti-automorphisms; centralizer tile sets; parity cancellation; axis-landing phenomena in triangular tilings. Dataset description The dataset contains precomputed index arrays and sign arrays, stored in NumPy binary format, for efficient multiplication of floretions up to order 8. For two order-n floretions x and y, the product z = x*y can be computed coefficient by coefficient. For each output basis vector z_r, the corresponding coefficient is obtained as an ordinary dot product between the coefficient vector of y and a signed, reordered version of the coefficient vector of x. The noncommutativity of floretion multiplication is encoded in the row-dependent index and sign arrays. For each output row r: the indices array specifies how the coefficients of x should be reordered; the signs array specifies which signs must be applied; the resulting signed and reordered vector is dotted with the coefficient vector of y. This replaces repeated symbolic basis multiplication by array indexing, sign changes, and standard dot products. Order-one example Write x = x[0]i + x[1]j + x[2]k + x[3]e and y = y[0]i + y[1]j + y[2]k + y[3]e. The coefficient of i in z = x*y is x[0]y[3] + x[1]y[2] - x[2]y[1] + x[3]y[0]. This can be computed as np.dot(np.multiply([1, -1, 1, 1], [x[3], x[2], x[1], x[0]]), [y[0], y[1], y[2], y[3]]). Here [1, -1, 1, 1] is the first row of the signs matrix, while [3, 2, 1, 0] is the first row of the indices matrix. For order one, the matrices are: Signs matrix [[ 1 -1 1 1][ 1 1 -1 1][-1 1 1 1][-1 -1 -1 1]] Indices matrix [[3 2 1 0][2 3 0 1][1 0 3 2][0 1 2 3]] File sizes and segmentation The number of basis vectors grows as 4^n, so the arrays become large quickly. To keep file sizes manageable: order 7 data is split into 4 segments; order 8 data is split into 64 segments. Each segment contains the corresponding portion of the precomputed index and sign data needed for multiplication. Related resources Current paper: Bitwise Triangular Coordinates for Central Products of Quaternion Groups: Floretion Base Vectors, Digitwise S3-Actions, and Centralizer Tiles Online Floretion Calculator Floretions at GitHub